TRANSLATIONS OF MATHEMATICAL MONOGRAPHS VOLUME
66
Theory of Limit Cycles YEYAN·QIAN Nanjlng University and
Cal Sui·Un Chen Lan-sun Huang Ke·cbeng Luo Dlng-jun
MaZbl-en Wang Er·nlan Wang Mlng-sbu YangXIn·an
Translated by
Cbl Y. Lo Mlcblgan State University
AmericanMathematicaiSociety · Providence · Rhodelsland
t&miF~ 11-t-J~i.f ~~
1:¥iH4*t1*:±1 Ji&tt tfHifi 1965, 1984 Translation edited by S. H. Gould and J. K. Hale 1980 Mathematics Subject Qassification (1985 Revision). Primary 34C05. SUMMARY OF CONTENTS. This book belongs to the "Modern Mathematics Series." Its first edition was published in 1965. Now this is the second edition, with many substantial changes in content. It contains tremendous results obtained during the past twenty years in China, and also introduces some great achievements from abroad. This book has eighteen sections, which can be divided into three parts. Part I (§§1-8) discUBBeB limit cycles of general plane stationary systems, including their existence, nonexistence, stability, and uniqueness. Part II (§§9-17) discusses the global topological structure of limit cycles and phase-portraits of quadratic systems. Part III (§18) collects some important results which either could not be included under the subject matter of the previous sections or appeared in the literature very recently. This book serves as a reference book for college seniors, graduate students, and researchers in mathematics and physics.
Library of Congress Cataloging-in-Publication Data
Yeh, Yen-chi'ien. Theory of limit cycles. (Translations of mathematical monographs, ISSN 0065-9282; v. 66) Translation of: Chi hsien huan lun, 2nd ed. Bibliography: p. 415 1. Differential equations. 2. Curves. I. Cai, Sui-lin. II. Lo, Chi Y. (Chi Yeung) III. Title. IV. Series. QA37l.Y413 1986 515.3'5 86-14070 ISBN 0-8218-4518-7 Copyright (£)1986 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. Copying and reprinting information can be found at the back of this volume. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability.
Contents Preface to the second edition
vii
Abstract
ix
Introduction
1
§1. Fundamental concepts, concrete examples, and several criteria to establish existence and nonexistence of limit cycles
5
§2. Multiplicity and stability of limit cycles
25
§3. Limit cycles in rotated vector fields
41
§4. General behavior of limit cycles depending on a parameter
77
§5. Existence of a limit cycle
91
§6. Uniqueness "'fa limit -cycle
117
§7. Existence of several limit cycles
155
§8. Structural stability of different systems
175
§9. Work of M. Frommer and N. N. Bautin
191
§10. Global structural analysis of some quadratic systems without limit cycles
219
§11. General properties and relative positions of limit cycles in quadratic differential systems
245
§12. Classification of quadratic differential systems. Limit cycles of equations of Class I
261
§13. Global structure of trajectories of equations of Class II without limit cycles
281
v
vi
THEORY OF LIMIT CYCLES
§14. Relative positions of limit cycles and conditions for having at most one and two limit cycles in equations of Class II 305 §15. Several local and global properties of equations of Class III
335
§16. Method of Dulac functions in the qualitative study of quadratic systems
359
§17. Limit cycles in bounded quadratic systems
371
§18. Appendix
401
Bibliography
415
Preface to the Second Edition
It has been eighteen years since the publication of the first edition of this book in 1964. Over these eighteen years, there has been tremendous progress in the theory of limit cycles, especially in quadratic differential systems, in China as well as in other countries, and many new results have been obtained. Some work in this area which was considered correct in the past has turned out to be wrong; some work which was considered important in the past now seems to be not worthy of further development. It is worthwhile to mention that, owing to research developments from other fields such as biology and chemistry, research on limit cycles, especially in the theory of polynomial systems, has become more and more important. In China, there are at least ten mathematicians who have produced good work in this area over the past twenty-some years. In other countries, besides the Russian mathematicians who have had a traditional interest and made solid contributions in this area, more people in the United States and France have shown strong interest in the study of limit cycles of quadratic systems. For these reasons, the author felt that this book should be completely revised and its second edition should be published without delay. Not having enough time to work on this project, and hoping to collect new ideas and valuable comments from others for the better presentation of the second edition, I have omitted §7, §9 and the latter part of §10 from the first edition, and concentrated on the overall rearrangement of material, final editing of the manuscripts and unification of the use of mathematical symbols in the new edition; except for the sections dealing with my own work, I entrusted the task of revision and supplementation to some of my colleagues, as follows: Wang Er-nian supplemented part of §3 and part of §6; Ma Zhi-en supplemented §3; Huang Ke-cheng supplemented §5 and wrote the new §7; Luo Ding-jun rewrote §8 and supplemented the first half of §9; Cai Sui-lin rewrote §10; Wang Ming-shu supplemented §11 and wrote the new vii
viii
THEORY OF LIMIT CYCLES
§15; Yang Xin-an supplemented §12 and wrote the new §17; and Chen Lan-sun supplemented the latter two-thirds of §14 and wrote the new §16. In my final editorial work I have made very few changes in some of their manuscripts while in other cases I have made considerable changes, but they all applied their expertise in this field and spent their time and energy to make the second edition reflect more completely the current development of the theory of limit cycles. It is obvious that I could not have taken on such a task all by myself. I therefore express my heartfelt gratitude to the above eight colleagues. Due to our limited time and ability, it is natural that there may still be some mistakes or important omissions in the new edition. We shall be grateful if our readers would kindly let us know. Ye Yan-qian Department of Mathematics Nanjing University August, 1982.
Abstract Limit cycles of plane autonomous differential systems appear in the very famous classical paper "Memoire sur les courbes definies par une equation differentielle" of H. Poincare (1881-1886). In the 1930s, van der Pol and A. A. Andronov showed that the closed orbit in the phase plane of a self-sustained oscillation occurring in a vacuum tube circuit is a limit cycle as considered by Poincare. After this observation, the existence, nonexistence, uniqueness and other properties of limit cycles have been studied extensively by mathematicians and physicists. Then, from the 1950s, very many mathematical models from physics, engineering, chemistry, biology, economics, etc., were displayed as plane autonomous systems with limit cycles. Also, due to the well-known paper of I. G. Petrovskil and E. M. Landis concerning the maximum number of limit cycles of all quadratic differential systems (the second part of Hilbert's 16th problem), the problem of limit cycles has become more and more important and has attracted the attention of many pure and applied mathematicians. The purpose of this book is to bring together in one place most of the main contributions in the theory of limit cycles. Aside from the introduction (a brief historical review), it divides into three parts. §§1-8 are concerned with limit cycles of general plane autonomous systems, and §§9--17 with limit cycles and the global topological structure of phase-portraits of quadratic systems. At the end of every section, a large number of reference papers are listed. The last part, §18, has the character of an appendix, in which we mention briefly results that either could not be included in the subject matter of the foregoing sections, or have appeared in periodicals very recently. We assume that the readers have a basic knowledge of the qualitative and stability theory of ODE. Here are the main contents of each section in the first two parts. §1 gives the fundamental concepts and examples of limit cycles, and also some criteria for the existence and nonexistence of limit cycles, including well-known ones ix
X
THEORY OF LIMIT CYCLES
as well as some new ones. §2 gives criteria for the determination of the stability and multiplicity of limit cycles. Aside from the classical ones, we introduce also results of V. F. Tkachev and M. Urabe. §3 deals with the theory of rotated vector fields due to G. F. D. Duff, and also many of its extensions and applications by X. Y. Chen and Z. E. Ma. We will use this theory very often in the second part. §4 is concerned with the variation of limit cycles with the varying of a parameter in the general case; the main contents are the classical formula of H. Poincare and contributions of M. Urabe and X. Y. Chen. §5 discusses the question of the existence of limit cycles. Aside from the well-known theorems of A. F. Filippov and A. V. Dragilev, we present here also contributions ofK. Z. Hwang, Z. J. Wu, and X. W. Zheng. This section is divided into six subsections according to the methods of proof. §6 is concerned with the problem of uniqueness of limit cycles. It also divides into seven subsections, in which we introduce methods of point-transformation due to H. Poincare, A. A. Andronov, and E. A. Leontovich, and also results of G. Sansone, J. L. Massera, Z. F. Zhang, L. A. Cherkas, and G. S. Rychkov. §7 deals with the problem of the existence of any given number of limit cycles. The main results almost all belong to Chinese mathematicians, among which the contribution of Z. F. Zhang is preeminent, solving completely the problem of the number and position of limit cycles of the equation x + p, sin x + x = 0. §8 is a short introduction to the well-known necessary and sufficient conditions for the structural stability of a plane autonomous system in a bounded domain, which we will use in the second part. It contains also some new results of G. T. dos Santos and D. J. Luo about polynomial systems. §9 deals with classical results of H. Dulac and M. Frommer on necessary and sufficient conditions for a quadratic differential system to have a center, and presents the corresponding phase-portraits due to Frommer. In this section we also give a detailed proof of an important result of N. N. Bautin concerning the maximum order of fineness of a focus of any quadratic system and the maximum number of limit cycles that can be generated from this focus. In §10, we analyze the global topological structure of phase-portraits of three types of quadratic systems without limit cycle, namely, the homogeneous systems (results of L. S. Lyagina, L. Markus, et al.), the system = x + h.o.t., iJ = + h.o.t. (result of A. N. Berlenskil) and, finally, the structurally stable quadratic systems without limit cycle (results of G. T. dos Santos and S. L. Zai). §11 deals with general properties and possible relative positions of limit cycles of quadratic systems, among which results of Y. Q. Ye, C. C. Tung, Y. S. Chin, M. S. Wang, and S. L. Shi are presented. §12 introduces the classification of quadratic systems due toY. Q. Ye, and proves a theorem on the existence, nonexistence and uniqueness of limit cycles of systems of type I, due toY. Q. Ye, Y. H.
x
y
ABSTRACT
xi
Deng, D. J. Luo, L. S. Chen, and X. A. Yang. §13 investigates the global topological structure of phase-portraits of a special system of type II, which contains two parameters a and m and has no limit cycle. We obtain global bifurcation curves in the (a, m) projective plane. §14 is concerned with the relative position (especially coexistence), uniqueness and the number of limit cycles of systems of type II containing only two second-degree terms in the first equation. These results are due mainly to M. S. Wang, K. T. Lee, S. X. Yu, N. D. Zhu, K. C. Chen, L. I. Zhilevich, and L. A. Cherkas. §15 discusses various interesting global properties of systems of type ill; especially, we give the detailed proof of a theorem (concerning system IDa=o) similar to that in §12, which was conjectured and partly proved by N. D. Zhu, and later completely proved by L.A. Cherkas, L. I. Zhilevich, and G. S. Rychkov. §16 discusses the Dulac function method used frequently by Chinese mathematicians in their research work on the qualitative investigation of quadratic systems, and uses this method to prove an interesting theorem of L. S. Chen and M. S. Wang, concerning the nesting of limit cycles surrounding just one focus. §17 introduces X. A. Yang's results on the uniqueness or nonexistence of limit cycles of bounded quadratic systems. These systems were first studied by R. J. Dickson and L. M. Perko, but the limit cycle problem remained open il' their papers.
Introduction
In the qualitative theory of differential equations, research on limit cycles is an interesting and difficult part. Following the discovery of the limit cycle by Poincare in his four-part paper Integral curves defined by differential equations (1881-1886) [1], it was immediately given special consideration by this famous mathematician. In order to determine whether there existed a limit cycle for a given differential equation and to study the properties of limit cycles, he first introduced such important theoretical concepts as the method of a topographical system, the successor function, the method of small parameter (which first appeared in his book New methods in celestial mechanics) and the Annular Region Theorem, and he constructed many examples to examine the effects of these methods. At the same time, he had already noticed the close relationship between the study of limit cycles and the solutions of the global structural problems of a family of integral curves of differential equations. In 1901 the Swedish mathematician Ivar Bendixson published a very important paper using virtually the same title [2]. In this paper he gave a rigorous proof of the Annular Region Theorem and extended it to the well-known PoincareBendixson theorem on the limit set of trajectories of dynamical systems in a bounded region. Moreover, he first applied Green's formula to establish the relationship between possession of a closed trajectory and the divergence of a plane vector field, and obtained a theorem for the nonexistence of a closed trajectory. This connection has been continuously extended and strengthened, and the relation between the value of the divergence integrated once around a closed trajectory and its stability, the relation between the value of a saddle point and the stability from inside of a singular closed trajectory passing the saddle point, and so on, have been obtained. In the same year as Bendixson's paper was published, Hilbert set forth his famous list of problems [3] at the International Congress of Mathematics. The second half of the sixteenth problem was: What is the maximum number of 1
2
THEORY OF LIMIT CYCLES
limit cycles for the equation dy Qn(x,y) dx = Pn(x,y)
(1)
(Pn and Qn are polynomials of real variables x, y with real coefficients and degree not higher than n), and what are their relative positions? Strangely, mathematicians have shown great interest in the other Hilbert problems, and have done intensive research on them; but very few people have studied this problem.(!) To our knowledge, in the first three decades of the twentieth century, only the French mathematician Henri Dulac produced good results on this problem. In 1923 he published a• very long paper of 140 pages [4], in which he proved that the number of limit cycles of equation (1) is finite. Moreover, he also studied the necessary and sufficient conditions for existence of a center for equation (1) when n = 2 (see [5]). It seems to us that he had already sensed a close relationship between these two problems. Dulac also produced some other fundamental results in the theory of limit cycles; readers can see these in §1. Later, the German mathematician Max Frommer published a paper in 1934 [6] giving necessary and sufficient conditions for the existence of a center for equation (1) (n = 2). At the same time, he also pointed out that the equation dy x + (1 + c)x 2 + 2xy- y2 -y + 2xy - y2 dx =
( 2)
has a limit cycle when c > 0 is sufficiently small. In fact, as we shall see later, limit cycles occur only for nonlinear equations, and they are the most common trajectories for these equations. The trend of development of mathematical theory is often guided by practical problems. For the field of differential equations, this situation is particularly clear. The driving force behind the study of the theory of limit cycles was furnished much more by practical problems than by great mathematicians. This is the situation: during the twentieth century, applied electronics had made rapid advancement; physicists invented the triode vacuum tube which was able to produce stable self-excited oscillations of constant amplitude, thus making it possible to propagate sound and pictures through electronics. However, it was not possible to describe this oscillation phenomenon by linear differential equations. In 1926, van der Pol [7) first obtained a differential equation, which was later named after him, to describe oscillations of constant amplitude of a triode vacuum tube:
x+ ~t(x 2 - I)x + x = o
(~t ~ o).
(3)
( 1 )The situation has changed since the first edition of this book appeared, and in reent years more people have become interested in this problem.
INTRODUCTION
3
After transforming this equation into an equivalent differential system in the phase plane, he used graphical methods to prove the existence of an isolated closed trajectory. Moreover, he used an averaging method (van der Pol's method), which was at that time without rigorous mathematical foundation, to obtain an approximate equation for the closed trajectory when IJ.tl was very small. It is obvious that he was not familar with the work of Poincare and Dulac on limit cycles. Three years later, the Russian theoretical physicist A. A. Andronov in a short paper [8] clarified that the isolated closed trajectory of the van der Pol equation was the limit cycle studied earlier by Poincare. Thus he established a close relationship between pure mathematical theory and electronic technology. From that time onward, the schools of Moscow and Gorki in the Soviet Union began to produce a tremendous amount of research work on electronic technology and the theory of limit cycles. In mathematical theory, they studied mainly the existence, uniqueness and stability of limit cycles and the problems of how the limit cycles were generated and disappeared. Most of their important and fundamental work can be found in the book Theory of oscillations by A. A. Andronov, A. A. Vitt, and S. E. Kh&"kin [104]. Although equations for nonlinear oscillations include time-dependent systems as well as stationary systems (or autonomous systems) which do not contain the time variable explicitly, most of the publications of Andronov and others on differential equations belong to steady-state systems. Hence we can say that their book is devoted entirely to the mathematical theory of limit cycles and its applications in physics. Naturally, its emphasis is on applications. As for other eountries, most of the research on limit cycles after van der Pol was produced after 1940, except some earlier work of the French engineer A. Lienard and the geometers E. Cartan and H. Cartan [9], [10]. Among the research done after 1940, we especially note that done by N. Levinson, G. F. D. Duff, S. P. Diliberto, G. Sansone, R. Canti, and M. Urabe. In China, since 1957, we have carried on deep and systematic research on problems of limit cycles of equations whose right sides are quadratic polynomials. (See [11]-[18]; in the earlier fifties, Chinese scholars had done some work on the existence, stability, and uniqueness of limit cycles.) The three main problems of research are as follows: 1. relative positions of limit cycles of equation (1) when n = 2, 2. quadratic algebraic curve cycles of (1) when n = 2 or 3, and 3. research on the number of limit cycles and the global structure of trajectories for a given equation (1) (n = 2). The first two problems were thought to be completely solved. However, since the conjectures of Petrovski1 and Landis [19], [20] have been disproved, the first problem is still a long way from being completely solved and research
4
THEORY OF LIMIT CYCLES
on the second is also still in progress. From abroad, in addition to the famous work of N. N. Bautin in 1952 [21}, we have found out that since 1960 there has been a seminar at the Belorussian State University to study the third problem, and many papers have been published. Surveying all the international research work on the limit cycle and its academic trend, our view is that although problems of limit cycles in the qualitative theory of differential equations are of primary importance, the workers in the field of differential equations do not pay enough attention to them. As a result, there has never been a treatise on limit cycles in the theory of pure mathematics, or a comprehensive report with special topics on limit cycles in an international journal. In China there was only one report by the present author, published in 1962. Engineers and physicists are still interested in these problems, but without strong support from mathematicians. (Recently this situation has improved.) Hence, results in this area are fragmentary and unorganized. The aim of this book is to collect important results on limit cycles for the past four decades from China and abroad, and to introduce them to beginning workers in the field, and at the same time to correlate this theory with the other areas of the qualitative theory of differential equations. Besides giving an account of more important and fundamental work in the main text of this book, we also introduce at the end of each section some secondary or deeper results; moreover, we provide enough exercises to help beginners better grasp the contents and methods of each section.
§1. Fundamental Concepts, Concrete Examples, and Several Criteria to Establish Existence and Nonexistence of Limit Cycles
We are given a system of differential equations dx dt
dy dt
= P(x, y),
= Q(x, y),
(1.1)
where x, y, and t are real variables, P and Q are continuous single-valued real functions of x, y, and uniqueness of its solution is guaranteed. DEFINITION 1.1. H the solution x =
6
THEORY OF LIMIT CYCLES
2°. The part of a normal line segment passing through any point P of r in the interior of U and including P is a transversal, i.e. all trajectories meeting this part do not touch it tangentially, and, as t increases, they all cross it in the same direction. 3°. Any closed trajectory in U meets the transversall from any point of r at only one point. Any nonclosed trajectory 1 in U meets l at an infinite number of points, which lie on the same side of r, and their order on l is the same as their order on 1· From this theorem we can see that when a closed trajectory r is an internal (external) limit cycle or periodic cycle, the nonclosed trajectory lying in the inner (outer) neighborhood of r can only be a spiral with one end approaching a closed trajectory at infinity, and the closed trajectory lying in the inner (outer) neighborhood of r can only be a closed trajectory which is wholly contained in r (orr is contained wholly in its interior). THEOREM 1. 2. Any closed trajectory r is either an external (interna~ limit cycle or an external ( internaQ periodic cycle. The former case can also be further divided into three different cases: 1. There exists a sufficiently small outer (inner) neighborhood of r such that all the trajectories in it are nonclosed, and they take r as w-limit set. Then r is called an externally (internally) stable cycle. 2. There exists a sufficiently small outer (inner) neighborhood of r such that all the trajectories in it are nonclosed, and they take r as o:-limit set. Then r is called an externally (internally) unstable cycle. 3. In any arbitrarily small outer (inner) neighborhood of r there exist both closed and nonclosed trajectories. Then r is called an externally (internally) compound limit cycle. DEFINITION 1.4. r is called a stable limit cycle, an unstable limit cycle or a periodic cycle if both inside and outside of r satisfy the corresponding conditions as shown in Theorem 1.2; r is called a semistable limit cycle if it is externally (internally) stable and internally (externally) unstable; r is called a compound limit cycle if it does not belong to any one of the above four cases. From the definitions of stability of trajectories, i~ is easy to establish THEOREM 1. 3. Externally (internally) compound limit cycles and periodic cycles are externally (internally) stable trajectories; externally (internally) stable limit cycles are externally (internally) asymptotically stable trajectories; and externally (internally) unstable limit cycles are externally (internally) asymptotically stable trajectories in a negative direction. Next, the following theorem is well known.
§1.
FUNDAMENTAL CONCEPTS
7
THEOREM 1.4. If closed trajectoreis f1 and f2 together form an annular region G, and there are neither singular points nor other closed trajectories in G, then all the trajectories in G take f 1 as their w-limit set and r 2 as their a-limit set, or vice versa. In other words, two adjacent closed trajectories (under the above conditions) possess different stability on their adjacent sides. DEFINITION 1.5. A closed trajectory f is positively (negatively) oriented if as t increases the point on r moves along r in a counterclockwise (clockwise) direction. It is easy to see that all the closed trajectories in the same family of periodic cycles have the same orientation. However, it should be noted that any two adjacent limit cycles may have different orientation. EXAMPLE 1. Consider the system of equations dr dt = (r- 1)(r- 3),
dt
=
dr dt dr dt
_ 0 dcp 4 h 3 - ' dt 7r w en r ~ ; dcp 4 - 0 - - - 1 when r _< 1. - ' dt - 11'
(r- 1)(r- 2)(r- 3) sec lr- 2li when 1 < r < 3; (1.2)
It is easy to see that the trajectories of this system of equations are: 1) a circle of radius greater than or equal to 3, negatively oriented; 2) a circle of radius less than or equal to 1, positively oriented; and 3) an annular region (1 < r < 3) completely filled by nonclosed trajectories; along each nonclosed trajectory dcpfdt < 0 when r > 2, dcpfdt = 0 when r = 2, and dcpfdt > 0 when r < 2. From this we can see that r = 1 and r = 3 are adjacent one-sided limit cycles. The former is a positively oriented, externally stable limit cycle, and the latter is a negatively oriented, internally unstable cycle. Now let us turn to some criteria to distinguish existence or nonexistence of limit cycles and two important concrete examples. THEOREM 1. 5. If a bounded region 0 contains at most a finite number of singular points and one positive semitrajectory "'(, then thew-limit set of 'Y can only be: 1) a unique singular point; 2) a unique closed trajectory; or 3) a countable infinite or finite number of trajectories which approach singular points as t -+ ±oo and those singular points. COROLLARY (POINCARE ANNULAR REGION THEOREM). If 0 is an annular region which does not contain any singular point, and any trajectory
THEORY OF LIMIT CYCLES
8
which intersects the boundary curve of n moves in the exterior-to-interior (interior-to-exterior) direction, then there exist at least one externally stable (unstable) limit cycle in n whose interior contains its inner boundary curve and one internally stable (unstable) limit cycle. These two one-sided limit cycles can both be two-sided cycles, or they may coincide to become one stable (unstable) limit cycle.( 1 )
REMARK. If the inner boundary of 0 shrinks into a negatively (positively) asymptotically stable singular point, or part of the inner or outer boundary curve becomes an arc of a trajectory of equations (1.1), on which there can be some saddle points or negatively oriented (positively oriented) asymptotically stable singular points, the above corollary can still be established. Next, from the theory of functions of a complex variable, it is easy to establish THEOREM 1.6. If P and Q in equations (1.1) are analytic functions of x, y, then any closed trajectory cannot be a compound cycle. Thus the outer boundary and inner boundary of the region which is completely filled by periodic cycles must contain singular points (maybe an infinite number of singular points). EXAMPLE 2. The system of equations dxjdt
= y,
dyjdt
= -x + ll-(1- x 2 )y,
(1.3)
which is equivalent to the van der Pol equation mentioned in the introduction, has a stable (unstable) limit cycle for p, > 0 ( < 0).(2) PROOF. It is easy to see that when IL > 0 the only singular point (0, 0) of (1.3) is a negatively oriented, asymptotically stable singular point. Hence, in order to prove the existence of limit cycles, we only have to construct a simple closed curve L whose interior contains the origin, so that when the trajectories of (1.1) intersect L, they all move in the exterior-to-interior direction. First draw the graph of the isocline of
= -x + IL(1- x 2 )y = 0, which has three branches and three asymptotes: y = 0, x = ±1. Q(x, y)
(1.4)
Now we take a point A(O, -yo) on the negative y-axis, and construct the trajectory of (1.5) dxjdt = y, (1 )Later on, in most cases, we often get a unique stable cycle, and this cycle is often a single cycle (see the definition in §2) and not a multiple cycle obtained from coincidence of two limit cycles. (2 )The case J.l < 0 can be changed into the case J.l > 0 by changing the signs of x and t. Hence we shall only prove the case J.l > 0.
§1. FUNDAMENTAL CONCEPTS
9
passing through A, which intersects the line x = -1 at B( -1, -yt). It is easy to see that Yt = 2JJ/3 + YO· Note that along AB we have
-x + JJ(1- x 2 )y _ JJ(1- x 2 )y = _.:_ < O; y
y
y
hence the trajectories of (1.3), when intersecting AB, always cross it from right to left (as indicated by the small arrows in Figure 1.1).
B'
FIGURE 1.1
Next, draw a circular arc with its center at the origin passing through B, and let it intersect the branch of the curve of (1.4) in the second quadrant at C after intersecting the negative x-axis. (The point of intersection always exists if the radius of the arc is sufficiently large.) Note that along fiG we have -X+ J.t(1 - x2)y X (1 2) Q --"""'---'--....:....:....+-=J.I. -x < ' y y and we know that the trajectories of (1.3) always cross it from left to right. Now we study the point of contact D(x2, Y2) of the trajectory of the equations (1.6) dx/dt = y, dyfdt = -x + J.tY and the curve (1.4). It is easy to prove that x2 should satisfy the equation 1 + (1 - J.t2)x2
+ 2J.t2X4 -
J.'2X6
= 0.
10
THEORY OF LIMIT CYCLES
When lxl = 1 the left side of the above equation is positive, whereas when lxl is sufficiently large the left side is negative; hence there must exist points of contact on that branch of the curve of (1.4) located in the second (and fourth) quadrant. We denote the rightmost (leftmost) of the contact points by D (D'). When Yo is sufficiently large, C lies on the left of D. Now we construct a trajectory of equations (1.6) passing through D, which intersects (definitely) the positive y-axis at E. Note that -x + JL(l - x 2)y -x + JLY 2 --.:.....:...=-JLX < 0• y
y
We know that the trajectories of (1.3) when intersecting DE (they both have positive slopes) always cross it from left to right. Moreover, the trajectories of (1.3) when intersecting CD of (1.4) always cross it from left to right. Since DE is fixed, and Yo can be taken sufficiently large, we assume A' (the point of symmetry of A with respect to 0) is situated above E. At the same time the trajectories of (1.3), when intersecting the line segment EA', cross it from left to right. Since the vector fields determined by equations (1.3), (1.5), and (1.6) and the curve (1.4) are symmetric with respect to the origin, the curve A' B'C' D' E' A (the curve of symmetry of ABCDEA' with respect to the origin) has similar properties. Thus the arcs of these two curves combined form the outer boundary curve of the required annular region. Because (1.3) is an analytic system, we can deduce from the corollary of Theorem 1.6 that one of these one-sided limit cycles, if they do not coincide, must be a stable cycle. In fact, in §6 we shall prove that (1.3) has a unique stable limit cycle. REMARK. When JL > 2, the origin is an unstable nodal point of (1.3); hence a singular point in the interior of a limit cycle need not be a focus. Moreover, when JL is larger, for example, JL > 10, the limit cycle need not be a convex closed curve. THEOREM 1. 7 (SYMMETRY PRINCIPLE). Suppose in equations (1.1) P(x,y) = P(-x,y),
Q(-x,y) = -Q(x,y),
(1.7)
and the origin is the only singular point on the- y-axis. If a tra;"ectory r starts from the positive y-axis and returns to the negative y-axis, then r is a closed trajectory. If all the trajectories near the origin possess this property, then the origin is a center.
PROOF. Since the vector field determined by (1.1) is symmetric with theyaxis, rand its curve of symmetry with respect to they-axis (also a trajectory) form a simple closed curve. Moreover, the two points of intersection of this simple closed curve and the y-axis are regular points; hence it is a closed trajectory.
§1.
FUNDAMENTAL CONCEPTS
11
REMARK. H we change conditions (1.7) to
P(x, -y) = -P(x, y),
Q(x, -y) = Q(x, y)
(1.8)
and assume the origin is the only singular point on the x-axis, then we have conclusions similar to Theorem 1.7. ·However, we should note that the conditions P(-x,y) = -P(x,y), Q( -x, y) = Q(x, y) will lead to P(O, y) = -P(O, y) = 0, i.e. the y-axis is a trajectory; thus even if f starts from the positive y-axis and returns to the negative y-axis, it can only be a singular closed trajectory. Similarly, the conditions
P(x, -y)
= P(x, y),
Q(x, -y)
= -Q(x, y)
are not applicable, too. THEOREM 1 . 8. Suppose the origin is the only singular point on the y-axis and all the trajectories start from the positive y-axis, move around the origin once and return to the positive y-axis. Then the fixed point of the topological mapping oJ the positive y-axis into itself, determined by the trajectory, lies on the closed trajectory. This theorem is obvious. COROLLARY. If the map determined by the trajectory carries some closed interval on the positive y-axis which does not contain the origin into itself, then the equations must have a closed trajectory. PROOF. From Brouwer's fixed point theorem in topology, we know this map must have a fixed point, and this fixed point is not the origin. Theorem 1.8 and its corollary are very useful for the study of limit cycles of piecewise linear equations which often appear in the theory of nonlinear oscillations, because at the same time the topological map can be given by means of analytic expression. This theorem is also useful for the study of limit cycles of systems whose right sides are discontinuous, because it only requires that there be a unique trajectory which passes through every point on the y-axis.( 3 ) EXAMPLE 3. Motion of a pendulum.( 4 ) For simplification, assume the mass of the pendulum is concentrated at its center of gravity; moreover, we only consider the constant damping produced by dry friction at its point of ( 3 )A limit cycler of discontinuous systems can be defined as follows: A closed trajectory r is a stable limit cycle if there exists a neighborhood of r such that a trajectory starting from any point in this neighborhood (it may intersect f) can enter and remain in any small neighborhood of r as t increases to 00. ( 4 )This example is taken from [104), Chapter 3, §5.
THEORY OF LIMIT CYCLES
12
'!I
FIGURE 1.3
FIGURE 1.2
suspension, but neglect air resistance (linear damping). Then the equation of motion is
x+ x = - /o
when
x > 0,
x+ x = /o
when
x < 0;
(1.9)
here x represents angular displacement, and we take gfl = 1. Changing (1.9) into an equivalent system of equations, it is easy to see that the trajectory in the upper half-plane (x,y = x) is a circular arc with 0"(-/0 ,0) as center, whereas the trajectory in the lower half-plane is a circular arc with 0'{!0 , 0) as center, whereas the trajectory in the lower half-plane is a circular arc with 0' {!o, 0) as center; the clockwise direction on the trajectory is the direction of increasing t. Every point on the line segment (-/o, /o) on the x-axis is a singular point, which corresponds to the equilibrium position; since at that time the restoring force is less than the frictional force, and also x = 0, the motion should stop. From this we can see if the spring does not supply enough energy to the pendulum, then its motion will cease after a finite number of swings. Its trajectory is indicated in Figure 1.2. The energy is supplied by the spring as follows: when the pendulum reaches its lowest point (x = 0), it receives an impulse given by the spring through its trigger mechanism such that it receives constant energy. To simplify our calculations, we assume in the following that the impulse occurs at x =- /o, y > 0. Then in the phase plane we have path of motion as indicated by ABCDE as shown in Figure 1.3, where AB is a circular arc with center 0"(-/o,O) and radius equal to Ro, BC = R1- Ro, and R~- ~ = h 2 = constant; CD is a circular arc with 0" as center and with radius R 1 ; and DE is a circular arc with O'{!o, 0) as center, and with radius R2 = R1 - 2/o. Let
§1. FUNDAMENTAL CONCEPTS
13
the coordinates of A and E be (-x 0 ,0) and (-x 1 ,0) respectively. Then it is easy to calculate the relation between xo and x1: (x1
+ 3fo) 2 -
(xo- fo) 2
= h2 •
(1.10)
This is the topological map of the negative x-a.xis into itself determined by the trajectory of the motion. The fixed point xi = x0 of this map can be determined from (1.11) that is,
x(j = h 2 /Bfo - fo. Of course we should assume x0 > fo, which is equivalent to h
> 4fo.
(1.12) (1.13)
When (1.13) is satisfied, not only is the uniqueness and existence of a closed trajec~ory guaranteed, but also it is not difficult to prove that it is a stable limit cycle. In fact, from (1.11) we can get
x0 = .jh2 + (x0 - fo)2- 3fo; hence, when fo
< xo < x0, we have 2 -+--:(,....xo---J,=-o)= 2 x 0 > x1 = Jr-:-h:::-
3fo
(1.14)
> xo,
(1.15)
where the first inequality can be seen from (1.10) and (1.14) and the second inequality can easily be deduced from (1.11). Similarly, when x 0 > x0, we have Xo < X1 < Xo. (1.16) Inequalities (1.15) and {1.16) indicate that when A =f. A*, E should be between A and A*. But the value of x0 is uniquely determined from (1.12); hence when t ---> oo, all motion should approach the closed trajectory passing through A•. Finally, we introduce several rules to distinguish nonexistence of closed trajectories. THEOREM 1.9 (POINCARE METHOD OF TANGENTIAL CURVES). Let F(x, y) = C be a family of curves, where F(x, y) is continuously differentiable. If in a region G the quantity
dF = aFdx + aFdy = paF +QaF dt ax dt ay dt ax ay (which represents the rate of change of the function F with respect to t along a trajectory of system {1.1)) has constant sign, and the curve PaF +QaF =O ax ay
THEORY OF LIMIT CYCLES
14
(which represents the locus of points of contact between curves in the family and the trajectories of (1.1), and is called a tangential curve) does not contain the whole trajectory of (1.1) or any closed branch, then the system (1.1) does not possess a closed trajectory which is wholly contained in G and does not have a singular closed trajectory with only one singular point on it.
PROOF. Suppose the theorem is not true. Then the system (1.1) contains a closed trajectory which is wholly contained in G, or a singular closed trajectory which contains only one singular point. Now integrate P8Ff8x + Q8Ff8y along the above trajectory r once in the direction of increasing t; we obtain
i (p~: +Q~:) i ~ dt=
dt.
Since F is single-valued, we know that the right side of the above formula is equal to zero; however, on the other hand, the integrand on the left of the inequality has a constant sign, but is not identically zero on r, and, along r' t monotonically increases; hence its value should be different from zero, a contradiction. THEOREM 1.10 (BENDIXSON). If the divergence 8Pf8x + 8Qf8y of (1.1) has constant sign in a simply connected region G, and is not identically zero on any subregion of G, then the system (1.1) does not possess any closed trajectory or singular closed trajectory which lies entirely in G. ( We assume P and Q have continuous partial derivatives.) PROOF. Suppose the theorem is not true. System (1.1) has a closed trajectory r. Since G is simply connected, rand its interior lie entirely in G. From Green's formula we have
i
Pdy- Qdx =/Is (
~: + ~~)
dxdy.
(1.17)
But Pdy = Qdx holds everywhere along r. Hence the left side of (1.17) is zero, while the integrand on the right has constant sign but is not identically zero in S; hence the double integral is not zero, a contradiction. When r is a singular closed trajectory, since dyfdx does not have a definite value at the singular point, the tangential direction of r may be discontinuous at that point, and the above proof is not rigorous. However, we can still prove the theorem after a slight modification. For this, let r have only one singular point 0 as in Figure 1.4. Use a smooth arc "Y = AB (dotted line) to replace a small section of the arc AOB which contains 0 on r and obtain a closed curve f'. Applying Green's formula to f' and its interior 8', we obtain
fr,
--
Pdy- Qdx
=
i v• ·
ds =
jfs, (~: + ~~) dxdy,
(1.18)
§1.
FUNDAMENTAL CONCEPTS
15
r 0
FIGURE 1.4 where v• represents the vector ( -Q, P) and ds represents the element of arc length of 1· We have P = Q = 0 at 0, and according to the continuity of P and Q we only have to choose 1 lying in a sufficiently small neighborhood of 0; then the absolute value of the left side of (1.18) is less than any small positive e. For the double integral on the right, its value should be very close to the value ofthe right side of (1.17); hence its absolute value cannot be less thane, a contradiction. REMARK. The proof of Theorem 1.10 mainly depends on the following fact: PROPOSITION. In the interiorS of any closed trajectory or singular closed trajectory of system (1.1), we have
/Is (~: + ~~)
dxdy = 0.
(1.19)
Readers can see easily that (1.19) has an obvious physical meaning, because the integrand represents the divergence of the vector field defined by (1.1). THEOREM 1.11 (DULAC). If there exists a continuously differentiable function B(x, y) in a simply connected region G such that 8(BP)/8x + 8(BQ)j8y has constant sign and is not identically zero in any subregion, then system (1.1) does not have any closed trajectory or singular closed trajectory lying entirely in G. PROOF. We follow the proof of Theorem 1.10, but use BP and BQ to replace P and Q respectively. We shall call B(x, y) the Dulac function, and the method of proving nonexistence of a closed or singular closed trajectory by Theorem 1.11 the method of Dulac functions. Theorems 1.10 and 1.11 can be extended to multiply connected regions as THEOREM 1. 12 (DULAC). If we change the region G in Theorem 1.10 or Theorem 1.11 to be n-multiply connected (i.e. G has one or several outer
THEORY OF LIMIT CYCLES
16
boundary curves, and n- 1 inner boundary curves), then system (1.1) has at most n - 1 closed trajectories which lie entirely in G. PROOF. From the proof of Theorem 1.10, we know that if there is a closed trajectory r of system (1.1) in G, then r should contain at least one inner boundary curve C of Gin its interior. Similarly, we also know if the interior of r also contains other closed trajectories r ..... ' r k' then the region in the interior of r but in the exterior of all the trajectories r 1' ... ' r k also contains at least one inner boundary curve C. Let C correspond tor; we can see that for different r their corresponding curves C are also different. Hence if the number of closed trajectories in G is more than n - 1, then the connectivity number of G must be greater than n. The theorem is proved. For the rules to distinguish nonexistence of closed trajectories other than these classical results mentioned above, there is some recent work of V. F. Tkachev and Vl. F. Tkachev [23], Yu. S. Bogdanov [24], Chen Guang-qing [25], Chen Xiang-yan [26], and Yang Zong-pei [27]. In the following we introduce a theorem in [26] which has more generality and has obvious geometrical meaning.(5 ) THEOREM 1.13. For system (1.1), assume there exist a simply connected region G and continuously differentiable functions M(x, y) and N(x, y) such that, in G,
E(x, y) = M(x, y)P(x, y)
+ N(x, y)Q(x, y)
aM aN F(x, y) = ay - ax ~ 0
~ 0,
(1.20)
(::; 0).
(1.21)
F(x, y) ¢ 0 in any subregion of G,
(1.22)
1°. If
then (i) if E(x,y)
=0 in G,
(1.23)
then system (1.1) does not have any closed and singular closed trajectory in
G; and (ii) if E(x,y) ¢0 in G,
(1.24)
then (1.1) does not have any positively (negatively) oriented closed trajectory or singular closed trajectory in G. 2°. If (1.25) F(x,y) 0 in G,
=
( 5 )The conditions in (23) are similar to those in Theorem 1.13, but much stronger; moreover, two sets of the conditions in (23) are incorrect. The proof given here is not the same.
§1.
FUNDAMENTAL CONCEPTS
17
then {1.1) does not have any closed trajectory or singular closed trajectory unless it is contained entirely in the point set of F(x,y) = 0.
PROOF. Suppose (1.1) has a positively (negatively) oriented closed trajectory r. From Green's formula, we have 1M dx
Jr
+ N dy =- f f
JJintr
(1.26)
F(x,y) dxdy.
Under condition 1°, the right side of (1.26) is negative (positive). On the other hand, the left side of (1.26) is equal to
+(-)loT E(x, y) dt,
{1.27)
where x = x(t), y = y(t) represent the solution of (1.1) along r, and T > 0 is the smallest period of the solution. Thus we know that when (1.23) holds, the left side of (1.26) is zero; hence (1.26) does not hold, i.e., r does not exist. Also when (1.24) holds, the expression (1.27) is nonnegative (nonpositive), so (1.26) still does not hold. For a singular closed trajectory, we only have to change JoT to The proof for condition 2° is similar. Now we would like to explain the geometrical meaning of Theorem 1.13. The vector field orthogonal to (M, N) is ( -N, M), and its corresponding system of equations is
L":o.
dxfdt = -N(x, y),
{1.28)
dyfdt = M(x, y).
Condition (1.20) indicates that on moving the vector field (P, Q) to the vector field ( -N, M) in a counterclockwise direction, the angle O(x, y) traversed satisfies 0 ~ O(x, y) ~ 1r (length of the vector is adjustable). Conditions {1.21) and (1.22) illustrate that (1.28) satisfies the conditions of Theorem 1.10, and therefore does not have a closed trajectory or a singular closed trajectory, and its divergence always keeps positive (negative) sign. Therefore, Theorem 1.13(1 °) indicates that under the above conditions (P, Q) is obtained from (-N, M) along the counterclockwise direction through an angle O(x, y); when (J 1r, (P, Q) does not have a closed or singular closed trajectory (obvious), and when(}¢. 1r, (P, Q) does not have a positively (negatively) oriented closed or singular closed trajectory.( 6 ) Theorem 1.13(2°) indicates that when {1.28) has a first integral, or M dx + N dy = 0 is a total differential, {1.1) does not have a closed trajectory or a singular closed trajectory, unless that closed (or
=
( 6 )However, a negatively (positively) oriented closed trajectory may still exist. For example, if (M,N) = (y,-x) and (P,Q) == (M,N), then for 9 = 7r:/2, (P,Q) has a family of negatively oriented closed trl\iectories x 2 + y 2 = C with div( -N, M) 2 > 0.
=
18
THEORY OF LIMIT CYCLES
singular closed) trajectory is also a closed (or singular closed) trajectory of (1.28). Readers will understand more about the above results after they become acquainted with the theory of rotated vector fields to be introduced in
§3. It is easy to see that if we take M = auI ax and N = auI ay in Theorem 1.13, we can deduce Theorem 1.9 (change F(x, y) in Theorem 1.9 to U(x, y)); and if we take M = BQ and N = -BP, we can deduce Theorem 1.12. Moreover, we still have the following corollary.
COROLLARY [25]. If there exist nonnegative C 1 functions Mo(x,y) and N 0 (x, y) and a C 1 function B(x, y) such that in any simply connected region G a a ay (MoP) - ax (NoQ)
a
a
+ ax (BP) + ay (BQ)
~0
(~ 0),
(1.29)
and the points where equality is attained do not completely fill any subregion ofG, then system (1.1) does not have any positively (negatively) oriented limit cycles in G. PROOF. This can be done by taking M = MoP+BQ and N = NoQ-BP in Theorem 1.13. Finally we note that if E(x, y) 0 and F(x, y) 0 in Theorem 1.13, then we cannot obtain any useful conclusions. At the same time we can prove
=
=
THEOREM 1.14. If P and Q of system (1.1) belong to C 1 in a simply connected region G, and aPiax + aQiay 0, i.e. the equation
=
Pdy-Qdx = 0
(1.30)
is a total differential equation, then (1.1) does not have limit cycles, and does not even have any one-sided limit cycles. PROOF. Let the general integral of (1.30) be ~(x, y) = C, which represents the equation of a family of trajectories of ( 1.1). Along each trajectory ~ takes constant value, and along different but adjacent trajectories ~ cannot take the same value. If there exists a limit cycle f of (1.1), then every trajectory in a sufficiently small neighborhood off takes r as w- or a-limit set; hence, by the continuity of ~(x, y), we know in this neighborhood ~(x, y) Cr, which is impossible. Also if f is a compound limit cycle, then in every arbitrarily small neighborhood of f there exist nonclosed trajectories which cover some open annular region near f, in which ~(x,y) should be identically equal to a constant. This is impossible. Similarly we can prove that if in some simply connected region the system (1.1) has continuous integrating factor ~-t(x,y), then (1.1) cannot have limit
=
§1.
FUNDAMENTAL CONCEPTS
19
cycles, foci or nodal points in this region. Conversely, if there exist limit cycles, foci or nodal points, then the integrating factor of (1.30) cannot be continuous. EXAMPLE 4. The system dyfdt = y,
dxfdt = x
takes (0, 0) as nodal point, and at the same time the equation x dy- y dx = 0 takes 1/x2 and 1/(x2 + y 2 ) as integrating factor, but these integrating factors are discontinuous at x = 0 or x = y = 0. EXAMPLE 5. The system
~;
= -y + x( x 2
+ y2 -
(1.31)
1),
+ y2 = 1, and at the same time the equation [-y + x(x 2 + y 2 - 1)] dy- [x + y(x 2 + y 2 - 1)] dx = 0
has limit cycle x 2
has integrating factor
J.L(x, y) = (x 2
y 2 ) 2 exp { -2tan- 1
:
~}
(1.32)
and general integral U(x,y) =
x2
+ y2 -
X
2
+y 2
1
{ -1 y } exp -2tan - = C, X
(1.33)
where J.L and U are discontinuous at (0, 0), i.e., they must have a point of discontinuity in any simply connected region containing x 2 + y 2 = 1. Starting from Theorem 1.10, since we want to apply Green's formula, we have to assume P(x, y) and Q(x, y) have continuous partial derivatives in the region G. N. P. Erugin [28] first pointed out that if we rewrite Green's formula as
£
P(x, y) dy- Q(x, y) dx =
JIs
dxP(x, y) dy
+ dyQ(x, y) dx,
(1.34)
where the double integral on the right is a Stieltjes integral, it could be understood as the limit of L[dxP(x, y)dy + dyQ(x, y)dx] and the requirement on P and Q could be kept the same without strengthening it. N. N. Krasovksil [29] had made use of the above rewritten Green's formula to study the global stability of the zero solution of some nonlinear secondorder differential equations. In that paper, Theorem 1.10 was extended to the case where P and Q did not necessarily have partial derivatives.
THEORY OF LIMIT CYCLES
20
EXAMPLE 6. Prove that if in the system
dxfdt
= ft(x) + h(y),
dyfdt =ax+ by,
(1.35)
ft(O) = /2(0) = 0, f1(x) and h(y) are continuous nonlinear functions, a# 0, and rp(x) = ft(x) + bx is a monotone function of x, then this system does not have a closed trajectory. PROOF. Apply the transformation X1 = -ax- by, Y1 = y. The original equations become
dxtfdt = -a[ft(x) + h(y)} + bx1, Suppose there exists a closed trajectory equation, we obtain 0 = £[-a(ft(x)
r.
Applying formula {1.34) to this
+ h(y)) + bx1} dy1 + x1dx1
=lis dx 1[-a(ft(x) + h(y)) + bx1} dy1 + dy 1X1 dx1 = -a
IIs
dx1 rp (- byl:
Xl) dyt.
This is a contradiction since, according to the assumption, the right side of the above formula should not be equal to zero. In addition, Krasovski1 [30] also obtained the following theorem. THEOREM 1.15. Let P and Q of system (1.1) have continuous partial derivatives, let the origin be the unique singular point of {1.1), and let the roots A1(x,y) and A2(x,y) of the quadratic equation of A
ap _A ax aQ ax
aP ay =0 aQ -A ay
have positive real parts at the origin and negative real parts outside the circle x 2 + y 2 = R 2 ; moreover, 00 ./
m(r) dr = oo,
Then system (1.1) has at least one externally stable cycle and one internally stable cycle, and they may coincide to form a stable cycle.
Not only is this theorem interesting in itself, but also in the proof the author used equations of orthogonal trajectories of the system (1.1). This method is rarely used in plane qualitative theory.
§1. FUNDAMENTAL CONCEPTS
21
Readers should note that if we want to extend the Poincare Annular Region Theorem to a multiply connected region in the plane, it is necessary to consider regions containing singular points, since according to the well-known theory of indices, if the outer and inner boundaries of any n-multiply connected region satisfy the conditions of the above theorem, then the sum of the indices of all interior singular points is 1 - n, but when n > 1, r must have singular points. Ye Yan-qian and Ma Zhi-en (31] extended the concepts of singular closed trajectories of Definition 1.1 and general concepts of singular points and then extended the Annular Region Theorem to regions which contain singular points and general n-multiply connected regions. Yu Shu-xiang (32] applied the results of (31] to study existence of closed trajectories of dynamical systems with singular points on the two-dimensional manifolds, and extended and improved the results of Sacker and Sell [33]; (32] also gave a new proof of • extended the theorem the theorem of (31]. In addition, Dong Zhen-xi (34] also of (31] to general two-dimensional manifolds. It is very interesting to consider the extension of the problems of limit cycles of the system in Example 1 on the annular surface. Tian Jing-huang [35] proved that the van der Pol equations
~: =
-siny + J.Lsin3x,
~~ = sinx
{0::::; x::::; 2rr, 0::::; y::::; 2rr)
on the annular surface have at least two limit cycles when 0 < IJ.LI « 1. As J.L -+ 0, these cycles approach two different nonzero periodic solutions of the corresponding equations. Similarly, Tian and Gao Long-chang (36] proved that the system
~:
=-siny+J.Lsin2x,
~~
=sinx
(O::::;x::::;2rr, o::::;y::::;2rr)
has at least one limit cycle when 0 < IJ.LI « 1. But in the above two papers, the properties of having at most one or two limit cycles have not been proved.
Exercises 1. Prove that the equations dxjdt = xy,
dyjdt = ao + a 1 x + a2x 2 + a3y 2
have two centers when ao > 0 and a2 < 0. 2. Prove that the equations dxjdt = x(x 2 + y2 - l)(x 2 + y 2 - 9) - y(x 2 + y 2 - 2x- 8), dyjdt = y(x 2 + y 2 - l)(x 2 + Y2 - 9) + x(x 2 + y 2 - 2x- 8)
have a unique (stable) limit cycle.
THEORY OF LIMIT CYCLES
22
3. Calculate the curvature of the orthogonal trajectory of equations (1.1), and show it is equal to
:x (,(P:+Q') + ~ ( ,(P Q+Q 2
2 ).
Thus prove that if equations (1.1) have a closed trajectory
r, then [37]
·~ H(x,y)dxdy=O. !Jintr 4. Let G be a simply connected region. If there exist continuously differentiable functions B1 (x, y) and F(x, y) such that the expression.
a a [ aF aF] ax (BtP) + ay (BtQ) + Bt Pax + Q ay keeps constant sign and is not identically zero on any subregion, then system (1.1) does not have any closed trajectory or singular closed trajectory which lies entirely in G [23]. Hint: take B(x, y) = B1 (x, y)eF(x,y) in Theorem 1.11. 5. Let G be a simply connected region. If there exist continuously differentiable functions M(x, y), N(x, y), F(x, y), and B(x, y) which satisfy
M(x,y) ~ 0,
N(x,y) ~ 0,
MP 2
-
2FPQ + NQ 2 ~ 0,
a a a a ay (M P)- ax (NQ) + ax [P(B +F)]+ ay [Q(B- F)] ~ 0 (~ 0), and the points where the above equality holds do not cover completely any subregion of G, then (1.1) does not have any positively (negatively) oriented closed or singular closed trajectory [27]. 6. Prove that the van der Pol equation is equivalent to the system of equations
dzjdt
= x,
dxjdt
= -z + p,(x- x3 /3),
and use this system to construct an annular region to satisfy the corollary of Theorem 1.6 following the method of Example 2. 7. Assuming we have proved the uniqueness and existence of the limit cycle r of Example 2, prove that any trajectory starting from any point different from 0(0, 0) will take r as w-limit set as t -+ oo. 8. Suppose in a region G that IRI < h/2 holds everywhere, where R(x, y) is the radius of curvature of a trajectory of system (1.1) at (x, y) and h is a diameter of some bounded set V in G. Prove that then G does not have any closed trajectory which contains V in its interior.
§1. FUNDAMENTAL CONCEPTS
23
9. Use Theorem 1.11 and the Dulac function similar to B(x, y) = xkyh to prove that when
the system
dxjdt = x(ax +by+ c),
= y(a1x + b1y + cl) when u = 0 the system has
dyjdt
does not have a closed trajectory, and integral but does not have a limit cycle. 10. Suppose in the system of equations
dxjdt
a first
= y- F(x), dyjdt = -g(x), F(O) = 0, g(x) is an odd function, xg(x) > 0 (x =I 0),
F(x) is an even function, g'(O) > 0, and F(x) and g(x) have continuous second-order derivatives. Prove that 0(0, 0) is a center [38].
§2 . Multiplicity and Stability of Limit Cycles In §1 we have defined the stability of limit cycles for a system of equations
dyfdt = Q(x, y).
dxfdt = P(x, y),
(2.1)
In applications only a stable limit cycle has practical significance, since every spiral sufficiently close to a limit cycle can approximately represent an oscillation of constant amplitude independent of initial conditions; and an unstable limit cycle fs similar to an unstable equilibrium position in mechanics, which in reality does not exist; hence, how to distinguish stability of limit cycles becomes a very important problem. First of all, if the existence of a limit cycle is determined from the fixed point of a point transformation, then no matter whether P and Q on the right sides of (2.1) are continuous or not, we can, under suitable conditions, distinguish whether the limit cycle is stable, provided that the point transformation is continuous. This is the often-used Konigs theorem in the theory of nonlinear oscillations. DEFINITION 2.1. Let 8 = f(s) be a continuous point transformation which carries some line. segment l into itself, and let s* be a fixed point of this transformation, i.e., s* = f(s*). If there exists a small neighborhood of s* (on l) such that for any point s inside it the sequence of points St
= f(s), 82 = /(81}, ... , 8n+l = f(sn), ...
always converges to 8*, then 8* is stable under this point transformation. Conversely, if in any small neighborhood of 8* we can find a point 8 such that the above sequence of points do not converge to 8*, then 8* is called unstable. (1) ( 1 )Note that under this definition, when the right sides of (2.1) are continuous and the uniqueness of its solution is assured, an unstable fixed point may correspond to a one-sided compound limit cycle.
2S
THEORY OF LIMIT CYCLES
26
THEOREM 2.1 (KONIGS). Lets= f(s) be a continuous point transformation of a line segment l into itself, and let s = 0 be a fixed point of this transformation. If the section of the arc near the origin in the curves= f(s) in the (s, s) plane lies in the angular region ls/sl :51- E (~ 1 +E), E > 0,
(2.2)
then the fixed point s = 0 is stable (unstable). PROOF. Suppose ls/sl :5 1- E = 6 < 1 near the origin, i.e., there exists a small neighborhood lsi :5 11 of s = 0 such that for all points s :f 0 we have lsi :5 6lsl < lsi. Then, provided that lsi :5 17, the sequence of numbers lsl,lsll,ls21, ... will satisfy the inequality lsnl :5 lsl6n. Hence lsnl- 0 as n-co, i.e., s = 0 is stable. > 1, i.e., Conversely, if for every lsi :5 11 we have ls/sl ~ 1 + E = lsi ~ elsl > lsi, then it is easy to see any sequence of points s, SI, 82, ... cannot converge to 0; hence s = 0 is unstable.
e
COROLLARY. I/ the functions= f(s) has a derivative at s = 0, then when lds/dsls=O
< 1 (> 1)
(2.3)
s = 0 is stable (unstable). From this we can see that for some system whose right side may not be continuous, if there exists a point transformation on some transversal of the closed trajectory r which satisfies condition (2.2) or (2.3) and the solution of the equation depends continuously on the initial condition in some small neighborhood of r' then r is a stable or unstable limit cycle. EXAMPLE 1. Consider the point transformation obtained in Example 3 of §1 on the negative x-axis x1
= Jh 2 + (xo- /o)2 -
which has a fixed point x(j = h2 /8/o we verify condition (2.3):
< (dx1)
3/o,
- / 0 , and we know that
x(j
> f 0 • Now
x(j - fo < 1. ..jh2 + (x(j - /o)2 Moreover, although every trajectory in the vicinity of the closed trajectory r overlaps r on the line x = -/0 , it is obvious that the solution varies continuously with respect to its initial condition. Thus, from the corollary of Theorem 2.1, we can prover is a stable limit cycle. If condition (2.2) or (2.3) is not satisfied (for example, if the point transformations= f(s) satisfies lds/dsls=O = 1), then there is no way to distinguish the stability of the corresponding limit cycle. 0
dXQ
=
:ro=Xo
§2 . MULTIPLICITY AND STABILITY
27
In order to further study this problem, we shall assume P and Q on the right sides of (2.1) have continuous partial derivatives of any required order. Suppose system (2.1) has a closed trajectory r, negatively oriented, whose equations are y = g(t), (2.4) X= f(t), where f and g are periodic functions of t with period T. Now introduce curvilinear coordinates (8, n) in a sufficiently small neighborhood of r, where 8 denotes arc length on r' measured from a fixed point on r' positive in the clockwise direction, i.e., the direction of increasing 8 is the same as the direction of increasing t, and where n is the length of the normal of r, whose outward direction is taken as positive. Suppose the equations of r with s as parameter are
x = rp(8),
(2.5)
For a point A in the vicinity of r, assume that A lies on the normal line of a point Bon r (Figure 2.1). Let the rectangular coordinates of B be (rp(s),.,P(s)). Then the formula connecting the rectangular coordinates (x, y) of A and its curvilinear coordinates (s,n) is(2)
x = rp(8)- nt/11 (8),
y = '1/1(8)
+ nrp'(s),
(2.6)
where
rp'( 8) = dxl = Po '1/1'(8) = dyl = Qo d8 8 JPJ+Qf d8 IJ JPJ +Qf Po and Qo being the values of P and Qat B, i.e. Po= P(rp(8),'1/1(8)),
(2.7)
Qo = Q(rp(8),'1/1(8)).
Substituting (2.6) into (2.1), we obtain
dy '1/1 1(8) + rp 1 (8)(dn/d8) + nrp11 (8) dx = -'-rp..,.-7'(-78)----''1/1":-:'-7-(87)(7-:d----'n/,. . ,d;- - ;8):----_----'n.,P'-:- :,: -(--7-8) 7, Q(rp(8)- nt/1 1 (8),'1/1(8) +nrp1 (8)) = P(rp(8)- nt/1 1 (8), '1/1(8) + nrp 1 (8)) · (2)We can see from a(x,y) a(s,n)
=
ltp'-m/l' '1/J'+ntp"
that, if we require P and Q to have continuous first partial derivatives, then tp1 , '1/J', tp 11 , and f/1 11 are continuous; hence, provided In I is sufficiently small, we have a(x, y)ja(s, n) > 0,
i.e., in a sufficiently small neighborhood of r, (2.6) represents a coordinate transformation which preserves its orientation. If we choose n as the arc length of the orthogonal trajectory of (2.1), measured from the point on r instead, then the coordinate transformation can even be global; but this fact is not needed here.
28
THEORY OF LIMIT CYCLES
FIGURE 2.1
From this we can solve the equation
dn ds
=
Qcp'- Pt/J'- n(Pcp" + Qt/J") Pep'+ Qt/J'
(2.8)
= F(s, n). Now we shall use equation (2.8) to study the stability of r.
Since cp and 1/J are periodic functions, we known (2.8) is a nonlinear equation with periodic coefficients, taking n as unknown function and s as independent variable. From (2.7), we can see that (2.8) has a zero solution, n = 0, which corresponds to the periodic trajectory r of (2.5). When P and Q have continuous partial derivatives of first order, F(s, n) also has first-order continuous partial derivatives with respect to n; hence (2.8) can be written as
dn/ds
= F~(s, n)in=O · n + o(n).
(2.9)
In order to compute F~(s, n)in=O• we note that
cp"(s) = '1/J"(s) =
(PJ
~oQ 5 )2 [P~Qxo + PoQo(Q11o- Pxo)- Q~P11 o],
(P~ :oQ5)2 [P~Qxo + PoQo(Q11o- Pxo)- Q~P11 o]
(2.10)
(where P 11o, Pxo, Q11o, and Qxo denote the values of the partial derivatives of P and Qat n = 0), and we know that P0 cp" + Q0 1/J" = 0. Noting again that P = Po and Q = Qo when n = 0, and using the above relation formula, we can easily calculate
F' ( n
s, n
)I _ n-0
= PJQ 11o- PoQo(P11 o + Qxo) (PJ + Q5)3/2
+ Q5Pxo
= H( )· 8 '
(2.1l)
here H(s) is the curvature of the orthogonal trajectory of system (2.1) at the point B. Hence the linear approximation equation is dn/ds = H(s)n,
(2.12)
§2 . MULTIPLICITY AND STABILITY
29
whose solution is
n = noexp
(lol
(no= n(O)).
H(s')ds')
(2.13)
From this we get THEOREM 2. 2. Let the arc length of a closed trajectory r of system (2.1) bel. Then, when
fo' H(s) ds < 0 r is a stable (unstable)
(> 0),
{2.14)
limit cycle.
PROOF. When (2.14) holds, we have from (2.13)
ln(l)l < lnol (> lnol). This shows that the zero solution of {2.12) is asymptotically stable {negatively oriented, asymptotically stable). From the well-known theorem of Lyapunov we know that the zero solution of {2.9) is also asymptotically stable (negatively oriented, asymptotically stable) and it is obvious that r is a stable (unstable) limit cycle. COROLLARY (DILIBERTO). If H(s) < 0 (> 0) holds everywhere along a closed trajectory r then r is a stable (unstable) limit cycle [39]. Now we rewrite (2.14) in a familiar form. Substituting ds = into (2.14), we ge~
.jPJ + Q~ dt
{T P. 2 1 Q2 [PJQyo- PoQo(Pyo + Qxo) + Q~Pxo] dt fo o + o = {T + Q _ PJPxo + PoQo(Pyo + Qxo) + Q~Qyo] dt fo xo. yO PJ + Q~ {T 1 j d(P.2 + Q2)
f' H(s) ds =
fo
[p
=
lo
(Pxa
+ Qyo) dt- 2!r
P} + Q5o
=loT (Pxo + Qyo) dt. Thus we obtain the well-known THEOREM 2.3. !fin a closed trajectory
iT (~= then
r is a stable (unstable)
+ ~~)
limit cycle.
r
dt < 0
of system (2.1}
(> 0),
(2.15)
THEORY OF LIMIT CYCLES
30
From the above two theorems, we know that when r is a periodic cycle, semistable cycle or compound limit cycle, we must have
{T
Jo
(8P8x + 8Q) {1 8y dt Jo H(s)ds =
= 0.
(2.16)
However, when r is an ordinary stable or unstable limit cycle, (2.16) may also hold, since (2.14) and (2.15) are only sufficient conditions, not necessary conditions. DEFINITION 2.2. When condition (2.14) or (2.15) holds, r is a single cycle or coarse cycle:(*) when condition (2.16) holds, r is a multiple cycle or
noncoarse cycle. In order to study stability of multiple cycles, we integrate (2.8) from s = 0 to s = l, and obtain w(no) = n(l,no)- no=
fa' F(s,n(s,no)) ds.
(2.17)
\ll(no) is called a successor function which denotes, when a point (0, no) in a neighborhood of r moves one round in the direction of increasing t (i.e., s) and arrives at the point
(l, n(l, no)) = (0, n(l, no)), the difference between the n-coordinates of these two points. Thus it is obvious \ll(no) = 0 is a necessary and sufficient condition for any trajectory passing through (0, n 0 ) to be a closed trajectory; noW(no)
< 0 (> 0)
r
is a necessary and sufficient condition for sufficiently small lnoli and w(no)
to be stable (unstable) for all
< 0 (> 0)
is a necessary and sufficient condition for r to be externally stable but internally unstable (externally unstable but internally stable) for all sufficiently smalllnol· Since we know that \11(0) = 0, we have THEOREM 2.4. If
w'(o) < o (> o) then
r
(2.18)
is a stable (unstable) limit cycle; if
w'(O)
=o
but
w"(o)
# o,
(2.19)
then r is a semistable limit cycle. (•)Translator's note: A single or coarse cycle is referred to in English journals as hyperbolic.
§2 . MULTIPLICITY AND STABILITY
31
Using a well-known theorem in differential calculus, we can in general establish THEOREM 2. 5. If, for a given closed trajectory
w'(O) = w"(O) = ... = wCk- 1)(0) = o,
but
r,
wCk)(O) < o (> o), (2.20)
when k is odd, then r is a stable (unstable) limit cycle. If
w'(o) = w"(o) = ... = wCk- 1)(0) = o, where k is even, then
r
but
wCk)(o) -1 o,
(2.21)
is a semistable limit cycle.
REMARK. If P and Q (hence 'IJ(no)) are analytic functions, then (2.20) or (2.21) is also a necessary condition for being stable (unstable) or semistable. DEFINITION 2.3. Any r satisfying condition (2.20) or (2.21) is called a k- fold limit cycle. From this definition it is easy to see if r is a k-fold limit cycle, then no = 0 is a k-multiple root of the equation 'IJ(n0 ) = 0. If we draw the linen= no and the curve n = n(l, no) in the (n 0 , n) plane, then the origin is a k-fold point of intersection of these two curves. In particular, 'if( no), which corresponds to the periodic cycle, is identically zero for all sufficiently small values of lnol· In the following we shall prove that for k = 1 Definition 2.3 and Definition 2.2 are the same, i.e. the conditions
w'(O) < 0 (> 0) and
fa' H(s) ds <
0 (> 0)
are equivalent. F6r this, we differentiate both sides of (2.17) with respect to
no, and obtain w'(no)
In order to compute equation
=fa' F~(s,n(s,no))n~0 (s,no)ds. n~ 0 (s,
! n~0 (s,
and the initial condition
no), we note that it satisfies the variational
no)=
F~(s, n(s, no))· n~ 0 (s, no)
n~ 0 (0, no)
= 1; then we obtain
1 ( ) ];" F~(s,n(.s,no)) ds nno s, no = e o ,
and from this ,y,/( '!I!' no
)-!o'F'( n s, n (s, no ))J.•·F~(r,n(r,no))drds e 0
0
J.'F~(s,n(s,no))ds - . l =eo [ J. ·F~(r,n(r,n 0 ))dr] =eo 1
0
32
THEORY OF LIMIT CYCLES
Note that n(8, 0)
= 0, and we know that
"\II'(O)
= ef~ F~(s,n)ln=ods _ 1 = ef~ H(s)ds -1.
(2.22 )
From this it is obvious "111'(0) < 0 (> 0) is equivalent to f~ H(8) d8 < 0 (> 0). In general, w(k)(O) can be obtained from the above method, but its representation formula becomes more complicated ask increases (see [40]). For example, "111 11 (0) = "111 111 (0)
fol F::2ef: F~(T,n)dT d8ln=O,
when "111'(0) = 0,
= fol F::~e 2 f: F~(T,n)dT d8ln=O,
w< 4>(o) =
when "111'(0) = "111"(0)
= 0,
fol F~!)e3 I: F~(T,n)dT dsln=O +2
1 1
F"' nse 2 J." o F~(T,n) dT
0
(2.23)
1 8
F"n2efo F~(T,n) dT dt d 8 In==O,
0
when "Ill., (0) = "Ill" (0) = "Ill"' (0)
= 0,
etc.
From the above formulas we see that in order to compute "111 11 (0), "111 111 (0), ... we first have to compute 2 (8, 0), F::~ (8, 0), ... , i.e., the coefficients of the Taylor series expansion of F(8, n) at n = 0. In the following, we introduce a method of the Japanese mathematician Minoru Urabe [41]; using this method we can also obtain, when r is a periodic cycle, the approximate representation formula for the period of a closed trajectory in the vicinity of r if we know that the period of r is T. Now we use the transformation (2.6), but still use t to denote the independent variable of the parametric equations of r j let T (time) be the independent variable of the parametric equations of a trajectory in the vicinity of r, with r(to) =to. Let x1(r),y1(T) be the point coordinates of this trajectory. Thus, referring to (2.4) and (2.6), we have
F::
r(to) =to,
x1(r) = f(t)
+ p(t)l(t),
y 1(r) =
g(t) + p(t)m(t),
(2.24)
where
= n(s,no), m(t) =
respect to t, we obtain
p
dr _ 1 dt -
dr
Po + l
dp dt
dl
+ Pdt'
dp
Ql dt = Qo + m dt
dm
+ P(JI•
(2.25)
§2 . MULTIPLICITY AND STABILITY
33
where P1 = P(x1(r),y1(r)) and Q1 = Q(x1(r),y1(r)). Let k = k(t) = (Qo dPo- Po dQo) (PJ dt dt
+Q~)-3/2.
Then from (2.10) it is easy to deduce that dlfdt
= kPo,
dmfdt = kQo.
Substituting in (2.25) and solving for dpfdt and dr/dt, we obtain dpfdt =
JP~ + Qg(1 + kp)(PoQ1- QoPt)(PoP1 + QoQt)-
1,
(2.26)
drfdt = (PJ + Qg)(1 + kp)(PoP1 + QoQt)- 1 •
If we let
a ax
a ay
D=l-+m-
and assume P and Q are analytic functions, then when p is sufficiently small we have p2 pn P1 =Po +pDPo + -D 2Po + ··· + -DnPo + ··· 2! n! ' pn Q1 =Qo+pDQo+···+---.,DnQo+···.
n.
Substituting in (2.26), we obtain two equations with t as independent variable and p as unknown function, satisfied by a trajectory in a neighborhood off:
dt = VP~ + Qg(1 + kp) · [p(PoDQo- QoDPo) +
~~ (PoD 2Qo- QoD 2Po)+···] 2
] -1
· [ PJ + Qg + p(PoDPo + QoDQo) + ~! (PoD 2Po+ QoD2Qo) + · · ·
,
dr dt = (1 + kp)[Po2 + Qo2] · [Po2 + Q02 + p(PoDPo + QoDQo) + · · ·]- 1 .
(2.27) From the first term in (2.27) we immediately obtain dn dp dt ds=dt.ds _ (1 k ) p(PoDQo- QoDPo) + (p 2/2!)(PoD 2Qo- Q0 D 2Po)+··· + p PJ + Q5 + p(PoDPo + QoDQo) + · · · (2.28)
THEORY OF LIMIT CYCLES
34
Expanding the right side of (2.28) into power series in p and comparing with the expansion formula of equation (2.8), dn ds
= F'n (s, 0)n + 2!1 F"nl (s, 0)n 2 + ...
(2.29)
we can easily determine F~ ( s, 0), F~, ( s, 0), . . .. In particular, we have
- QoDPo = H( ) Fn' (s, O) = PoDQo p,2 Q2 S ' 0
+
0
which is the same as (2.10). Also, from the second term of (2.27) we obtain( 3 ) dr = 1 + dt
(k _PoDPoPJ ++Q5 QoDQo) ... p+
_ 1 + 2PoQo(Pxo- Qyo) + (Q5- PJ)(Pyo + Qxo) + ... (PJ + Q5)3/2 p • Integrating both sides from to to to + T, we get r(to
+ T)- r(to) = T + ro+T
lto
2PoQo(Pxo- Qyo) + (Q~- PJ)(Pyo + Qxo) (t) dt + ... (PJ + Q5)31 2
p
'
which is an approximate representation formula for the period of the closed trajectory (x1(r),y1(r)) in the vicinity off. In the following we shall study a special type of limit cycle, and a method of distinguishing such cycles. DEFINITION 2.4. If F(s,n) in (2.8) keeps a constant sign in the outside and the inside of a small neighborhood of r, then r is called a monotonically close cycle [42]. THEOREM 2. 6. A necessary and sufficient condition for a limit cycle
r
of analytic system (2.1) to be a monotonically close stable (unstable) cycle is that i = 1, 2, ... 'k - 1, but O'k
< 0 (> 0)
where k is odd; a necessary and sufficient condition for r to be a monotonically close semistable cycle is that O'i = 0, i = 1, ... , k- 1, but O'k =/= 0, where k is even. e)The right sides of both formulas of (2.27) do not contain the unknown function Tj hence, if we obtain p(t) from the first formula, then we obtain T(t) after integrating the second formula.
§2 .
MULTIPLICITY AND STABILITY
35
PROOF. From the Taylor expansion of F(s,n) with respect ton, we can see that in order to keep a constant sign both outside and inside a small neighborhood of r it is necessary and sufficient that there exist an integer k such that for all s we have (k)( s,O ) Fnk
=/:0.
(2.30)
But from the right side of (2.28) we can see that the above conditions are in fact equivalent to O't = 0'2 = · · · = O'k-1 = 0 but uk =I 0. When conditions (2.30) hold, we can obtain the conclusion about stability from (2.29). COROLLARY. On a monotonically close multiple limit cycle, H(s) = 0 holds everywhere. In other words, a monotonically close multiple cycle is one branch of the curve H(s) = 0. But we should note that the real branch of H(s) = 0 is not necessarily a trajectory of system (2.1); even if it is a trajectory, it is not necessarily closed. EXAMPLE 2. Study the system
~~
=-y+x(x2+y2-1)2,
~~
=x+y(x2+y2-1)2.
After transforming the system into polar coordinates, it is easy to prove that it has a unique limit cycle r: x 2 + y 2 = 1; moreover, it is semistable. Calculating the curvature of its orthogonal trajectory, we obtain (x2 + y2)(x2 + y2- 1)[(x2 + y2- 1)5 + 5(x2 + y2 - 1) + 4] H(s) = {(-y + x(x2 + y2 - 1)2]2 + [x + y(x2 + y2 - 1)2]2}3/2 Thus we can see that the locus of H(s) = 0 comprises the origin, the unit circler, and the circle x 2 + y 2 = 8 < 1; the latter circle is not a trajectory of the system. According to the definition of a limit cycle, it is easy to get the misconception that monotonically close cycles are perhaps more commonly seen. If this were true, then to distinguish the existence of multiple cycles would become easier. But, in fact, a monotonically close cycle is just a very special limit cycle. We shall see in §11 that when P and Q of the system (2.1) are quadratic polynomials, its limit cycles are, generally speaking, not monotonically close. Finally, we shall study the Lyapunov stability of a periodic solution bn a limit cycle. As is well known, this periodic solution cannot be asymptotically Lyapunov stable; hence we can only study whether it is stable or unstable. For nonlinear stationary systems, since the first integral exists, there only exist periodic cycles in its phase-plane. Generally speaking, the period always changes with its amplitude; hence the periodic solution in this kind of periodic cycle is always Lyapunov unstable; but for limit cycles of nonstationary systems, the following theorem holds.
THEORY OF LIMIT CYCLES
36
THEOREM 2. 7. If r is a single stable cycle of system (2.1), then any periodic solution on r must be Lyapunov stable. PROOF. Let x = r,o(t), y = 1/J(t) be a periodic solution of (2.1} on period T. From the hypothesis we have
h=
loT [Px(r,o(t), 1/J(t)) + Qy(r,o(t), 1/J(t))} dt < 0.
r, with (2.31)
Substituting x = r,o(t) + u and y = 1/J(t) + v in (2.1), we can get a system of equations with periodic coefficients satisfied by u and v:
du/dt = Px(r,o, 1/J)u + Py( r,o, 1/J )v + .. ·, dv/dt
= Qx(r,o, 1/J)u + Qy(r,o, 1/J)v + ···,
(2.32)
whose linear approximate equations have characteristic exponents 1 and eh. Hence the Jordan form of the characteristic matrix is
C=
(
eh
0
If we take
B= (h/T 0
0) 1
.
0)
0 ,
then eTB =C. Now let X(t) denote the fundamental solution matrix of linear approximate equations of (2.32}. We may as well assume X(t) satisfies the relation
X(t + T} = X(t)C. We know the linear approximate equations have a periodic solution
(r,o' (t}, 1/J' (t)). Therefore X(t)
= (Ut(t) Vt ( t)
(3r,o'(t)) {31/J' (t)
1
f:. 0, and (u 1 (t}, v1 (t)) is a solution which is linearly independent of (r,o'{t), 1/J'(t)). Let Z(t) = eTB x- 1 (t). It is easy to see that Z{t) = Z(t + T),
where (3 and
z-•(t) = X(t}e-TB =
-ht/T
(
Ute' Vte ht/T'
(3r,o'(t)) (31/J'(t) ,
IZ- 1 (t)l-# o. Now in {2.1} we make the change of variables
+ r,o(s}, e-hsfT Vt(s)z + ,P(s),
x = e-hs/T Ut(s)z y =
(2.33)
§2 . MULTIPLICITY AND STABILITY
37
which is periodic with respect to s. We get at once
!~ = cp'(s) + [~e-hs/T u1(s) +
!
(e-ha/T u1 (s))] z + R1 (s, z),
where R1 ( s, z) has period T with respect to s, and is an infinitesimal of second order with respect to z. Rearranging the terms, we get e-hs/T u 1(s) [
!: - ~]
+ [cp'(s) + z! (e-ha/T u1(s))] (
~: -
1)
= R1 (s, z). (2.34)
Similarly, from the representation formula for dy / dt we can deduce that e-ha/T v 1(s) [
!: - ~] +
[,P'(s) + z! (e-hs/T v1 (s))] ( ~:-
1) = R2(s, z). (2.35)
H we consider (2.34) and (2.35} as a system of linear algebraic equations with the unknown functions dzjdt-hz/T and ds/dt-1, then the determinant of its coefficients when z = 0 is IZ- 1(s)l # 0; hence, when lzl is very small, it is also not zero. Thus we can solve
dz dt
h
= Tz+q1(s,z),
ds dt
= 1 + q2(s,z),
(2.36)
where Qi(s, z) has period T with respect to s, and is a second-order infinitesimal with respect to z. The above formulas can also be rewritten as
dz ds
h
= Tz+p1(s,z),
dt ds = 1+P2(s,z),
(2.37)
where Pi(s,z) h~ the same properties as Qi(s,z). Now suppose the solution of the first equation of (2.37) to be z(s; zo, so). We shall show that when lzo I « 1 we have 0
> so, k > 0, so
and k are constants, and
(2.38)
zo is the value of z when
8 =so. In fact, from the first equation of (2.37), we have z(s; zo, so)= zo exp(h(s- so)/T) exp
(1:
z- 1 p1(r, z) dr),
(2.39)
where z = z(r; z0 , s0 ) under the integral sign on the right. Since (ojoz)p1(s, z) is continuous and periodic with respect to s, and Pl ( 8, 0) = 0, we know that, as z--+ 0, z- 1p 1(s, z) approaches zero uniformly with respect to s. That is to say, for € > 0, there exists 0 = o(e) > 0 such that, when lzl ~ o, lz- 1pl(s, z)l < € for all 8. Now we take 0 < e < -h/T and lzol ~ ~o(e)e"o(h/T+e). Then, using (2.39) and the method of contradiction, we can easily deduce that
iz(8; zo, 8o)l < o(e)
THEORY OF LIMIT CYCLES
38
for all s ~ s0 • Based on this, we use (2.39) again to deduce (2.38) with k = e-so(h/T+e) and o: = -h/T- ~. Substituting z(s; zo, so) into the second equation of (2.37) and integrating, we get
t- to= s- so+["' P2(s, z(s; zo, so)) ds.
(2.40)
so
According to (2.38) and the properties of P2(s, z) we know that when lzol is sufficiently small, we can solve for s from (2.40) and get s = s(t; z0 , s0 ); and we also have
ls(t; zo, so) - tl ~ Iso-to! + llzol 2,
(2.41)
where l > 0 is a constant. Inequalities (2.38) and (2.41) show that the solution of (2.36) is Lyapunov stable. But the solution of (2.36) under the transformation (2.33) just happens to correspond to the solution x = rp(t), y = .,P(t) of (2.1). Hence this shows that the latter is a Lyapunov stable solution. The theorem is completely proved. It is easy to see if r is a single unstable limit cycle of equations (2.1), then any periodic solution on r must be negatively oriented and Lyapunov stable. For the problem of Lyapunov stability of periodic solutions on multiple cycles, we refer to the work of V. I. Zubov [44]. His results are very interesting but will not be introduced here. Finally, if a singular closed trajectory containing a finite number of saddle points is an internal limit cycle, then there are simpler methods to determine the stability of its trajectory, which are very useful for the study of some polynomial stationary systems. We shall introduce these methods in the next section. Exercises 1. Let
f( Prove that
) _ Q(rp(s)- n.,P'(s), .,P(s) + nrp'(s)) s, n - P(rp(s)- nt/l'(s), .,P(s) + nrp'(s)) ·
j~(s,O)
= 0 and F::(s,O) = 0 are equivalent, and
F~(s,O)
=
aojanln=O, where()= arctan(Q/P). 2. Calculate F::(s, 0) and '11"(0). 3. Let o:(s, n) denote the difference between the angle of inclination of any point B on r and the angle of inclination of the trajectory of a point A(s, n) lying on the normal of r of B. Prove that r is a monotonically close semistable cycle if and only if for all s, sino: takes an extreme value at n = 0
[45].
§2 . MULTIPLICITY AND STABILITY
39
4. Show that the system
= yf../2 + x(2x 2 + y 2 dyfdt = -..f2x + y(2x 2 + y 2 dxfdt
1) 2 ,
1) 2
has a nonmonotonically close, semistable cycle, but that this semistable cycle can be made into a monotonically close cycle through an affine transformation. 5. Let P(x, y) and Q(x, y) have second-order continuous partial derivatives in a simply connected region G, and assume that P(x, y) = 0 can be represented by y = f(x), and Q(x, y) = 0 can be represented by x = cp(y). Let the functions
a ( 1 aQ) A(x,y) =ax Q ay
I
a (1 aP) B(x,y) = - ay Pax
have the same signinG, never change sign, and assume that in any subregion at least one of A, B is not identically zero. Also assume that near the upper (lower) half of y = f(x), Band p-tapjax have opposite (same) sign, and near the right (left) side of x = cp(y), A and Q- 1 aQjay have opposite (same) sign. Then {2.1) does not have multiple cycles in G [23]. 6. Apply the above exercise to prove that the van der Pol equation has a single cycle.
§3. Limit Cycles in Rotated Vector Fields In the previous two sections we made a preliminary study of the existence, nonexistence and stability of limit cycles; in this section and the following one we shall study differential equations containing a parameter, and the situation of limit cycles following the variation of parameters. Of course, these two problems are related just as the real roots of a quadratic equation involving a parameter. H a root is simple, then as the parameter varies slightly the root also varies slightly, but it does not disappear; if the root is a double root, then no matter bow slightly the parameter varies, the double root may disappear or split into two simple roots. Since it is in general not possible to write down the equation of a limit cycle (except for some artificially constructed ones), its position can only be approximately determined, and when the parameter of the equation varies, it is difficult to ascertain the generation, disappearance, and change of form of the limit cycle. Hence, people naturally hope to study first the simple case with rather regular variation; and this is the theory of rotated vector fields to be discussed in this section. H the vector fields following the variation of the parameter always rotate in the same direction, independent of the position of the point on the plane, then we can prove that variation of limit cycles is most regular, i.e., monotonically expanding or contracting. This situation is similar to the familiar fact that if a smooth plane curve y = f(x, a) translates along they-axis as a varies, then its point of intersection with the x-axis (i.e., the real root of f(x, a) = 0) translates either to the left or to the right. The theory of rotated vector fields was first established by Duff in 1953 [46]; and later Seifert [47] and Chen Xiang-yan [48]-[50] successively weakened Duff's condition, but still guaranteed that the main conclusions held. In 1978 Ma Zhi-en [291] also discussed how singular points and singular closed trajectories vary in a family of rotated vector fields in which a singular point can move. To simplify our presentation, we introduce the fundamental theory
41
THEORY OF LIMIT CYCLES
42
of rotated vector fields according to Duff's paper, but in the last part we briefly discuss the results of [48], [50], and [291]. Consider the equations
dxj dt = P(x, y, o:),
dyjdt = Q(x, y, o:),
(3.1)
where P and Q are continuous functions of x and y and the parameter o:, and satisfy the following conditions: 1. P and Q satisfy Lipschitz conditions in any bounded region: IP(x2,y2,o:)- P(Xt.Y1,o:)l :5 M(lx2- x1l
+ IY2- Y11),
IQ(x2, Y2, o:) - Q(xt. Yt. o:)l :5 M(lx2 - x1l
+ IY2- Y11).
(3.2)
2. 8Pj8o: and 8Qj8o: exist and are continuous. 3. The plane vector field F(o:) defined by (3.1) has only isolated singular points. DEFINITION 3.1. If when o: varies in [0, T] the singular points of F(o:) remain unchanged, and at all regular points p
8P ao:
Q (3.3)
8Q > 0, ao:
and if, moreover, there exists a positive function k(x, y) such that, for all (x, y), T > 0 is the smallest positive number for which
P(x, y, o: + T) = -kP(x, y, o:), Q(x, y, o: + T) = -kQ(x, y, o:),
(3.4)
then we say that F(o:) (for all o:, 0 :5 o: < T) forms a complete family of rotated vector fields. Let 0 be the angle which the vector (P, Q) makes with the x-axis. From (3.3) and (3.4) it is easy to see that for any 0:1 and 0:2 with lo:1- o:2l < T the following inequality holds: 0
< IO(x, y, o:t)- O(x, y, o:2)l < 1r.
(3.5)
Now we shall explain the geometrical significance of the above definition. First, from (3.3) it is easy to see that
80 ao:
= _i_ tan-1 Q = ao:
P
p
p2
1
+ Q2
Q
8P
8Q > 0
ao:
ao:
(3.6)
at all regular points. This shows that the vector at any regular point of the plane vector field defined by (3.1) rotates in a counterclockwise direction as o:
§3.
ROTATED VECTOR FIELDS
43
increases. From (3.4) and (3.5) we know that O(x, y, o: + T) = O(x, y, o:) + 71'j moreover, when the parameter varies from o: to o: + T, its corresponding vector completes exactly half a rotation, but does not necessarily keep the same length; and when o: increases to 2T, the vector completes exactly one round and returns to its original direction. This is the meaning of the S
dyfdt
= Q(x, y),
(3.7)
we can construct a system of equations containing o:, dxfdt =.Pcoso:- Qsino:,
dyfdt
= Psino: + Qcoso:.
(3.8)
It is easy to see that when o: varies in [0, 11'), (3.8) forms a complete family ofrotated vector fields; moreover, the angle ofrotation from (3.7) to (3.8) is just o:, and the length of the vector remains unchanged. Hence we call (3.8) a uniform family of rotated vector fields.
THEOREM 3 .1. The index of a singular point of the complete family F( o:) does not vary with o:. PROOF. In a neighborhood of a singular point 0 of F(a) we construct a simple closed curve with interior which contains 0 and no singular point other than 0. From (3.5) and the well-known property of indices of a closed curve with respect to the vector fields, we see at once that this closed curve with respect to the two vector fields has same index, i.e., the index of the singular point in the interior of this closed curve does not vary with o:. THEOREM 3. 2. Closed trajectories of different vector fields of a complete family do not intersect. PROOF. Let the closed trajectory r 01 of F(at) and the closed trajectory of F(o:2) meet at a point A, and, as t increases, suppose r 01 at A moves in the exterior-to-interior direction. Then from conditions (3.3) and (3.4) we know the trajectories of F(o:t) all move in the interior-to-exterior direction at all the points of F(o:t). Thus r 01 after entering into r 02 cannot run out of it again; therefore it cannot be a closed trajectory. This contradicts the assumption.
r
02
THEOREM 3. 3. If an externally stable cycle r 00 is positively (negatively) oriented, then for any sufficiently small e > 0 there exist at < ao (at > ao) such that for any a between a1 and ao there exist at least one externally stable cycle r and one internally stable cycle r~ in an outer €-neighborhood Q
44
THEORY OF LIMIT CYCLES
FIGURE 3.1
off ao (these cycles may coincide); moreover, there exists an external8(~ e)neighborhood of r a such that it is covered completely by the closed trajectories of F(a) (a between ao and at). PROOF. We shall only prove this theorem for the case when r 00 is positively oriented. Although an application of the Annular Region Theorem would make the proof simpler, we prefer to use Duff's original proof because it makes extensive use of knowledge and technique in mathematical analysis. This is very important to researchers in the qualitative theory of differential equations. As in Figure 3.1, suppose the boundary curve of an outer ue-neighborhood of rOo and the normal line ' througl a point A of r ao meet at B (here g > 0 is sufficiently small and 0 < u < 1). The positive semitrajectory 'Y+(B, ao) of F(a0 ) through B meets AB again at B 11 and the positive semitrajectory 7+(B, a) of F(a) (a< ao) passing through B meets BBt again at B2(a). Now we prove that when a decreases from a 0 (and u is sufficiently small) we can make some B2(a) coincide with B, and hence 'Y+(B, a) becomes a dosed trajectory of F(a) and lies entirely inane-neighborhood of r ao· Under the assumptions of system (3.1) in this section, any trajectory is a differentiable rectifiable curve. Let s be the arc length of 'Y(B,o:o), positive when measured from B in the direction of increasing t; let n denote the length of an outward normal of this trajectory. Suppose the parametric equations of 7(B, o:o) are x =
/(s),
y = g(s).
(3.9)
45
§3. ROTATED VECTOR FIELDS
Using the curvilinear coordinates (s, n), let the equation of 1+(B, a) (a< ao) ben= h(s, a). It is easy to see that the rectangular coordinates of any point and its curvilinear coordinates can be related by
x- f(s) = h(8)g'(s),
y- g(8) = -h(s)f'(s).
{3.10)
Hence
h(s) = J(x- /{s)) 2
+ (y- g(8))2.
Using {3.10) to find h'(s), we get
h' (8)
= g' (8) ~:
-
I' (8) ~~ .
(3.11)
Let Sa be the arc length of 1+(B, a), and let Pa and Qa be P(x, y, a) and Q(x, y, a) respectively. Then
dx dx dsa Pa (1 + k*h)JP£ + Q~JP~ 0 + Q~ 0 - = - - = --;:;:~~...:....._-~~~~~....=..!!...-~ ds dsa ds J P~ + Q~ PaPa 0 + QaQa 0 _ Pa(1 + k*h)JP~0 + Q~ 0 PaPa 0 + QaQao
dy ds
(1 + k*h)JP~ + Q~JP£0
Qa
+ Q~ 0
-=--;:x:=~.:....._--~==:...._-7-::....!;:,--="--..:...~
+ Q~ PaPa + QaQao Qa{1 + k*h)JP~0 Q~ 0 PaPao + QaQao JP~
_ where
k*
=
0
(P.
ao
dQao _ Q dPao) ds ao ds
/{P2 + ao
Q2 )
a0
'
P010 and Qa 0 denote the values of P and Qat (xo, Yo, ao) respectively, and the curvilinear coordinates of (xo, Yo) are (s, 0). Moreover, we note that g'2 +/' 2 = 1; hence, substituting in (3.11), we get dh ds
= =
(g'(s)Pa- f'(s)Qa)(1 + k*h)JP2o PaPa 0 + QaQao
+ Q~o
(1 + k*h)y'P~ + Q~JP~ 0 + Q~o . •J. s1n '~'• PaPa 0 + QaQao
(3.12)
where 1/J is the angle between the angle of inclination of 1+ (B, a) when its arc length iss and the angle of inclination of 1+(B, a) at (s, n), i.e., O(s, 0, ao)0(8, n, a). Since P and Q satisfy a Lipschitz condition, it is easy to see that (J also satisfies a Lipschitz condition with respect to n. Moreover, from (3.3) we know that ao;aa has a positive lower bound p > 0 inane-neighborhood
46
THEORY OF LIMIT CYCLES
of r ooo i thus we get 1/J = O(s, 0, o:o) - O(s, n, o:o)
+ O(s, n, o:o) - 0( s, n, o:)
~ -Mh + ( ~:) oo• (o:o- o:) ~p(o:o- o:)- Mh,
(3.13)
where M > 0 is a constant and o:• is between o:o and o:. According to the conditions satisfied by P and Q, it is easy to see that the coefficient of sin 1/J on the right side of (3.12) can be greater than 1/2 provided that lo:- o:0 1 and c are sufficiently small. Hence, combining (3.12) and (3.13), we get dh 1 . 1 ds > 2 sm'l/J > '4[p(o:o- o:)- Mh]. Integrating this inequality from 0 to L, where Lis the arc length of 1+ (B, o:o) from B to B 11 we obtain h(L, o:) ~ it(o:o- o:)[1- e-ML/ 4 ]
= K(o: 0 -
o:),
where K > 0 is a constant. From this we can see that h(L, o:) > uc provided that o:o- o: > uc/ K, i.e., B2(u) has moved off of B1B to make B lie between B1 and B 2(o:). Since B2(u) moving along lis a continuous function of o:, we know there exists 0:2 < o:o such that lo:o- o:2l $ uc/K, and B2(o:2) = B, i.e. 1+(B, o: 2) is a closed trajectory of F(o:2). It is easy to see that if we prescribe a sufficiently small u, we can make this closed trajectory lie in an c-neighborhood of r ooo· Moreover, since o:2 - o:o as u - 0 and A is an arbitrary point on r ooo, for a fixed c > 0 we have a fixed 0:1 < O:o such that there exists an outer neighborhood Ne of r ooo which is covered completely by the closed trajectories of F(o:), and the whole of Ne lies in the outer €neighborhood of r ooo. According to Theorem 3.2, different closed trajectories of F(o:) in Ne do not intersect; for definiteness, assume the outer boundary curve of Ne is a closed trajectory r 001 ofF( o:l).(l) To prove that for every fixed o:, 0:1 < o: < o:o, there exist an internally stable cycle and an externally stable cycle (they may coincide) in the neighborhood Ne we only have to note that F(o:) in this neighborhood must have an innermost trajectory r 00 , which must be an internally stable cycle; for if it were internally unstable, then between r 00 and r ooo, we would have roo', where o:o > o: > o:'. Both roo' and r ooo together form an annular region; the ( 1 )We note that to prove the existence of limit cycles in the following Theorem 3.15, we only use the existence part of this theorem. Hence the proof of the stability of a limit cycle in this theorem can be established by applying the conclusions of existence in Theorem 3.15.
§3.
ROTATED VECTOR FIELDS
47
trajectories of r a, while meeting the annular region, cross it in the exterior-tointerior direction. Thus from Theorem 3.15 (see the footnote to the preceding paragraph) we know that in this annular region there will again exist a new closed trajectory of F(a) which should lie in the interior of r a· This contradicts the definition of r a· Similarly, inN£, F(a) must have an outermost closed trajectory r a', which must be an externally stable cycle. The theorem is completely proved. Please note: again from Theorem 3.2 and Theorem 3.3 we can only assert that a complete family of rotated vector fields defines on the transversal l a single-valued function o = a(n) such that for a point B(n) of l with coordinate non l, there exists one and only one closed trajectory r a(n) of F(o(n)) passing through it. Existence of the function a(n) is assured from the latter half of Theorem 3.3; the fact that a(n) is single-valued is assured by Theorem 3.2, and the continuity of a(n) can be obtained without difficulty from these two theorems by the method of contradiction. However, we have no way to be sure that there exist a sufficiently small outer or inner neighborhood of r ao and some a1 sufficiently close to ao such that for every a between ao and Ot, F(a) has a unique closed trajectory in this neighborhood. In fact, the following example is good enough to explain that this neighborhood may not exist. EXAMPLE 2. The equations
~: = -y + xtan [(r- r
0) 3
(sin r
~ ro + 2) J = P(x,y),}
~~ =x+ytan[(r-r0 ) 3 (sinr~ro dx dt = -y,
when r =I ro,
+2)] =Q(x,y)
dy dt = x,
when r = ro (3.14) (where r0 > 0 and r = y'x 2 + y2 ) have an unstable limit cycler= r0 • From this we construct a complete family of uniform vector fields F(a):
dxjdt = P(x, y) cos a- Q(x, y) sin a, dyjdt = P(x, y) sin a+ Q(x, y) cos a.
(3.15)
We shall study whether F(a) in a neighborhood of r = r0 has closed trajectories. Let V = x 2 + y 2 • Along the trajectory of (3.15) we compute dV/dt, and obtain
~ = 2r
2
cosa [tan{(r-r0 ) 3
(sinr~ro +2)} -tana].
48
THEORY OF LIMIT CYCLES
a
ro
r
FIGURE 3.2
From this we can see that, when tan [(r- ro) 3 (sin r
~ ro + 2)] = tano:,
or r• is a root of the equation
(r- ro) 3 (sin - 1 - + 2) = o: r- r 0
(lo:l
«
1),
(3.16)
then F(o:) has a closed trajectory r = r•. The graph of curve (3.16) in Figure 3.2 (we only consider the case r > ro) is between the two curves h: (r-ro) 3 = o: and l2: 3(r- r 0 ) 3 = o:. Considering o: as a function of r, we study the roots of do:fdr = 0. Since the roots of do: = 3(r- ro) 2 (sin - 1- + 2) - (r- ro) cos - 1- = 0 dr r- ro r- ro and the roots of cos(1/(r - ro)) are very close when lr - rol is sufficiently small, we know that near ro + 1/(2k7r + 7r/2) (k = 0, 1, 2, ... ) the function o:(r) has maximum value o:k, which is nearly equal to 3/(2k7r + ?r/2) 3 ; and near r 0 + 1/((2k- 3)7r + 1rj2) the function o:(r) has minimum value o:k, which is nearly equal to 1/ ((2k - 3)7r + 1r /2) 3 • When k is very large, it is obvious that O:k > o:k. This shows that every line o: = o:• which is parallel to and sufficiently close to the r-axis intersects the curve (3.16) at at least two points. Every circle r = r• in the (x, y)-piane, where the radius r• is the abscissa of the point of intersection, is an isolated closed trajectory of F(o:*), and hence is a limit cycle. To distinguish stability of this limit cycle, we only need rewrite
~L=o• =2r
2
coso:* [tan{(r-ro) 3
(sinr~ro +2)}
-tan { (r* - ro) 3 (sin r•
~ ro + 2)}] ,
§3. ROTATED VECTOR FIELDS
49
and we can see that if the point of intersection (r*, a*) lies in the rising section of the arc of the curve (3.16), then dVfdt < 0 for r < r* and dVfdt > 0 for r > r* (r is close to r*), i.e., the circle r = r• is an unstable limit cycle of F( a•). Conversely, if (r•, a•) lies in the falling section of the arc of the curve (3.16), then r = r• is a stable limit cycle of F(a*). Moreover, when (r•, a•) is an extreme point of the curve (3.14), r = r• is a semistable limit cycle of F(a*). In exact analogy with Theorem 3.3, we can prove THEOREM 3. 4. If an internally stable limit cycle f ao is positively oriented (negatively oriented), then for any sufficiently small e > 0 there must exist 0:2 > o:o (a2 < ao) such that for all a, ao < o: ~ a2 (a2 ~ a < ao), in an inner E-neighborhood off ao, there must exist an externally stable cycle and an internally stable cycle of F(a) (these two cycles may coincide). Moreover, there exists an inner 6-neighborhood (6 ~e) off a which is completely covered by closed trajectories of F(a). We can also write two theorems about unstable limit cycles similar to Theorems 3.3 and 3.4, but when the orientation of r ao is fixed, the direction of variation of a should be opposite to that of the above two theorems. When the inverse function n = n(a) of the above single-valued continuous function o:(n) is also single-valued, a(n) or n(a) is obviously monotonic. Hence we can say that the limit cycles in the rotated vector fields, according to different orientation and stability, monotonically expand or contract following the increase in a. For stable and unstable cycles, their behavior following the increase in o: can be given in the following table: Orientation positive positive negative negative Stability unstable stable unstable stable Motion contracts expands expands contracts THEOREM 3. 5. When a varies in a suitable sense, a semistable limit cycle transforms into at least one stable cycle and one unstable cycle, lying in the interior and exterior of the semistable cycle respectively. Moreover, if a varies in the opposite sense, this semistable cycle disappears. PROOF. From Theorem 3.3, Theorem 3.4, and the above table we know that for a positively oriented, externally stable but internally unstable r ao' its outer 61-neighborhood should be covered completely by all limit cycles r a corresponding to a 1 < a < o:o. If there exists some a such that F(a) has a series of periodic cycles in an outer at-neighborhood of r ao, whose outer boundary r a is an externally stable cycle and whose inner boundary r~ is an internally stable cycle, then we can consider the annular region bounded by
THEORY OF LIMIT CYCLES
50
r a and r~ as a generalized stable limit cycle ofF( a). Conversely, if the region between r a and r~ is not covered completely by closed trajectories of F(a), then it is easy to see that there exists at least one stable cycle or generalized stable cycle of F( o:) in the region. Finally, if there is one and only one closed trajectory of F(o:) in the outer neighborhood of r ao• then this cycle must be stable. Similarly, an inner 82-neighborhood of r ao is also covered completely by the limit cycles r a for every o: (o: 2 < o: < o: 0 ), and for every o: there exists at least one unstable or generalized unstable cycle in it. This means that when 0: decreases from o:o, r ao transforms into at least one stable cycle (in its outer neighborhood) and one unstable cycle (in its inner neighborhood). On the other hand, if o: increases from o:o, then according to the nonintersecting property (Theorem 3.2) we know that F(o:) cannot have closed trajectories in the neighborhood of r ao. The theorem is completely proved. According to the orientation and the internal or external stability of semistable cycles, we can form the following table: Orientation Stability Variation of o: when cycle disappears
positive internal stable
positive external stable
negative internal stable
negative external stable
o: decreases o: increases o: increases o: decreases
THEOREM 3. 6. If, when o: varies, the region covered by r a is R, then the inner and outer boundary curves of R have singular points (the point at infinity is considered a singular point). PROOF. Let B be a boundary point of R which is not a singular point. Let {Bn} be a sequence of points in R, Bn -+ B as n -+ oo. Each point Bn must have a cycler a,. passing through it. From the property (3.4) of a complete family we know we can assume that all the O:n are between 0 and T, and thus the sequence of numbers {o:n} has at least one limit point a. We now prove that {an} has only one limit point. Otherwise, let {an} have subsequences on -+ a and o:~ -+ a' =/: a. Then, because of the continuity of rotated vector fields with respect to (x, y, a), from a subsequence Bn of Bn which converges to B, we know that F(on) in the direction defined by Bn approaches F(a) in the direction defined by B; from the subsequence B~-+ B, we know F(a~) in the direction defined by B~ approaches (Fa') in the direction defined by B. But F(a) and F(o:') define different directions at B; hence, when n is sufficiently large, r a,. and r a:. should intersect near B. This contradicts Theorem 3.2.
§3. ROTATED VECTOR FIELDS
51
FIGURE 3.3
We now study the trajectory ')'(a) of F(a) which passes through B. Since B is not a singular point, ')'(a) exists and is unique. It is easy to see all the points on ')'(a) should be boundary points of R, and the points in the w- and a-limit sets of ')'(a) are also boundary points of R. Thus, if ')'(a) has an w- or a-limit point at infinity, the theorem can then be proved. Now suppose ')'(a) is bounded. Then from Theorem 1.6 of §1 we know that either ')'(li) itself is a closed trajectory, or ')'(a) takes another closed trajectory fa as w- or a-limit set, or the limit set of ')'(a) has singular points. In the last case, the theorem is proved. Now we prove that the first two cases cannot occur. First suppose 1(a) is a closed trajectory of F(a). We may as well assume ')'(Ci) is an inner boundary curve of R. Thus from Theorem 3.3 or some similar theorem, as a varies in a suitable sense from a, there will still exist a closed trajectory of F(a) in the interior of ')'(li); hence ')'(Ci) is not an inner boundary curve of R, which contradicts the assumption. Next let fa be a limit cycle of ')'(a), positively oriented (or negatively oriented) and close to F(a). Then fa is also a boundary curve of R. Hence in an arbitrary neighborhood of it there exist closed trajectories ofF( an) (n very large), and an -+a. These closed trajectories do not intersect, and do not contain each other, as shown in Figure 3.3. When n is sufficiently large, fa,. can be arbitrarily close to fa, but it cannot contain fa in its interior if ')'(a) approaches fa from outside. Thus, as in Figure 3.3, we cannot find a small neighborhood in the exterior of fa such that the lin which are sufficiently close to a will satisfy the requirements of Theorem 1.1 in §1.
THEORY OF LIMIT CYCLES
52
In the following, assume P and Q in {3.1) have first-order continuous partial derivatives with respect to :z:, y, and a, and when a= 0 the system (3.1) haa 0 (0,0) as its elementary singular point. Thus in the vicinity of 0 (3.1) can be written as
=
dxfdt
= a(a)x + b(a)y + P(:z:, y, a),
dyfdt = c(a)x + d(a)y + (J{x, y, a),
(3.17)
where P and (J are higher-order infinitesimals with respect to x and y, and ~(0)
= a{O)d(O) -
b(O)c{O) :f: 0.
(3.18)
From Theorem 3.1 we know that the index of 0 does not vary with a; hence ~(a) cannot change sign. Now suppose ~(a)>
0,
a(O)
+ d(O) :f: 0.
(3.19)
Then 0 is a coarse focus or nodal point of F(O). Note that
= Px{O,O,a + T) = -kPx(O,O,a) = -ka(a), d(a+T) = Q11 {0,0,a+T) = -kQ11 (0,0,a) = -kd(a).
a( a+ T)
We also know that a(T)+d(T) and a(O)+d(O) have different signs. Hence there exists at least one a= ao in the interval (0, T) such that a(ao) + d(ao) = 0. At the same time, 0 is a center of the first approximate equation of F(o:o), but 0 is not necessarily a center of F(ao). If 0 is a center of F(ao), then from Theorem 3.2 we know that for a¢ ao (modT), F(a) does not have a closed trajectory in a neighborhood of 0 which is completely covered by periodic cycles of F(a0 ). If 0 is not a center of F(ao), then it is a stable or unstable fine focus of F(a0 ). At the same time, we can prove the following important theorem (a restatement of the well-known Hop£ bifurcation theorem in plane stationary systems, but with weaker conditions). It will prove very useful in the latter half of this book. THEOREM 3. 7. Suppose ~(ao) > 0, a(ao) +d(ao) = 0, and 0 is a stable focus of F(ao). Moreover, suppose a( a)+ d(a) > 0 when a> ao (i.e., 0 has become an unstable focus of F(a)). The when a increases from ao, there exist at least one externally stable cycle and one internally stable cycle of F(a) in the vicinity of 0 (they may coincide to become a stable cycle). PROOF. From the hypotheses, there exists a small neighborhood U of 0 which is completely covered by the spirals of F(ao) (Figure 3.4), and these spirals all take 0 as their unique w-limit point. We take a point A# 0 in U; then the positive semitrajectory "Y+(A,ao) of F(o:o) starting from A will meet F(ao) at a point Bon the normal line segment of A, and the point 0 should
§3.
ROTATED VECTOR FIELDS
53
FIGURE 3.4 lie inside a simply connected region surrounded by a spiral AB on -y+(A, ao) and the normal line segment AB at the point A. By the continuity of a solution with respect to its parameter, we know that, as long as a1 > ao and a 1 - ao is sufficiently small, then for the a between ao and a1 the positive semitrajectory -y+(A, a) of F(a) starting from A will meet AB at a point Bo. near B; moreover, AB is a transversal with respect to F(a), i.e., all the trajectories of F(a), after crossing AB, will enter the above-mentioned simply connected region. On the other hand, 0 is also an unstable focus of F(a); hence the proof of this theorem is obtained by the Annular Region Theorem. REMARK. For Theorem 3. 7, conditions of rotated vector fields only guarantee that, as a varies from 0 toT, the elementary singular point whose index is +1 must change its stability; hence, for values of a in some interval, there exist limit cycles in a neighborhood of this singular point, all of which cover a region. If as a varies the system (3.1} does not form a family of rotated vector fields, but only its elementary singular point 0 can change its stability (naturally still passing the fine focus), for example, when a = ao, then from the proof of Theorem 3.7 we know(2) that if 0 is a stable focus for a= ao, then, for those a (in the vicinity of ao) which change 0 to an unstable focus, in a neighborhood of 0 the system (3.1) must have a stable cycle; conversely, if 0 is an unstable focus for a = o:o, then for those a (in the vicinity of ao) which make 0 become a stable focus, (3.1) must have an unstable cycle in a neighborhood of 0. Next we shall discuss the behavior of the separatrices of saddle points and the separatrix cycles formed by separatrices in the rotated vector fields. In ( 2 )Here
we use only the part asserting that 0 changes its stability at oo.
THEORY OF LIMIT CYCLES
54
0
0
FIGURE 3.5
FIGURE 3.6
this area, except for the work of Duff, Andronov, and Leontovich in the sixties, we rely on recent work of Ma Zhi-en [51]. In addition, we shall introduce some conclusions which do not require F(o:) to be a family of rotated vector fields. THEOREM 3.8 [291]. Let F(o:) be a complete family of rotated vector fields, 0 :5 o: < T, and let the sector AOB be an arbitrary hyperbolic region of F(o:o) at the singular point 0 (Figure 3.5). Then as o: increases from o:o, the two sides 6A and OB of this hyperbolic region will continuously rotate around 0 in the counterclockwise direction, and when o: changes to o:o + T, OB will turn into the position of OA. PROOF. With 0 as center, draw a sufficiently small circular arc Jn, which forms from the hyperbolic region a bounded region G whose interior does not contain singular points of F(o:) and is covered completely by the hyperbolic trajectories of F(o:o). Because of the rotational property of vector fields, when o: > o:o the trajectories of F(o:), when meeting the two sides OA and OB of G, always enter the region G from the positive direction. Let M be an arbitrary point on Then 1+(M, o:) (o: > o:o) after entering C can neither run out of G from AO, nor enter the point 0, for otherwise it would touch tangentially some trajectory of F(o:o) in G, which is impossible. Hence 1+(M,o:) must run out of the region G from some point N on As the point M moves continuously to 0 and approaches 0, the point N will move along BA in the direction of A, but it cannot cross the point A; hence there must exist a limit point N. It is easy to prove that any negative semitrajectory of F( o:) through N must enter the singular point 0. Now we prove that as o:---+ o:o, ON will come close to and coincide with OB. For this we choose a monotonically decreasing sequence of parameters {O:n} which approaches o:o. Let a negatively oriented trajectory of the vector field
- -
OB.
AB.
-
-
-
55
§3. ROTATED VECTOR FIELDS
-
F( an) in G entering 0 meet the circular arc AB at N n (if there is more than ~ one such trajectory, then we choose the one closest to OB as ON n)· From (3.3) and (3.5}, it is easy to prove that, as n --+ oo, Nn will monotonically approach B along the arc AB, but it cannot pass B; hence Nn must have a limit position, which is denoted by N•. We now prove that N" = B. Suppose not. Then N• should be on the right of B. At the same time, the negative semitrajectory 1-(B, ao) of F(ao) which passes through a point B between the point B on BA and N" should arrive at a point C in the vicinity of Ao (Figure 3.6). By the continuity of a solution with respect to its parameter a, when n is sufficiently large (that is, lan- aol « 1), 1-(B, o:n) will also reach the vicinity of the point C. But this is impossible since F(an) already has a negatively oriented trajectory, which passes through N n and enters 0, to separate B from C. Hence we must have N" = B. Again we note AB is an arbitrary circular arc in a neighborhood of 0 which intersects the two sides of the hyperbolic region, and we know OB will rotate in the counterclockwise direction with respect to a. Since F(ao + T) and F(ao} possess similar trajectories, but with opposite directions, and since (3.5} holds when ao :::; a< ao + T, it follows that OB rotates to the position of 6A if and only if a = ao + T. The theorem is completely proved.(3}
-
-
COROLLARY. Let F(a) be a complete family of rotated vector fields, 0:::; o: < T. Then, as a varies, the four separatrices of the saddle point 0 all rotate around 0 in the counterclockwise direction.( 4 ) When a varies from 0 toT, the separatrix leaving (entering) the point 0 just turns into the position of the separatrix entering (leaving) 0.
From this corollary, we immediately know that, in the family of rotated vector fields F(o:}, when a varies from a 0 , the separatrix cycle of F(a 0 ) passing through a saddle point must split.( 3 ) When the separatrix cycle splits, a limit cycle may appear in its vicinity. To study this question, we first introduce a more classical theorem to distinguish the stability of a separatrix cycle, but use the method of proof in [51]. THEOREM 3.9. Let the system (2.1) have a separatrix cycle r 0 passing through its unique saddle point N, and let the other two separatrices passing (3)Refer to footnote 4. it is required that the rotated vector fields in a bounded interval of the parameters form a complete family, and that they rotate everywhere in a neighborhood of 0; otherwise this assertion does not hold. For an example, consider the system dx/dt = y, dy/dt = x + ax 2 y, which for 0 :5 a< oo forms a complete family, but whose vector fields on the two lines x = 0 and y = 0 do not rotate. At the same time, each trajectory entering (leaving) 0 has fixed slope ±1, no matter what value a takes. ( 4 )Here
THEORY OF LIMIT CYCLES
56
through N lie in the exterior of r 0 • If P and Q are continuously differentiable, and at the point N
aP ~ aQ < 0 (> 0),
(3.20)
ax 8y then fo is internally stable (unstable).( 5 )
PROOF. In the following we consider only the case (Px + Qy)N < 0. We may as well assume fo is negatively oriented. Construct a complete family of rotated vector fields
F(o:):
dxfdt
= P(x, y, o:),
dyfdt
= "Q(x, y, o:),
such that (2.1) becomes F(O). For example, we can construct a family of uniform rotated vector fields starting from (2.1). Let 8() f8o: > 0. Since (P x(x, y, 0) + "Qy(x, y, O))N < 0, from the continuity property we know there exist a a-neighborhood of N and T/ > 0 such that, when (x, y) is in this neighborhood and lo:l < TJ,
P x(x, y, o:) + "Qy(x, y, o:) :::; -a < 0.
(3.21)
We first prove that in a sufficiently small inner neighborhood of r 0 there does not exist a closed trajectory of F(O). Suppose not. On the two separatrices of fo in a a-neighborhood of N we take regular points At and A 2 respectively (Figure 3. 7). The time corresponding to the section of the trajecis finite, say To, whereas the time corresponding to the sections tory NAt and A;N of the trajectory is infinite. Since Px + Qy is continuous, it is bounded in the interior of f 0 • Let Px + Qy < M. By virtue of the continuity of a solution with respect to its initial value, we can take very small neighborhoods f3t of At and fJ2 of A 2 such that the time from any point A~ on the closed trajectory of F(O) in fJ2 to any point A~ in f3t is greater than 2MTo/a. Moreover, let r meet the boundary of a a-neighborhood of Nat Bt and B2 respectively. Thus
A'JZA2
j(Px+Qy)dt= f_..._ (Px+Qy)dt+ f-..(Px+Qy)dt
1B1BB~
fr
+
f,__(Px
1A;A;
1.B~A~
+ Qy)dt +
f_..._(Px
1A;Bl
+ Qy)dt.
We note that the second and fourth integrals on the right side are negative, and, from the above estimate, we get 2MTo (3.22) (Px + Qy)dt < MTo- a· - - < 0. r a
i
( 5 )This theorem has been generalized to the case of a simple closed trajectory passing through more than one saddle point [162].
§3. ROTATED VECTOR FIELDS
FIGURE 3.7
57
FIGURE 3.8
If in an arbitrary inner neighborhood of fo there exists a closed trajectory r of F(O), then r is a periodic cycle or one of the sequence of limit cycles. In the former case, we should have
i
(Px
+ Qy)dt =
0,
which contradicts (3.22). In the latter case, any limit cycle adjacent tor does not satisfy (3.22); hence in a sufficiently small inner neighborhood of fo there does not exist a closed trajectory of F(O). Now we prove Theorem 3.9. From the above analysis we know that fo is either internally stable or internally unstable. Supposing r 0 is internally unstable, we shall obtain a contradiction. Draw a sufficiently short transversal l of F(O) at an arbitrary regular point A of fo such that a positive semitrajectory of F(O) passing thrugh a point Bon l will meet l again at C, where B lies between A and C. When 0 < a « 1, l is also a transversal of F(a), and a positive semitrajectory of F(a) passing through B will meet l again at a point D between Band C (Figure 3.8). On the other hand, by Theorem 3.8 we know that the separatrix of F(a) leaving N intersects lata pointE in the exterior of fo. Hence the trajectories NE, fiF, and ED and sections of the transversals DB and FE form an annular region which satisfies the corollary of Theorem 1.6; hence in this region there exist one externally unstable cycle and one internally unstable cycle of F(a) (they may coincide). It is easy to see
THEORY OF LIMIT CYCLES
58
that they should lie in the interior of r 0 . Hence, along this cycle, we should have
f
(P x(x, y, o:) + Qy(x, y, o:)) dt
~ 0.
On the other hand, by virtue of the continuity of a solution with respect to its parameter and its initial value, as long as B and A are sufficiently close, and !o:l is sufficiently small, the above integral should be less than zero, a contradiction. Thus, r 0 must be internally stable. The proof is completed. REMARK. The proof of this theorem does not use all the properties of saddle points. In fact, as long as (3.20) holds at the unique singular point N, and ro is a singular closed trajectory passing through N, with no other trajectories in the interior of ro passing through N (that is, F(O) in the interior of ro is a hyperbolic region), then the conclusion of Theorem 3.9 is established. We should point out if Px + Qy = 0 holds at N, then the stability of any separatrix cycle r o through N cannot be determined. EXAMPLE 3. Consider the system of equations dx
dt
= 2y = P(x, y),
It has a separatrix cycle 0(0, 0); here
~; = 2x- 3x2 + o:y(x 3 -
r 0 : x3 -
x 2 + y2
Px(O,O)
x2 + y2 )
= Q(x, y).
(3.23)
= 0 passing through the saddle point
+ Qy(O,O) =
0.
But take V(x, y) = x 3 -x 2 +y 2 • It is easy to calculate that along the trajectory of the equation dV fdt = 2o:y2 (x3 - x2 + y2 ). We can see that ro is internally stable when o: < 0, and ro is internally unstable when o: > 0. In the following we discuss the problem of generating limit cycles from separatrix cycles passing through saddle points. First we consider the case when F( o:) is a family of rotated vector fields. THEOREM 3 .1 0 [52]. Suppose the system (3.1) forms a complete family of rotated vector fields with respect too:, andro is a separatrix cycle of F(o:o) passing through a saddle point N. If at N
oP(x, y, o:o)
ax
+
oQ(x, y, o:o)
oy
<
O (
O)
> '
(3.24)
then when o: varies from o:o in a suitable sense, a unique stable (unstable) limit cycle of F(o:) will be generated close to the inside of ro, and when o:
§3. ROTATED VECTOR FIELDS
59
FIGURE 3.9
varies in the opposite sense, cycle in its neighborhood.
r o disappears
and there does not exist a limit
PROOF. We only prove the case in which ro is internally stable. As in Figure 3.9, let ro be positively oriented. Take any point A on r 0, and draw a normal line segment through A; when a point B on l is sufficiently close to A, the positive.semitrajectory "f+(B, ao) through B will meet l again at a point C between A and B. Let a increase from ao to at, lat - aol « 1, so that the positive semitrajectory "f+(B,at) of F(at) through B meets l again at a point D between B and C, and the line segment B D on l still remains as transversal. On the other hand, F( at) does not have any separatrix cycle, the separatrix r1 of F(at) leaving N always stays in the interior of ro, and the separatrix r2 entering N always stays in the exterior of r 0. Let them meet l at E and F respectively, and let EF remain as transversal to F(at). Similarly to the proof of the preceding theorem, we obtain an annular region in which there exist at least one externally stable closed trajectory and one internally stable closed trajectory (they may coincide). In fact, similarly to the proof of Theorem 3.9, we can show that the one generated is a unique stable limit cycle. According to Theorem 3.4, we know that, following the increase in a, the stable cycle generated by the separatrix cycle r 0 will monotonically contract inwards and completely cover an inner neighborhood of ro. Hence, by Theorem 3.2, we also know that as a decreases from ao, there does not exist any
THEORY OF LIMIT CYCLES
60
closed trajectory of F(o:) in this neighborhood. The theorem is completely proved. REMARK. If at the singular point N 8P(x, y, o:o) 8x
+
8Q(x, y, o:o) _ 0 8y - '
(3.25)
then even if r 0 is internally stable (internally unstable) in a family of rotated vector fields, the uniqueness of a limit cycle of F(o:) (o: 'I o:o} generated by r 0 may not be guaranteed. EXAMPLE 4. Multiply the right side of each equation in (3.14) by a factor [(x- r 0 ) 2 + y 2 ]. The equations thus obtained have one more higher-order singular point (ro, 0); therefore r = ro also changes into a separatrix cycle. It is easy to see that condition (3.25} holds at (ro, 0). Outside the circle r = ro, both the new and the old equations have the same trajectories. Hence, according to the analysis of Example 2, we know that when vector fields rotate uniformly, the separatrix cycle at o: = 0 can generate more than one limit cycle when o: -:J 0. Similarly to Theorem 3. 7 on generation of limit cycles by a fine focus, we can give a sufficient condition for the generation of separatrix cycles passing through a fine saddle point; namely, that at this point Px + Q 11 = 0 (the condition is not confined to rotated vector fields). THEOREM 3.11 [51]. Let there be given equations containing a parameter o:
F(o:):
dx
dt = P(x, y, o:),
dy dt = Q(x, y, o:),
(3.1)
where P and Q are continuously differentiable with respect to x, y, and o:. Let F(o:) have a saddle point N which does not vary with o:. Moreover, when lo:- o:ol « 1, let F(o:) always have a separatrix cycle r a passing through N (its position can vary with o:). If
[Px(x, y, o:o) + Q 11 (x, y, o:o)]N = 0 and
r ao
is an internally stable (unstable) separatrix cycle of F(o:o), and for
0 < lo:- o:ol
«
1 lPx(x, y, o:)
+ Q11 (x, y, o:)]N > 0
( < 0),
(3.26)
§3. ROTATED VECTOR FIELDS
61
FIGURE 3.10
then in an inner neighborhood of r 0 there must exist an externally stable (unstable) limit cycle and an internally stable (unstable) limit cycle of F( a) (they may coincide). PROOF. Here we only prove the case when r 00 is internally stable and the divergence Px + Q11 is greater than zero at N. We may as well assume fo is negatively oriented. At any regular point A on r 00 construct a transversal I of F(ao). From the internal stability of r ao and the continuity of F(a) with respect to a, we know as before that when 0 < a - a 0 « 1, the positive semitrajectory "Y+(B, a) of F(a) through B (which lies on l insider ao and is sufficiently close to A) will meet l again at a point D between A and B. It is easy to see that r a must lie in the exterior of the region surrounded by -y+(B, a) and DB (Figure 3.10). Let the point of intersection of r a and l beR. On the other hand, since for 0 < Ia- aol « 1 the inequality > holds in (3.26), r a is internally unstable, and then in a sufficiently small inner neighborhood of r a the positive semitrajectory "Y+(E, a) of F(a) passing through a point E on l sufficiently close to R will meet l again at a point F between E and D. Thus we apply the Annular Region Theorem in the annular region formed by "Y+(B, a), "Y+(E, a) and transversal segments BD and EF, and obtain the conclusion of this theorem. REMARK. The proofs of Theorems 3.11 and 3.10 are very similar, but now the problem is whether all the assumed conditions of Theorem 3.11 can be
THEORY OF LIMIT CYCLES
62
established. This does not seem as obvious as for Theorem 3.10 or Theorem 3.7. The following is a concrete example mentioned in [51]. EXAMPLE 5. Investigate the equations dx dt
)
(l
= - 5ax+
17a- 5 2 2(a- 1) 2 6a y 3a2 x y
+ 36a2xy2 -
6ay4
-
192ax3 y + 192x5 + 24x2y3
= P(x,y,a), dy dt
l)
(
(3.27)
= 2 3a - y + = Q(x,y,a),
16(1- 4a) 2 2(a- 1) 2 3a x 3a2 xy
containing a parameter a. It is easy to verify that when a > 1/2 the origin 0 is a saddle point of (3.27). Let F(x, y, a) = 8x3 + y 3 - 6axy; the total derivative of F along the trajectory of (3.27) is
: = (8x 3 + y 3 -
6axy) [(a -1) ( 1- : 2 xy)
+ (24x2 -
6ay) 2].
It is easy to see that when a > 1/2, (3.27) always has a separatrix cycle passing through the saddle point 0,
r a: 8x3 + y 3 -
6axy = 0.
(3.28)
Since Px(O,O,a) +Q11 (0,0, a)= a-1, we know from Theorem 3.9 that r"' is internally stable for 1/2 1, and the critical case appears for a = 1. At the same time,
+ y 3 - 6xy)(24x 2 - 6y) 2; interior of r 1· It is not difficult to see here that r 1
dF /dt = (8x 3
thus dF I dt < 0 in the is internally unstable. Hence, according to Theorem 3.11, we know that as a decreases from 1, an unstable limit cycle of F(a) will be generated in an inner neighborhood of r a• In fact, when a = 1, equations (3.27) can be changed to dxfdt
= -4x + 2y2 + 6(8x3 + y 3 -
6xy)(4x 2
-
y),
dyfdt = 4(y- 4x 2 ).
These equations, besides the saddle point 0, have a singular point M(1/2, 1), which lies in the interior of the separatrix cycle r 1 , and is also a stable coarse focus. Hence, when lo:- 11 « 1, the interior of r a has only a unique singular point, which is still a stable coarse focus. But when o: < 1, since r a is
§3. ROTATED VECTOR FIELDS
63
internally stable, there must exist an unstable cycle r a in the interior of r a, which contains the singular point Min its interior. When a increases to 1, a changes into r 1, and the stability of r 1 is changed. We should note that in the family ofvector fields F(a), as a varies, besides the limit cycles which may be generated by singular points and separatrix cycles, limit cycles may also be suddenly produced by the coalescing of trajectories. This fact can be seen from Theorem 3.5. Since a semistable cycle r ao may disappear as a varies in a fixed direction, if a varies in the opposite direction this is naturally the situation; originally F(a) does not have a limit cycle, and when a = ao a semistable cycle r ao suddenly appears; then r ao splits into at least two limit cycles. In the next section we shall prove that this suddenly generated limit cycle must be a multiple cycle; when F(a) does not form a family of rotated vector fields, this multiple cycle is not necessarily a semistable cycle. How to distinguish whether an equation involving a parameter can generate a multiple cycle when its parameter varies (even if this equation forms a family of rotated vector fields) is a very important, but very difficult and not yet solved problem. In the following we introduce some problems on the generalization of the theory of rotated vector fields. At the beginning of this section we mentioned that several persons have weakened Duff's conditions: for example, not requiring that P and Q have partial derivatives, letting inequality (3.3) become an equality, not requiring (3.4) to hold, etc.; and hope the main conclusion of Duff, i.e., the nonintersecting theorem (Theorem 3.2), can still be established. Chen Xiang-yan uged a more geometrical approach to consider this problem. In order to guarantee that Theorem 3.2 can be established, the key problem is to prove the following lemma, which seems obvious from direct observation, but is yet quite difficult to prove rigorously. Since he finally obtained a proof of the lemma, his generalization of Duff's theory seems better than others.
r
LEMMA (50]. Let Lo be a piecewise smooth simple closed curve whose parametric equations are x = rp(t) andy = .,P(t). At the points where rp' and 1/J' are defined, let
9(t) = rp'(t)Q(rp(t), .,P(t))- .,P'(t)P(rp(t), 1/J(t)). Suppose that when t increases Lo is positively oriented, and 9(t) ~ 0. Then there do not exist points on Lo where the trajectory of the equations dxfdt = P(x,y),
dyfdt = Q(x, y)
(3.29)
runs into the exterior of Lo when t increases, and there do not exist points on Lo where the trajectory of (3.29) enters the interior of Lo when t decreases. If
THEORY OF LIMIT CYCLES
64
we change Lo to be negatively oriented, or change 8(t) ::::; 0, then the conclusion is just the opposite. As usual, we assume P and Q are continuous and the uniqueness of a solution of (3.29) is guaranteed. PROOF. The condition 8(t) 2:: 0 is equivalent to the folowing: any trajectory of equations (3.29) passing through a smooth point of L 0 when t increases either enters the interior of Lo or touches Lo tangentially at that point. This property is obviously independent of the parameter of £ 0 . Hence without loss of generality we can assume that the points where rp'(t) and 1/J'(t) are defined always satisfy r.p'2 + 1/1'2 'f. 0; for example, we can take t as the arc length of the curve. Let the period of r.p(t) and 1/J(t) be T; moreover, except at the n points t = ti (i = 0, 1, ... , n- 1), to < t1 < t2 < .. · < tn = to+ T, the functions r.p'(t) and 1/J'(t) exist and are continuous everywhere. We now prove that for any point t E (ti-1 1 ti) the solution (xo(t), y0 (t)) of (3.29) which passes through (r.p{t), ,P(i)) when t = i cannot leave Lo and run to the exterior of L 0 as t increases. To get a contradiction, suppose there exists a Ot > 0 such that when t E {t, t + Ot), (xo(t), Yo(t)) lies in the exterior of £ 0 • We now get a contradiction as follows. Consider an auxiliary system of equations containing a parameter a:
dxfdt
= P(x,y)- a,P'(t),
dyfdt = Q(x, y) + ar.p'(t),
t E (ti-l! ti), a > 0.
(3.30)a
Assume its solution is x = xa(t), y = Ya(t) when it satisfies x = r.p(i) and y = ,P(i) at t = t. Since r.p12 (i) + ,P12 (i) > 0, there must exist a 62 > 0, 82 < min(ti - t, t- ti-t), such that for t', t" E [t- 62, t + 82] we always have
r.p' (t')r.p' (t")
+ 1/1 t'),P' (t") > 0. 1(
Moreover, since the functions on the right sides of (3.30)a are continuous, there exists a 83 > 0 such that, when 0
e(l) + a[rp' 2 (l) + ,P12 (t)J > 0.
Hence, as t increases from i, (xa(t), Ya(t)) must enter into the interior of Lo. If for some tt E (t, t+t5], the solution curve and L0 meet again at (r.p(t 2), ,P(t2 )), i.e., Xa(ti) = rp(t2) and Ya(h) = 1/J(h), then t2 E (t- 62, t + 62), and at the point of intersection we should have
65
§3. ROTATED VECTOR FIELDS
FIGURE 3.11
However, on the other hand, y~(tt)
+ a[cp'(tt)cp'(t2) + 1/J'(tt)1/J'(t2)]
>0, a contradiction. This proves that for 0
a~o
lim Ya(t) = Yo(t),
a~o
tE(t,i+6],
uniformly, and hence, when t E [t, t + 6], (x 0 (t), Ya((t)) cannot run to the exterior of Lo. This contradicts the assumptions. Since it has been proved that any trajectory of (3.29) passing through a point in the vicinity of (cp(t,), 1/J(t,)) when t increases cannot run to the exterior of Lo, we know, from the continuity of a solution with respect to its initial value, that its trajectory passing through the point (cp(t,),1/;(t,)) when t increases also cannot run to the exterior of Lo. The lemma is completely proved.(6 ) Now we can generalize the corollary of Theorem 1.6 (the Annular Region Theorem) to the following: THEOREM 3. 12. Let Lt and L2 be piecewise smooth simple closed curves, Lt :::> L2, and let the parametric equations of L, be x = cp,(t), y = 1/J,(t); ( 6 )Ifwe consider separately the (closed) point set consisting of all the points on Lo which make 9(t) = 0, and all the comer points of Lo, and the (open) point set of all the points on Lo which make 9(t) > 0, and the property of a trajectory of (3.29) passing through these Points, then the proof of this lemma can be simplified, and there is no need to introduce equations (3.30) ...
THEORY OF LIMIT CYCLES
66
suppose there are no singular points of (3.29) on Lt and L2 and in the annular region R bounded by them. Assume, moreover, at the points where IPi and tPi have derivatives, the inequalities cp~ (t)Q(cpt (t), tPt (t))
- '1/J~ (t)P(cpt (t), tPt (t)) ;::: 0 (~ 0),
cp~(t)Q(cp2(t), 'I/J2(t))- .,P~(t)P(cp2(t), 'I/J2(t)) ~ 0
(;::: 0)
always hold. Then when Lt and L2 are positively oriented there exist stable (unstable) limit cycles of (3.29) in R, and when Lt and L2 are negatively oriented there exist unstable (stable) limit cycles in R. This result is better than that in [53]. Next, Theorem 3.2 in this section can be generalized to THEOREM 3.13. If for the two systems of equations
dxjdt = Pi(x, y),
dyjdt = Qi(x, y)
(i = 1,2)
(3.31)
we have Pt(x,y)Q2(x,y)- P2(x,y)Qt(x,y);::: 0
(~ 0),
then their trajectories either coincide or do not intersect. PROOF. Let Li be a closed trajectory of (3.31)i, whose equations are x = IPi(t) andy= tPi(t) (i = 1, 2). If Lt and L2 intersect but do not coincide, then there are two possibilities: 1) Lt and L2 are tangent to each other externally or internally; 2) L2 has points in the exterior of Lt and points in the interior of Lt. In the former case, at their point of contact (if their points of contact cover a section of arc, i.e., if they have a common section of arc, then this arc will be understood as one point), as t increases or decreases L2 will enter respectively the interior or exterior of Lt. In the latter case, there must exist points of L2 on Lt entering the interior of Lt as t increases, and also points of L 2 on Lt running to the exterior of Lt as t increases. But cp~(t)Q2(1Pt(t),'I/Jt(t))-
.,pat)P2(1Pt(t),'I/Jt(t)}
= [PtQ2- QtP2]x=
(~ 0}.
From this we know that the above two cases cannot happen. The proof is complete. It is obvious that the necessary and sufficient condition for these two systems to have a common closed trajectory is that the equation
has closed branches, and these closed branches are closed trajectories of the above equations.
§3.
ROTATED VECTOR FIELDS
67
Since nonintersecting Theorem 3.2 can be generalized as above, the definition of rotated vector fields can be generalized correspondingly. DEFINITION 3.2. Consider a family of vector fields (3.1) with a parameter. If for any arbitrary fixed point (xo, Yo) and ao there always exist a positive number b'(xo, Yo, ao) such that when a E [ao - 6, ao + b'] the inequality (Q(xo, Yo, a)P(xo, Yo, ao)- P(xo, yo, a)Q(xo, Yo, ao)} sgn(a- ao) ~ 0 (:::;; 0}
(3.32} always holds, then the vector fields in (3.1) are called generalized rotated vector fields in the plane. By the methods of proof of the previous several theorems, except those involving the conclusion of equality (3.4), other results under the definition of 3.2 still remain valid. Of course, now we add a new possibility; that is, vector fields with different parameters may have the same closed trajectory. In subsequent sections we shall see the application of the theory of rotated vector fields in the study of uniqueness of limit cycles and the problems of limit cycles in quadratic differential systems (in which the right sides of (3.29) are quadratic polynomials of x andy). Now we first prove two theorems, which explain how this theory can lead to the Annular Region Theorem and also yield the converse of Theorem 1.9 in §1 [48]. In order to make a simpler statement, we make a stronger assumption on the boundary curves of the annular region in Theorem 3.14. THEOREM 3.1_4. LetP(x,y) andQ(x,y) insystem(3.29) be continuously differentiable, and let there exist an annular region G which does not contain any singular point, and whose inner and outer boundary curves are twice continuously differentiable curves, and let the trajectory of (3.29) when it meets the boundary curve of G always move in the exterior-to-interior direction. Then system (3.29) has an external stable cycle and an internal stable cycle in G (they may coincide). PROOF. Let the inner and outer boundary curves of G be 01 and 02 respectively. Now we take the counterclockwise direction on 01 and the clockwise direction on 02 as positive. We first construct a system of equations which possess closed trajectories 01 and 02. It is easy to see if we take any point (x, y) on oi and denote by cp(x, y) the angle required to tum from the positive direction of the trajectory of (3.29) (i.e., the direction of increasing t) to the positive direction of the tangent line of oi. then the system of equations dx/dt = P(x, y) coscp(x, y)- Q(x, y) sincp(x, y) = P*(x, y), dyjdt
= P(x, y) sincp(x, y) + Q(x, y) coscp(x, y) = Q*(x, y)
(3.33)
THEORY OF LIMIT CYCLES
68
FIGURE 3.12
will have C1 and C2 as their closed trajectories (Figure 3.12). However, system (3.33) is not defined outside Ci. Hence we attempt to extend cp(x, y) continuously to G1 :J G. Let N; (i = 1, 2) be an €-neighborhood of Ci; N! and N'f have no common points and do not contain singular points of (3.29). The conditions of the theorem guarantee that we can introduce curvilinear coordinates (s,n) in a neighborhood of Ci. Let cp(x,y) = ~(s,O), where (x, y) E C,. We redefine
-c
cp(x, y) =
{
11" • 2 mr mr , wen h ( x,y ) E Ni£' 2sm 2 +cps, 0) cos2 2 11"
c
c
when (x, y)
2,
rt N;.•
(3.34)
It is easy to see that cp(x, y) is continuous in G1, and, except at Ci, is always greater than zero. Now we construct a new system of equations
dxjdt = P(x,y,a) = P(x,y) cos a+ P*(x, y)sina, dyjdt = Q(x, y, a)= Q(x,y) cos a+ Q*(x, y) sin a.
(3.35)
It is easy to calculate P (J P oPjoa a"Qjaa = P*
Q
cosa
Q* . -sin a
sinal cos a
= PQ*- QP* = (P2 + Q2 ) sincp;?: 0, since 0 ~ cp ~ 1r. Hence the system (3.35) forms a complete family of generalized rotated vector fields with respect to a E [0, 1r). With a = 1rj2 (3.35) becomes the system (3.33), and at the same time it has a positively oriented closed trajectory C 1 and a negatively oriented closed trajectory C2 . Since C 1 and C2 have opposite orientation, the region G cannot be covered by the closed trajectories of (3.33), for otherwise C 1 and C 2 would have the same
§3.
ROTATED VECTOR FIELDS
69
orientation. Thus, (3.33) has a largest positively oriented closed trajectory Ci (which may be Ct) in G. According to Theorem 3.3, when a varies in a suitable direction, Ci will expand. Since there is no singular point in G, by Theorem 3.6 the region R, which is covered completely by all the trajectories of (3.35) (with respect to all a), should contain the whole region G; that is • to say, Ci can expand and arrive at C2.(1) But during the process of expansion the orientation cannot change; hence when a = 1r/2 - 1r = -1r /2 or a= 1rj2+1r = 37r/2, it expands to C2. This is because, at a= -1rj2 or 37r/2, (3.35) becomes dxfdt = -P*(x, y),
dyjdt = -Q*(x,y),
which takes C2 as positively oriented closed trajectory. But from 1r/2 to -1r /2 it must pass 0, from 1rj2 to 37r/2 it must pass 1r, and when a= 0 or 1r, (3.35) becomes (3.29) or dxfdt
= -P(x, y),
dyfdt
= -Q(x, y).
This shows (3.29) must have closed trajectories in G, among which the outermost one must be externally stable, and the innermost one must be internally stable. Of course, there may be only one limit cycle in G, which must be stable. The theorem is completely proved. In the following we prove that under suitable conditions the Poincare theorem (Theorem 1.9 in §1), which distinguishes nonexistence of closed trajectories, has a converse. THEOREM 3.15 [48]. If system (3.29) has a unique elementary singular point 0 whose index is +1 in some simply connected region G, but there does not exist a closed trajectory in G, then there exist a subregion a• of G and a continuously differentiable function h(x, y) defined in G such that the total derivative with respect to (3.29) keeps constant sign, but is not identically zero in a· (the point 0 is an exception). PROOF. First we define the required region G* and construct a system of equations starting from (3.29) such that it takes 0 as center and G• is completely filled by its closed trajectories. To do this, we construct from (3.29) a family of uniformly rotated vector fields dxfdt = P(x, y) cos a- Q(x, y) sin a, dyjdt = P(x, y) sin a+ Q(x, y) cos a.
(3.36)
From the remark following Theorem 3. 7 we know that all the closed trajectories of the complete family (3.36) cover a region G, and its inner boundary ( 7 )If (3.33) has another negatively oriented closed trajectory C~ between C1 and C:z, we then take the innermost c~ to replace
c2.
THEORY OF LIMIT CYCLES
70
FIGURE 3.13
is the point 0, and its outer boundary must have singular points. Hence G gets out of Gat least in some direction. In G, take the set formed by all the closed trajectories in the interior of Gas G* (Figure 3.13). Now we define the value of a*(x, y) at a point (x, y) in G* to be the value of the corresponding parameter a of the closed trajectory of the complete family passing through this point. Thus the system
.
-
dt = P(x, y) cosa*(x, y)- Q(x, y) sina*(x, y) = P(x, y, a*),
dy
dt = P(x, y) sina*(x, y)
+ Q(x, y) cosa*(x, y) = Q(x, y, a*)
(3.37)
has a system of closed trajectories which completely cover G*, and thereby takes 0 as its center. In the following we shall prove that by means of the orthogonal trajectories of (3.37) we can construct the function h(x, y) which satisfies the requirements of the theorem. As in Figure 3.13, we construct an arbitrary orthogonal trajectory l of (3.37) starting from 0, and let h be the arc length of l, measured from 0. Now define the function h(x, y) in G* as follows: for a point A in G* assume that the closed trajectory r of (3.37) passing through A meets l at B(x 0 , y0 ); we then take the arc length ho of l at Bas the value of h(x, y). We now prove that h(x, y) has continuous first-order partial derivatives. Suppose the parametric equations of l are x = p(h) andy= '1/J(h). The equations of the family of closed trajectories of (3.37) are x = x(s, h) and y = y(s, h), where s denotes the arc length of the closed trajectory measured from a point l with clockwise direction as positive. Thus, it is easy to see that x(O, h) = p(h) and y(O, h) = '1/J(h). Since l and r are orthogonal at the point of intersection, it
§3.
follows that ax ax as ah ay ay as ah
=
71
ROTATED VECTOR FIELDS
P(xo,Yo,a*(xo,yo)) Q(xo, Yo, a*(xo, yo))
cp'(ho)l 1/J'(ho)
s=O h=ho
1
Jp2 + Q2
::f:.O.
From xo = x(O, ho) and Yo = y(O, ho), we can solve for ho as a continuously differentiable function of (xo, Yo). On the other hand, by virtue of the continuous differentiability of a solution with respect to its initial value and the well-known formula ay ax
a::
= exp
(l:(Px +
Qy)dt)
> 0,
ayo we know that x 0 and y0 have continuous first-order partial derivatives with respect to x and y. Hence ho = h(xo, Yo), as a composite function of x and y, also has continuous first-order partial derivatives with respect to x and y. Finally, we prove that dhfdt has constant sign. To do this, we merely note that the trajectories of (3.37) are generated by rotated vector fields (3.36), but when a varies, the rotated vector fields (3.36) will strictly rotate, and hence the trajectory of (3.29) cannot be tangent to the closed trajectory of (3.37); that is, the closed trajectories of (3.37) are the arcs without contact of the rotated vector fields. Hence, along the trajectory of (3.29) dh ah ah dt = axP+ ayQ
> 0 (< O).
In the above formula, whether > or < holds depends on whether 0 is an unstable or stable singular point of (3.29). The theorem is completely proved. Before we conclude our discussion of rotated vector fields, it is interesting to study some properties of rotated vector fields in which the singular point can move. To make things clear, we now assume the singular point of a vector field moves following the variation of the parameter a, and can disappear, or split. But we require that the number of singular points after splitting be at most finite, and that this singular point not coincide with the original one. THEOREM 3. 16. A singular point which can move or disappear in rotated vector fields can only be one whose index is equal to zero and whose neighborhood does not contain an elliptic region. [291] PROOF. Suppose that for a1 ::f:. ao the singular point 0 of F(ao) moves to become a singular point 0' of F(al). Since 0 is a regular point with respect to F(at), there exists a sufficiently small circle C with 0 as center such that
72
THEORY OF LIMIT CYCLES
there are no singular points of F(at) on C and in its interior, and there is a unique singular point 0 of F(ao) in it. It is obvious that the index of C relative to F(at) is zero, but the angle between the vector of F(at) and the vector of F(ao) is less than 1r. Hence the index of C relative to F(ao) is also zero, i.e. the index of 0 is zero. Similarly we can prove that the index of 0' in F( at) is also zero. Now suppose a neighborhood of the singular point 0 contains an elliptic region. For any arbitrary a1 #- a 0 , let 0 move to become a singular point 0' of F(at). Then we can always find a sufficiently small elliptic region G that does not contain any singular point of F(at). Since the boundary trajectory of G is a trajectory starting from 0 and returning to 0, the trajectory of F(at) passing through any point P on the boundary of G after entering G (when t increases or decreases) can only get out of G from 0. Since P is arbitrary, 0 is a singular point of F(at), which contradicts the assumption. COROLLARY. An elementary singular point in rotated vector fields cannot move following the variation of parameter. REMARK. If a family of rotated vector fields is not defined in the whole plane, then the elementary singular point on the boundary of the domain of definition can move following the variation of parameter. EXAMPLE 6. For the equations with 6 as parameter, dxjdt =ox- y + mxy- y 2 ,
it is easy to calculate that p
Q
8P/88 8Q/88 =-x 2 ( 1 +ax). Hence the equations on the two sides of the line 1+ax = 0 form two families of rotated vector fields with different directions of rotation. It is not difficult to verify that the positions of the two elementary singular points (whose indices are +1 and -1 respectively) on 1 + ax = 0 will move up and down following the variation of 8. Similarly to Theorem 3.2, the following still holds: THEOREM 3 .1 7. In the family of rotated vector fields whose singular point can move following the variation of the parameter a, let ri be the singular closed trajectory or closed trajectory of F(ai) (i = 1, 2). Then, when a1 #- a2, r 1 and r 2 do not intersect.
73
§3. ROTATED VECTOR FIELDS
Corresponding to Theorems 3.3 and 3.4, we have THEOREM 3.18. In a family of rotated vector fields, let the singular point
0 move continuously following the variation of parameter a, but let it not split into several. Let r (ao) be an externally stable (externally unstable) singular closed trajectory of F(ao). Then when a varies from ao in a suitable sense, in the outer neighborhood of r 0 there exists at least one closed or singular closed trajectory of F(a) which contains fo in its interior; moreover, if the singular point of F(a) does not move out of the exterior of fo, then the outside of fo must have a closed trajectory of F(a).
The proof is omitted. See [291]. On the theory of rotated vector fields, a very important question, worthy of study, is: what kind of additional condition is required in order to guarantee that for every F(a) there exists a unique limit cycle? In this area Chen Xiangyan [50] obtained some results, but the conditions were rather too strong, and the scope of application was too small. After reading through Theorem 3.9, people naturally wonder: if there are several saddle points on a singular closed trajectory, and the divergence at these points can be positive or negative, then how are we going to distinguish the stability of this simple closed trajectory? In this area, L. A. Cherkas did some work in [54]. His main result was as (ollows: Let r be a positively oriented singular trajectory, and let A{ (j = 1, 2; i = 1, ... , n) be two characteristic roots of the i th saddle point on r, in which At > 0 > A~; let Ai = -AU At. Then for ll~=l Ai > 1 (< 1), r is stable (unstable).
Concerning limit cycles generated by the singular point in plane vector fields containing several parameters, there is some recent work by Takens [56] and others.
Exercises 1. Use the Annular Region Theorem to prove Theorem 3.2. 2. Prove that formula (3.12) and the first formula of (2.26) in §2 are the same when k* and k are the same function. 3. Starting from the equations dr 2 dt = -(r- 1)(r- 3) ,
dcp
di =
{ (r- l)(r- 2)(r- 3) seclr- 2111/2 when 0 < r < 4, but r 4/7r when r = 1, - 4/7r when r = 3,
f= 1, 3,
THEORY OF LIMIT CYCLES
74
form a complete family F(o:) of uniformly rotated vector fields. Study the behavior of limit cycles following the variation of parameter o:, and indicate the region covered by the limit cycles. 4. Compare the behavior of the variation of limit cycles in rotated vector fields and the behavior of a change of real roots following the variation of the parameter C in the equation (~)
where the equation aoxn + a 1 xn-l +···+an = 0 does not have real roots and K is a constant. Describe what kind of behavior of rotated vector fields corresponds to the equation(~) at K = 0. 5. Prove that the point transformation x1 = f(x, y), Yl = g(x, y) with positive functional determinant o(xl, yt)fo(x, y) will transform a complete family ofrotated vector fields into a complete family [46]. 6. Let F1 (P, Q) and F2 ( P*, Q*) be vector fields which satisfy condition 1 (i.e., formula (3.2)) and condition 3 at the beginning of this section, and let them have common singular points. Then a necessary and sufficient condition for existence of a complete family F(o:) such that F(O) = F1 and F(7r/2) = F2 is that the inequality PQ*- QP* > 0 holds at all regular points [46]. 7. Let G be an open annular region which is completely covered by all the trajectories of system (3.1) (for all o:) containing the singular point. (0 is excluded in the region G.) In G define a function o:*(x, y) as in the proof of Theorem 3.16. Prove that if P and Q have continuous first-order partial derivatives, then o:*(x,y) also has continuous first-order partial derivatives in G, and satisfies a first-order quasilinear equation [55]:
oa• Q(x, y, o:*) oy
+ P(x, y, a*)
oa• ox = 0.
8. Use the fact that 0 is the center of
dxfdt
= P(x,y,o:*(x,y)),
dyfdt
= Q(x, y, o:*(x, y))
to prove that if r is a closed trajectory of system (3.1), then
i
H(s)ds = -
i ~: 0~ 0
ds,
where H (s) is the curvature of the orthogonal trajectory of (3.1) and oo.* I on represents the directional derivative of o.*(x, y) along the direction of the outward normal of r [37]. 9. Prove that if the equations
dxfdt
= P(>..x, y),
dyjdt = >..Q(>..x, y)
75
§3. ROTATED VECTOR FIELDS
or
dyfdt = Q(x, )..y)
dxfdt = )..P(x, )..y),
form generalized rotated vector fields with respect to ).. E (0, oo), then the system (3.29) does not have a closed trajectory. 10. Use the following equations to verify the conclusion of Theorem 3.18
[51]: dxfdt = {[y + x(l- x2 - y 2)] cos a- [-x + y(l- x2 - y 2)] sin a} . [(x- g(a))2 + y2], dyfdt = {[y + x(l- x 2 - y 2)] sin a+ [-x + y(1- x 2 - y2 )] cos a} . [(x- g(a))2 + y2], where i) g(a) = v'l +tan a; ii) g(a) = v'l +a; iii) g(a) = v'1 + tan2a. 11. Use the viewpoint ofrotated vector fields to realize the significance of Theorem 1.13. 12. Prove that the two separatrices on the left side of x = 1/2 which pass through the saddle points (1/2, Yl > 0) of
dxfdt
= -y +ex+ xy- y 2 ,
dyfdt
= x(l- 2x).
(6 > 0)
must surround the origin 0. Suppose we kriow when D = 0, 0 is a stable focus. Prove for some D > 0 the neighborhood of 0 should have at least two limit cycles. 13. Apply Exercise 10 in §1 and the theory of rotated vector fields to prove the following criterion for nonexistence of limit cycles [38): Suppose in the equations dxfdt = y- F(x)- Q(x), dyfdt = g(x), where F(x) and g(x) are the same as in Exercise 10 of §1, Q(x) is odd and xQ(x) ~ 0 (or~ 0). Then there does not exist a closed trajectory of the equations. 14. Let the singular closed trajectory r have two saddle points Nt and N2, and let div Ni be the value of the divergence at the point Ni. Prove that when div N1
div N2
~>~ 1
r
2
( ) or<
is stable (or unstable). This condition can also be rewritten as follows: when div N1 div N2 > (<))..~ div N2 + )..~ div N1
r
is stable (or unstable).
§4. General Behavior of Limit Cycles Depending on a Parameter
Now that we have studied the behavior of rather regular variation of limit cycles of rotated vector fields, we proceed to study the general behavior of variation of limit cycles of the differential equations
dxfdt
= P(x, y, o:),
dyjdt = Q(x, y, o:)
(4.1)
following the variation of the parameter o:. There are three main topics to be discussed in this section: I. Suppose r is a limit cycle of (4.1)a=O· Does the system (4.1) still have limit cycles in the vicinity of r for o: =I 0 and sufficiently close to zero? If the answer is yes, is the limit cycle.unique? II. Study, for the case when 0:1 =I 0:2, lo:tl and lo:2l very small, whether the limit cycles of (4.1)a 1 and (4.1) 02 intersect in the vicinity ofr. Ill. Suppose (4.1)a=O has a series of periodic cycles with which a certain annular region is completely filled. Study, in the case when o: =I 0 and lo:l is very small, whether (4.1) has limit cycles. If yes, which one of the periodic cycles of (4.1)a=O is its limit position as o:--+ 0?(1) In 1954, the Japanese mathematician Minoru Urabe used Lyapunov's method mentioned in §2 to study these three problems in detail [41]. Since we do not have access to the journal containing his paper Chen Xiang-yan [57] used the method of Tkachev to obtain independently his results, and got better results in the third problem. Now we discuss these problems, following Chen's method. Suppose P and Q in (4.1) have all the required continuous partial derivatives with respect to x, y, and o: to be needed later, and suppose that when (1 )It is clear that if we include a = 0, then the study of whether the limit cycles corresponding to different a intersect is meaningless. If we exclude a = 0, then the problem becomes problem II. 77
THEORY OF LIMIT CYCLES
78
a= 0 the system (4.1) has a closed trajectory r, whose equations are x = cp(s),
(4.2)
where s is the arc length measured from a fixed point on r, whose positive direction is clockwise. As in §2, we move the coordinate system on r, obtaining
dn/ds
= F(s,n,a),
(4.3)
and then define a successor function
\ll(no, so, a)
= n(l +so, no, so, a) -no =
l
l+so
(4.4)
F(s,n(s,no,so,a),a)ds,
so
where l is the perimeter of r, and n = n(s, no, s0 , a) represents the solution of (4.1)a when s = so and n =no. It is obvious that n = n(s, no, so, a) is a closed trajectory of (4.l)a if and only if
\ll(no, so, a) and it is easy to see that n( s, 0, so, 0)
= 0,
(4.5)
=0 and \11(0, so, 0) =0.
THEOREM 4 .1.
\11~0 (0, so, 0) = exp (fo' H(s) ds) \11~(0, so, 0)
= exp
(
{ 1
)
[ 1
(
Jo H(s) ds Jo exp -
{4.6)
1,
ls+so ) ao( s + s0 ) so H(s) as aa ds. (4.7)
PROOF. (4.6) has been obtained in §2 (note that its value is independent of so). We now prove (4.7). Differentiating both sides of (4.4), we obtain
\ll~(no, so, a) = 1:+so ( F~ + F~ ::) ds.
(4.8)
Note that anjaa satisfies the variation equation
+ F' ds aa naa Q and the initial condition anjaals=so = 0, and that .!!.._ an = F' an
: : = exp
(l: F~
ds) l:+so exp (
-l: F~
ds)
F~ as.
Substituting in (4.8) and then integrating by parts, we obtain
rl+so
111~(n 0 , s0 , a)= exp ( lso
F~ ds
r
)
) rl+so ( lso exp - lso F~ as F~ ds.
(4.9)
79
§4. GENERAL BEHAVIOR
In §2 we have already computed F~(s,O,O) =
(4.10)
H(s),
which represents the value of the curvature of the orthogonal trajectory of (4.l)a=O on r. Similarly, it is easy to compute F~(s,O,O) = [P(cp(s),-,P(s),O)Q~(cp(s),t/J(s),O)
- P~(cp(s), 1/J(s), O)Q(cp(s), 1/J(s), 0)] · [P2 (cp(s), 1/J(s), 0) + Q2 (cp(s),tjJ(s),O)t 1 = ao( s)1ao:,
(4.11)
where ao( s) 1ao: represents the value of the partial derivative of the inclination 8 of the trajectory of (4.1) with respect too: at o: = 0, n = 0, and s = s (the arc length s of r). Let no= o: = 0 in (4.9). Note that n(s,O,so,O) 0. Substituting (4.10) and (4.11) into it, we obtain
=
w~(O, so, 0) = exp
[)()( ds. (! l+so H(s) ds) !l+so so exp (- isso H(s) as) a;;8)
so
It is easy to see that, along r, H(s) is a periodic function of s with period l. We make the change s = so + s' of the integration variable, and then letting s' be s we get (4.7). The theorem is proved. The methods oi studying our problems mainly apply the implicit function theorem under suitable conditions to (4.5) to obtain that no is a single-valued function of o: and so, or that o: can be solved to be a single-valued function of no and so. Hence we should first note that the results obtained are local. Next we note that in problem I the effective constituent is '11~ 0 (0, so, 0), since when it is not zero we can solve from (4.5) to obtain no= no(so, o:), which is single-valued and differentiable for very small lo:l. This shows that for every sufficiently small lo:l there exists a unique no which satisfies {4.5), i.e., the system (4.1)a has a unique closed trajectory in a neighborhood of r which passes through (so, no). Conversely, if W~ 0 (0, so, 0) = 0, then the existence or uniqueness of the closed trajectory of (4.1 )a cannot be guaranteed. In problem II, the effective constituent is '11~(0, s0 , 0); when it is not equal to zero, we can solve from (4.5) to get o: = o:(so,no), which is single-valued and differentiable for very small Ino I· This shows that the trajectory of (4.1) passing through the point (so, no) is closed only when o: = o:(so, no). Hence trajectories corresponding to different o: cannot intersect. Conversely, if w~ {0, s 0 , 0) = o, then the property of not intersecting is questionable.
THEORY OF LIMIT CYCLES
80
From Theorem 4.1, we easily get THEOREM 4. 2. If system (4.1) forms a family of rotated vector fields with respect to a, then the rate of expansion (or contraction) of limit cycles is nowhere equal to zero. PROOF. From the condition indicated in the theorem, we know that 89f8a ~ 0; hence from (4.7) we see that w~(O,so,O) > 0; hence (4.5) can be solved for a single-valued function a= a( so, no), and there exists
dno
I
da no=O
= I~ exp (- Isso+so H(s) as) (89(s + so)/8a) ds' exp (-I~ H(s) ds) - 1
(4.12)
whose value never equals zero for all values so. If r is a single stable cycle (assumed to be negatively oriented), then the denominator of the right side of (4.12} is greater than zero; hence
dno/ dalno=O > 0, i.e.,
r
expands as a increases. Conversely, if r is a single unstable cycle, then
dnofdalno=O < 0, i.e., r contracts as a increases. These conclusions conform with those provided by the table following Theorem 3.4. If r is a multiple cycle, then
dnofdalno=O = oo, which also suitably reflects the results in Theorem 3.5. Extending the conditions of Theorem 4.2, we can get THEOREM 4.3. If, for every so, w~(O,so,O) # 0 and keeps constant sign, then in a sufficiently small neighborhood of r, closed trajectories of (4.1) corresponding to different a must not intersect. PROOF. From the conditions of the theorem, we know that for every so (0 ~so~ l), on the normal line segment off passing through (so,O) equation (4.5} defines a as a single-valued function of no whose domain is a nonzero interval and which satisfies a(s 0 , 0) = 0. Hence, according to the Weierstrass theorem, a( so, no) has a minimum interval of no which is not equal to zero and is suitable for all s0 • This shows that there exists a neighborhood of r which is filled with the closed trajectories of (4.1). These closed trajectories do not intersect, and this situation has been explained in Theorem 4.2.(2) (2)Rigorously speaking, from the conditions of the theorem we know that when lnol and lal are sufficiently small, for every so we have lll~(no,so,a) > 0 (< 0). Now suppose fa 1 and fa~ intersect at (s 0,n0). Then llf(n 0,s0,al) = llf(n0,s0,a2) = 0. Thus there exists i'i between a1 and a2 such that llf~(n0 ,s 0 ,a) = 0, which contradicts the previous formula.
§4. GENERAL BEHAVIOR
81
However, we should note that, although o: is a single-valued function of n 0 , its inverse function need not be single-valued. In particular, if for some so the range of values of o: degenerates to a single point o: = 0, then obviously for every s 0 the range of values of o: is o: = 0. At the same time, r is one of the system of periodic cycles of {4.1)a=O• and, by the property of nonintersection, system {4.1)a=O does not have a closed trajectory in a small neighborhood of r. We should also note that the nonintersecting property assured by Theorem 4.3 only applies to o: with very small absolute value; its corresponding trajectory not only lies in a small neighborhood of r, but also takes r as its limit position as o: -+ 0. We call this kind of closed trajectory the first type. It is highly possible that system (4.1) has a closed trajectory f* for some o: with rather large absolute value which intersects r, but as o:-+ 0 the limit position off* is another closed trajectory rl of (4.1)a=O· Then Theorem 4.3 is not applicable to both r• and r. We call r• (with respect to r) a closed trajectory of the second type. IT we require lo:l to be sufficiently small, then in a suitably small neighborhood of r there do not exist closed trajectories of the second type-otherwise it would contradict the continuity of solution with respect to its parameter. In order to guarantee that the conditions of Theorem 4.3 can be established, aside from assuming that system (4.1) forms a family ofrotated vector fields with respect to o:, there are some other methods. For this we first prove: LEMMA 4.1. Iff is not a single cycle, then, for 0 has the same sign. PROOF. Let so+ s =
~so~
l,
\11~(0, s 0 ,0)
s' in (4.7), and note that J~ H(s) ds = 0. We obtain
1s' H(s) ds) -80(s') ds' 8so so 89(s') = exp (Joro H(s) d1i) 1so+l so exp (- Jor' H(s) d1i) fia ds'.
\11~(0, s 0 , 0)
= 1so+l exp (-
0:
(4.13)
It is easy to see that 89(s')/8o: has period l with respect to s', and from the equality
lo s' +I H(s) d1i = is' H(s) d1i + ls' +I H(s) d1i
1
os'
0
=
;~
H(s) d1i + lo H(s) d1i
r'
= lo
H(s) ds,
THEORY OF LIMIT CYCLES
82
we know that exp(- J;' H(s) as) also has period l with respect to (4.13) can be rewritten as
111~(0, s0 , 0) = exp = exp
(loro H(s) d8) lo{ exp 1
(fo
80
H(s)
(
-
s'.
Thus,
89(s') ds' lot' H(s) d8) ---a;;-
d8) 111~(0, 0, 0).
The proof is completed. According to this lemma, from Theorem 4.3 we can immediately obtain THEOREM 4.4. Ifr is not a single or periodic cycle, and 111~(0, so, 0) =f. 0 for some so, then, when a varies slightly to one side or to both sides, system (4.1) has a closed trajectory in a neighborhood of r; moreover, closed trajectories corresponding to different a do not intersect. COROLLARY. If for a= 0 the system (4.1) has a system of closed trajectories which completely fill a small neighborhood U of r, then, for a =f. 0 (but lal very smalQ, a necessary and sufficient condition to have a closed trajectory in U is that 111~(0, so, 0) 0.
=
This corollary is very useful in studying the third problem. Corresponding to Theorem 4.3, we have the well-known THEOREM 4.5. Ifr is a single cycle (i.e., 111~ 0 (0,so,O) =f. 0), then when is sufficiently small, system (4.1) has, in a neighborhood of r, a unique single cycler Q of the first type, which has the same stability as r.
lal
PROOF. From 111~ 0 (0, so, 0) =f. 0, we know that when Ia I is sufficiently small (but does not degenerate to a point o: = 0), equation (4.5) defines no to be a single-valued continuously differentiable function of o:, satisfying the condition n 0 (0, so)= 0; hence the existence and uniqueness of r a is proved. Moreover, since the value of H(s, a) ds varies continuously with o: and the position of r a, if Jr. H(s, o:) ds =f. 0, then, when lal is very small, H(s, a) ds has the same sign with it, i.e., r a is also a single cycle and also has the same stability as r. The proof is completed. Theorem 4.5 shows that when a varies slightly from 0 on any side a single cycle cannot disappear, and can keep its stability. But Theorem 4.4 shows that when r is a multiple cycle (but not a periodic cycle) and 111~(0, s0 , 0) =f. 0 for some a 0 , at least when o: varies slightly in one direction, the limit cycle does not disappear, but cannot keep its multiplicity, and its uniqueness is not assured. As an example, a semistable cycle in the rotated vector fields splits into two.
J
Jr ..
§4. GENERAL BEHAVIOR
83
THEOREM 4.6. If w~(O,so,O) ::/: 0 hold3 for some so, then a necessary and sufficient condition for r to be a semistable cycle is that a( no, so) takes a strict extremal value at no = 0; that is, when a varies from 0 to some direction, there appear limit cycles simultaneously on both sides of r for system (4.1), and when a varies in the other direction, these limit cycles disappear. PROOF. We may as well assume that W'~(O,s 0 ,0) > 0 at s0. Thus there exists o > 0 such that when lal < and lnol 0, and from \11'~(0, s* ,0) ::/: 0 we know that equation (4.5) defines a= a( no, s0) as a single-valued continuous function of no on the normal line segment -€ ~ no ~ € passing through the point (s;), 0) of r. Now let a take its minimum value at no = 0. Then there exists a small interval of no, [-17, 17] ~ [-c, c], such that a(no, s 0) > 0 for all no ::/: 0. Now we take 11 so small that 11 < o, and a( no, s 0) < o when lnol ~ 11· Thus, if no E (0, 17), then w~(n 0 , s 0, a( no, s0)) > 0.
o
But we already know that W'(no, s0, a( no, s0))
= 0; hence
W'(no,s 0,0) < 0;
that is, r is externally stable. Similarly, if no E ( -17, 0) we can still derive llf(no, s 0,0) < 0, that is, r is internally unstable. Combining all this, we see that r is semistable. Conversely, if r is externally stable but internally unstable, then, for all no ¥- 0 in a sufficiently small interval (-17, 17],
W'(no, s0,0) < 0,
w~(no, s;), a)
> 0,
\II'( no, s0, a( no, s0))
= 0.
Altogether we can see that a( no, s 0) > 0 for no ::/: 0; that is, a takes its minimum value at no = 0. Since a semistable cycle is a multiple cycle, under the condition of Theorem 4.6, if r is semistable, then not only does a take its extreme value at n 0 = 0, but also the closed trajectories which fill completely a neighborhood of r do not intersect. It is obvious that this theorem is a generalization of Theorem 3.5. We can also generalize Theorem 4.4 to THEOREM 4. 7. Let f be a k-fold cycle of equations (4.1}<., 1 < k < oo, and let there exist so such that \11(0, so, a) does not take an extremum at a= 0. Then when a varies in one or both directions, r does not disappear; moreover, for sufficiently smalllal, system (4.1) does not have more thank limit cycles in the vicinity of r. PROOF. By the hypotheses, for an arbitrary c > 0 we can find a1 < 0 < a 2 , lo:il < c, such that ll'(O, so, o:t) and ll'(O, so, o:2) have opposite signs. Thus,
84
THEORY OF LIMIT CYCLES
there also exists 6 > 0 such that for all n 0, lnol < 6, 111(n0, so, at) and 'll(no, so, a2) have opposite signs. Hence for any no which satisfies lnol < 6, we can at least find an a( no), between at and a2, to make 111(n0, s0 , a( no)) = 0. This shows that system (4.1)a(no) has a closed trajectory passing through the point (so,no). If all the a(no) =: 0, then f is a periodic cycle of (4.1)a=O• which contradicts the hypothesis. Hence the range of a(no) is a nonzero interval which may have a = 0 as an interior point or an end-point. Note that a(n0 ) is not necessarily a single-valued function; that is, limit cycles corresponding to different a may intersect. Next we assume there exists a sequence of numbers ai - 0 such that the system (4.1)a, always has k + 1 or more limit cycles of the first type in a neighborhood of r. Then for each ai there exist at least k + 1 numbers n~ (j = 1, ... , k + 1) such that
(j = 1, 2, ... , k + 1; i = 1, 2, ' .. ). Thus from Rolle's theorem we know there must exist nb such that (i=1,2, ... ),
where nb is among the n~. Let i - oo in the above formula; then ai - 0 and nb - 0. Hence we obtain w
r
However, we should note here that even if r is a stable or unstable cycle of (4.1)a=O• system (4.1)a#O may still have more than one limit cycle in a neighborhood of r. Hence it is meaningless to talk about the stability of the limit cycle. Summarizing the above theorems, we conclude that as long as the following two conditions hold simultaneously, the limit cycle r may disappear when a varies from 0 to either side: 1. w(no, so, 0) has an extremum at no = 0 for all so (that is, r is a semistable cycle or some compound limit cycle). 2. 111(0, so, a) has an extremum at a= 0 for all so. We should also note that in the above theorems, the conclusions obtained on nonintersection are all local: that is, they only assure that under suitable conditions two limit cycles r 0<1 and r a~ of the first type (relative
§4.
GENERAL BEHAVIOR
85
tor) (a1 =/:- a2, lail ::::; 1) do not intersect, but they do not ascertain whether two arbitrary cycles r a 1 and r a 2 (a1 =/:- a2) also do not intersect. For global results we only have THEOREM 4. 9. If system (4.1) forms a family of generalized rotated vector fields with respect to a, then any two cycles rOil and r 012 (a1 =/:- a2) either do not intersect or completely coincide. PROOF. We may as well assume that a 2 > a 1 and r 011 is positively oriented. From Definition 3.2 in §3 and the Finite Covering Theorem, we know that for every point (x,y) on r a 1 there exists a common positive number 01 such that when a E [a1, a1 + 81] the inequality
P(x, y, a1)Q(x, y, a) - P(x, y, a)Q(x, y, a 1 )
~
0
always holds; that is, the above formula holds everywhere on r 011 • This shows that when a increases from a1 in the interval [a 11 a1 + 8], the vector determined by the vector field (P(x, y, a), Q(x, y, a)) at every point r 011 either is stationary OJ:. rotates slightly in the counterclockwise direction. Let a1 + 81 = a~. We know as before that for every point (x, y) on r a 1 there exists a common positive number~ such that when a E [a~, a~+ 62] the inequality
P(x, y, a~)Q(x, y, a) - P(x, y, a)Q(x, y, a~) ~ 0 always holds. As before, we know that when a increases from a~ in the interval [a~, a~+ 62], the vector defined by the vector field (P(x, y, a), Q(x, y, a)) at every point (x, y) on r 011 either is stationary or continues to rotate in the counterclockwise direction. According to the Finite Covering Theorem, we can divide the interval [at. a2] into a finite number of small intervals: [at. a~], [a~, a~ = a~ + 82], [a~, a4], ... , [a~, a2] such that in every subinterval the vector field at every point on r 011 either is stationary or rotates in the counterclockwise direction. Thus finally we obtain that the inequality
P(x, y, at)Q(x, y, a2) - P(x, y, a2)Q(x, y, a1)
~
0
holds everywhere on r a 1 • Then from Theorem 3.13 in §3 we obtain the results we wished to prove . . In the following we study problem Ill. Let system (4.1)a=O have a system of closed trajectories which completely fill a neighborhood of the closed trajectory r. By the corollary of Theorem 4.4, we can assume that '11~(0, s 0 , 0) = 0 for all so, for otherwise when a i- 0 and lal is very small the system (4.1)
THEORY OF LIMIT CYCLES
86
cannot have a closed trajectory in a neighborhood of condition
w~(O,O,O) =
r.
lo' (-los H(s)as) :: exp
ds
Now we study the
= 0,
(4.14)
for the case s = 0. Change the integration variable to the time variable t, and note that from §2 we have exp
(-los H(s) as) = (-lot exp
=
+ Qyo) dt + ~ dlog(PJ + Q~))
v' p,2 + Q2 ( JPJ(;) + Q~(O) exp -
where Po = P(f(t), g(t), 0) and Qo are the equations of r' and
ds = JPJ
(Pxo
r
Jo (Pxo
)
+ Qyo) dt ,
= Q(f(t), g(t), 0); x = f(t)
and y
= g(t)
+Q~dt,
We obtain w~(O,O,O)
=
JPJ(O/+ Qg(o) loT exp
(-lot
(Pxo
+ Qyo) dt)
(PoQao- QoPao) dt
=0. (4.15) LEMMA 4. 2.
( exp
The vector function
(-lot
(Pxo
+ Qyo) dt,- exp
(-lot
(Pxo
+ Qyo) dt)
evaluated along r is a periodic solution of the adjoint equations of the variation equations of (4.1)a=O with respect to its periodic solution x = f(t), y = g(t). PROOF. The adjoint equations mentioned in the lemma are
d6/dt = -Px(/(t), g(t), 0)6 - Qx(/(t), g(t), 0)6, d6/dt
= -Py(f(t), g(t),0)6- Qy(f(t), g(t},0}6.
(4.16)
To prove this lemma, let {t = exp
(-lot (Pxo +
Qyo) dt) Qo,
6
= - exp
(-lot (Pxo +
Qyo) dt) Po
and substitute in (4.16) to verify it directly. Now suppose the equations of a system of the closed trajectories of (4.1) 0 ,0 are X= fh(t) = f(t, h), Y = Yh(t) = g(t, h), (4.17)
§4. GENERAL BEHAVIOR
87
where h is the parameter of the family of closed trajectories, fh is its corresponding closed trajectory, and 6h(t) and 6h(t) are the corresponding periodic solution of the adjoint equations of its variation equation. Making use of (4.15) and Lemma 4.2, we can rewrite the corollary of Theorem 4.4 as
4.10. In a family of closed trajectories {fh} of (4.1)a=O• for the fh which can generate a closed trajectory of (4.1)a at a f. 0, its corresponding parameter h must satisfy THEOREM
At(h) =
J [~th(t)Pao(/(t, h), g(t, h), 0) + 6h(t)Qao(/(t, h), g(t, h), 0)] dt =
!r"
0.
(4.18)
A necessary condition for a trajectory to be written in this form is that
(4.18) be extendable to suit nonstationary systems of equations containing a parameter inn-dimensional space.( 3 ) In order to study further the existence, number, and stability of limit cycles of system (4.1) when a f. 0, we shall move the coordinate system to fh. Suppose l is a transversal of the family of closed trajectories of (4.17) (this transversal exists; for example, we can take l as the orthogonal trajectory of (4.1)a=o). Take the point of intersection of any fh and las the starting point to measure the arc lengths. Thus through the transformation of coordinates x =
y = 1/J(s, h)+ n
(where x =
(4.19)
dn/ds = F(s, n, a, h).
Then we introduce a successor function( 4 )
{'"
'lll(no, a, h)= lo
where
lh
is the perimeter of
'lli~(O,O,h) =
fh.
F(s, n, a, h) ds,
As before, we can prove that
h) Jof'" exp ( - Jor H(s,h)d:s) oO(s 0 ~ ds =
A 1 (h).
(4.20)
Now let system (4.1)a have a closed trajectory La in the vicinity of some rh, and let the point of intersection of La and l lie on fha. Then it is easy to see that 'lli(O, a, ha) = 0. Hence the problem of studying whether system ( 3 )Refer
to Chapter 6 in [60]. no r" is a single cycle, the initial value of s can be fixed as will not be indicated in 111. ( 4 )Since
so
= 0; hence it
THEORY OF LIMIT CYCLES
88
(4.1) has a closed trajectory is now changed to the problem of whether the equation (4.21) '11(0, a, h) = 0 has a real solution. It is obvious that '11(0, 0, h) - 0; hence, expanding '11(0, a, h) in a power series of o, we have on
'11(0, a, h)= -
n.1
[An(h)
+ oAn+l(h, o)]
(n
~
1),
(4.22)
where An(h) is the first coefficient not identically equal to zero. THEOREM 4.11. When a varies, the necessary condition that fh" does not disappear (it means there exists a closed traiectory L"' of (4.1)a such that L"'-+ rh" as a-+ 0) is (4.23) An(h*) = 0. If (4.23) holds and An(h) does not have an extremum at h*, then as a varies, fh" does not disappear. Moreover, if An(h*)
= A~(h*) = · · · =
A~k-ll(h*) = 0,
then when lol is sufficiently small the system (4.1) 01 cannot have more thank limit cycles in the vicinity of fh".
PROOF. If An(h*) =f. 0, then from (4.22) we can see that when lol =f. 0, lol and lh- h* I sufficiently small, '11(0, a, h) =f. 0; that is, fh" does not generate any closed trajectory of system (4.1) 01 • Next suppose An(h*) = 0, but is not an extremum of An(h); then for any s > 0 there always exist h1 < h* < h2, lhi - h*l < s, such that An(hl) · An(h2) < 0, and so there exists 6 > 0 such that when lol < 6 we have '11(0, a, hl)'lt(O, a, h2)
< 0.
Thus for any fixed o, lol < 6, there exists at least one ha such that '11(0, a, ha) = 0, where h1 < ha < h2; that is, system (4.1)a has a closed trajectory, which intersects l on fha . Finally, suppose there exist ai -+ 0 (i -+ oo) such that when a = Oi system (4.1) has k + 1limit cycles in the vicinity of fh". Then there exist h{ (i = 1, ... , k + 1) such that .
An(hn
.
+ aiAn+I(h~, ai) =
nl
.
~w(o,ai, hn Qi
Thus, by Rolle's theorem, we know there exists 'It~~) (0, Oi, hi) = 0; that is,
hi
= o.
among the
+ Ot.ahk ak A n+l (h*i, a,·)A n
h{
such that
§4.
GENERAL BEHAVIOR
89
Now let i
--+ oo; then ai --+ 0, hi --+ h*, and from the above formula we get A~k)(h*) = 0, which contradicts the assumption. The theorem is completely
proved. COROLLARY. If An(h*) = 0 but A~(h*) :/: 0, then unique limit cycle of system (4.1) 0 •
fh"
can generate a
Using Theorem 4.11 and its corollary, we can derive some well-known classical results. THEOREM 4.12. The van der Pol equation
X+ J.L(x 2 -1):i: +X= 0 has a unique periodic solution when IJ.LI is sufficiently small. (Translation of t is not counted.) PROOF. The equivalent system of equations is
dxfdt = y,
dyfdt = -x + J.LY(1- x 2 ).
When J.L = 0, (4.24) has a system of closed trajectories x 2 for system (4.24) we have Qo = -x,
Po= y,
P:xo + Qyo = 0, Pp.o = 0,
(4.24)
+ y2 = h2 •
Since
Qp.o = y(1- x 2 ),
condition (4.18) becomes
A 1 (h)
= Jr, 1 y 2 (1- x2 ) dt = 0.
(4.25)
Let x = h sin t and y = h cost be substituted in the above formula. We obtain
A 1 (h)
= fo 2
7r
h 2 cos 2 t(1- h 2 sin 2 t) dt = 1rh 2
(
1-
h:)
= 0.
But h :/: 0; hence h = 2. It is easy to compute that A~ (2) :/: 0. Hence from the corollary of Theorem 4.11 we know that system (4.24) has a unique closed trajectory when IJ.LI is sufficiently small. This closed trajectory approaches the circle x 2 + y 2 = 4 of radius 2 as J.L--+ 0. We can prove (see Exercise 1) that when J.L > 0 the limit cycle is stable, whereas when J.L < 0 it is unstable. Next we consider the perturbed system of the Hamiltonian system
dx 8H dt =- 8y here we assume p(x, y, 0) of equations
+ p(x, y, a:),
dy 8H dt = ax
+ q(x, y, a:);
(4 ·26 )
= q(x, y, 0) = 0, and the general integral H(x, y) =
dxfdt = -8Hj8y,
dyjdt = 8H/8x
h
THEORY OF LIMIT CYCLES
90
is a system of closed trajectories rh. Now Po= -oHfoy,
Pao
= p~(x, y,O),
Qo = oHfox, Qao
Pxo + Qyo
=0,
= q~(x, y,O),
and condition (4.18) becomes At(h)=
£,. [~~p~(x,y,O)+ ~~q~(x,y,o)]
dt
1 p~(x,y,O)dy-q~(x,y,O)dx !r" = [p~a(x, y, 0) + q~0 (x, y, 0)] dx dy = 4>(h) = 0,
=
J£,.
where Gh is the interior region of rh. Thus, Theorem 4.10 and its corollary become THEOREM 4.13. If 4>(h*) = 0 and 4>'(h*) =I 0, then when lal is very small, system (4.26) has a unique limit cycle in the vicinity offh". Moreover, if
then system (4.26) cannot have more thank limit cycles in the vicinity ofrh".
The first half of this theorem is a result of Pontryagin [58], and the second half is the result of Zhang Zhi-fen. The problem of whether the k limit cycles mentioned in Theorem 4. 7 can be realized has been studied in detail by Andronov and Leontovich [59]. Since their results are more complicated, we do not discuss them here. Please refer to the original paper or Qin Yuan-xun's book [67], Part II, Chapter 4. Exercises 1. Prove, under the condition of the corollary of Theorem 4.11, that if rh expands ash increases, then for an A~(h*) < 0 (> 0) the unique limit cycle generated by rh" is a stable (unstable) cycle. 2. Prove that ifr is a single cycle and "llf~(O, s 0 , 0) 0, then 80f8ala=O 0 (along r). 3. Explain why the coupling theorem obtained by exchanging the positions of a and no in Theorem 4.6 is obvious, and meaningless. 4. Prove, under the condition of Theorem 4.7, that if k is odd, then in a neighborhood of r there exists at least one limit cycle of system (4.1) 0 which has the same stability as r. 5. What is the coupling problem of problem III (with the same meaning as in Exercise 3)? What is the corresponding condition of (4.18)?
=
=
§5. Existence of a Limit Cycle Since van der Pol studied the equation
X+ J.t(x 2 - 1):.i: +X= 0
(J.t > 0)
(5.1)
in 1926, and proved that the system of first-order differential equations in the corresponding phase plane
dxjdt = y,
dyjdt = -x + J.t(1- x 2 )y
(5.2)
had limit cycles, his results have been extended by many workers in physics and mathematics. Up to now, people not only have studied the problems of limit cycles of systems of second-order nonlinear equations more general than
(5.1):
x + f(x):.i: + g(x) =
0
(5.3)
and
x + f(x, :.i:):.i: + g(x) = 0,
(5.4)
but also have extended (5.2), and studied problems of limit cycles of systems of equations which cannot necessarily be transformed to second-order nonlinear equations, such as systems of the type
dxjdt = h(x) + h(y),
dyjdt = g1(x) + Y2(Y)
(5.5)
dyjdt = Q(x, y),
(5.6)
and the most general equations
dxjdt = P(x, y),
among which there is a considerable amount of research for the case where P and Q are quadratic or cubic polynomials. We shall give a deep analysis and detailed study of (5.6) in the second half of this book. In this section we shall mainly introduce methods of proving the existence of limit cycles for equations of the types (5.3), (5.4), and (5.5). As to this problem, although in essence we use the Annular Region Theorem to ascertain the existence of a limit cycle, we still apply different techniques to different types of equations. We shall 91
92
THEORY OF LIMIT CYCLES
mainly introduce several typical theorems for different types of equations and different techniques. Then in the end we enumerate some recent articles for the convenience of our readers' reference. I. Equations (5.4) and (5.3) have obvious physical significance in the theory of nonlinear oscillations. Hence, other than the hypotheses which guarantee the existence and uniqueness of their solutions, such as f and g being continuous (these will not be mentioned again), we shall assume that xg(x)
> 0 when x =F 0
(5.7)
(hence necessarily g(O) = 0) and
(5.8) /(0) < 0 or /(0, 0) < 0. (5.7) means that there exists a unique equilibrium position x = :i: = 0, and the restoring force is opposite to the direction of displacement; (5.8) means that the oscillation system has negative damping near the equilibrium position. Under these conditions, (5.3) is generally called the Lienard equation, the extension of equation (5.1) which was first studied by Lienard [9]. In order to study equation (5.3), in general, let G(x)
=fox g(~) d~,
F(x)
=fox/(~) d~,
(5.9)
and proceed to study the system of equations(!) dxfdt
= y- F(x),
dyfdt
= -g(x),
(5.10)
which is equivalent to (5.3). THEOREM 5 . 1 (A. V. D RAGILEV) . Let the following conditions hold:
1) xg(x) > 0 when x =F 0, and G(±oo) = +oo. 2) xF(x) < 0 when x =F 0 and lxl is sufficiently small. 3) There exist constants M > 0 and K > K' such that F(x) ~ K when x
>M
and F(x) ~ K' when x < -M.
Then system (5.10) has stable limit cycles. PROOF.(2) Let
A(x, y) = y /2 + G(x), A1 (x, y) = (y- K) /2 + G(x), -(1)Here we let y = :i; + f(x), which is not the same as taking y = :i; in (5.2). 2
2
This transformation is called the Liinard transformation; the (x, y)-plane is called the Liinard plane. The advantages are that two nonlinear functions are put separately into two different equations, and, moreover, we get to replace /(x) by its integral F(x), which has better smoothness properties. (2)Here we adopt a simpler method of proof proposed by Huang Ke-cheng [61] which makes the estimate of the interval of existence of limit cycles much easier.
§5. EXISTENCE
93
A2(x, y) = (y- K') 2/2 + G(x).
We compute their total derivatives with respect to t separately, and obtain
dA 8Adx 8Ady dt = ax dt + ay dt = y( -g(x)) + g(x)(y- F(x)) = -g(x)F(x), dAt dt
= g(x)(K- F(x )) ,
dA2 dt
= g(x) ( K
I
(5.11)
- F ( x )) .
From condition 1) we know that A(x,y) = C (C > 0) is a family of closed curves containing the origin. We call the curve the equivalent energy curve, and A(x, y) the energy function. From condition 2) and (5.11) we know that dA/ dt > 0 in the vicinity of the origin. Hence, for every closed curve A(x, y) = C in the vicinity of the origin, every trajectory intersecting it always crosses it in the interior-to-exterior direction. In order to apply the Annular Region Theorem, we only have to construct an outer boundary curve r of the annular region such that every trajectory intersecting r always crosses it in the exterior-to-interior direction. First, tiUce Yt > 0 such that Yt > K, -y1 < K', and for all the points in the region R+ = {- M :::; x :::; M, y ~ Yt} we have
y- F(x) > 0,
-g(x) K -K' y _ F(x) < 2M ,
and for all the points in the region R_
y- F(x) < 0,
(5.12)
= {-M:::; x:::; M, y:::; -yt} we have
-g(x) K- K' y-F(x) < 2M .
(5.13)
Let l = max((y1 - K') 2 /2, (-y1 - K) 2 /2]. We inspect the closed curve r = A"BCDA (Figure 5.1 corresponds to (-y1- K) 2 /2 > (y1- K') 2 /2), where BC is a closed curve At (x, y) = l + G(M) lying to the right of the line x = M, AD is a closed curve A2(x, y) = l + G( -M) lying to the left of the line x = -M, and AB and C15 are line segments. Let the point of intersection of the lines x = M and y = K be Ot and the point of intersection of x = - M and y = K' be 02. From the construction of r it is easy to prove that Yo :::; -y.,
YA ~ Yt,
l"lhVI = I02DI,
IOtBI = I02AI.
II DC and AB and DC lie in R+ and R_ respectively, with slope (K- K')/2M. From conditions 1) and 3) and (5.11)-(5.13), we know
Thus AB
/3
=
that all the trajectories intersecting r must cross it in the exterior-to-interior direction. The theorem is completely proved. The above method of proof can be extended to obtain sufficient conditions for existence of limit cycles of equations more general than (5.10):
dx/dt = h(y) - F(x),
dyjdt
=
-g(x),
(5.14)
94
THEORY OF LIMIT CYCLES '!/
D
o:--M
FIGURE 5.1
where g(x) does not necessarily have only one zero, and yh(y) > 0 when y # 0, h( +oo) = +oo, h( -oo) = -oo. For simplicity, we only consider here the case where h(y) y to prove one theorem. As to (5.14), we refer the readers to
[61].
=
THEOREM 5. 2. Let xF(x) < 0 (or> 0) when lxl # 0 .and lxl is sufficiently small, and let there exist constants M > 0, Xt > 0, and X2 > 0 such that 1) xg(x) > 0 when -x2 < x < 0 and 0 < x < Xti 2) F(x) ~ -M (or F(x) ~ M) when 0 < x < Xt, and F(xt) ~ M + ..,fii (or F(x) ~ -M- v'2l); 3) F(x) ~ M (or F(x) ~ -M) when -x2 < x < 0, and F(-x2) ~ -M- v'2l (or F(-x 2 ) ~ M + J2I), where l = max[G(xt),G(-x2)]. Then system (5.10) has stable (unstable) limit cycles. PROOF. Let
= !(Y + M) 2 + G(x), .X2(x,y) = !(y- M) 2 +G(x).
.X1(x,y) Thus it is easy to compute
= -g(x)[M + F(x)], d.X 2 jdt = g(x)[M- F(x)].
d.Xtfdt
(5.15)
The construction of the inner boundary curve of the annular region is the same as in Theorem 5.1. Now we are going to construct the outer boundary curve. In conditions 2) and 3) we only consider the case outside the
§5. EXISTENCE
0
95
s
FIGURE 5.2
parentheses. Inspect the closed curve r = ABCDEFA (Figure 5.2 corresponds to the case G(xt) > G( -x2)) in which AB is part of the curve ..\1(x,y) =(2M+ v'2i) 2/2; taking A(O,M +..til) as left end-point, it is easy to see that YB > M; AF is part of the curve ...\2(x, y) = l, and YF ~ M. DE is part of the curve ...\2(x, y) = (2M+ v'2i) 2/2, taking D(O, -M- ..til) as right end-point, and it is easy to see that YE < -M; fiG is part of the curve ..\1(x, y) = l, and Yo ~ -M; 1JlJ and EF are line segments. It is obvious that YB < YA = M +..til, YE > YD = -M- ..til. From the conditions of the theorem and (5.15) we know that all the trajectories of system (5.10) intersecting r always cross it in the exterior-to-interior direction. The theorem is completely proved. Since all the additional conditions on F(x) and g(x) in Theorem 5.2 are local, when the singular point of system (5.10) is not unique, i.e., g(x) has more than one zero, or the conditions G(±oo) = +oo do not hold, Theorem 5.2 is still applicable. EXAMPLE 1. The system of equations dxfdt = y- x 3
+ 3x,
dyfdt = x 2
-
6x
{5.16)
has limit cycles. (This system has singular points (0, 0) and (6, 198).) PROOF. Taking M = 2 and x 1 = x2 = 3, we can verify that the conditions of Theorem 5.2 hold. Applying the technique in Theorem 5.2 several times, we can give sufficient conditions for system (5.10) to have at least n limit cycles. For example, we
THEORY OF LIMIT CYCLES
96
can prove that the systems of equations dxjdt
= y + x 2 sin x,
dyjdt = -x,
and dxjdt = y + x 2 sinx,
all have at least n limit cycles in the strip system of equations, 2x g(x) = (1 + x2)2,
lxl
~
mr + 1rj2.
x2
G(x)
= 1 + x2,
G(±oo)
For the second
= 1.
II. The second method for proving the existence of limit cycles of system (5.10) is quite different from the methods used in the previous two theorems. We now introduce this method, using a method of A. F. Filippov [62] as its representative. THEOREM 5.3. Suppose that in (5.10) g(x) satisfies
xg(x)
>0
when xi= 0,
G(±oo) = +oo;
(5.17)
and suppose that after a change of variables
fox g(~)d~=z1 (x), fox f(~)d~=F(x)=Fl(zl)
whenx>O,
(5.18)
fox g(~) d~ = z2(x), fox f(~) d~ = F(x) = F2(z2)
when X< 0,
(5.19)
the functions F 1(z) and F2(z) satisfy the following conditions: 1) For small z (0 < z < o) we have F1(z) ~ F2(z) but F1(z) ¢. F2(z); and F 1 (z) < a.fi and F2(z) > -a.fi, where 0 0 such that
fozo (F1(z)- F2(z)) dz > 0, and, when z > zo, F 1(z) ~ F2(z), F1(z) > -a.fi and F2(z) Then system (5.10) has stable limit cycles.
< a.fi.
After changing x to z 1 and z2 and then eliminating t, (5.10) becomes
= F1(zl)- y, dz2jdy = F2(z2)- y,
dzi/dy
> 0, when z2 > 0. when Zt
(5.20) (5.21)
The key point of Filippov's method is this: through the transformations (5.18) and (5.19) we change the trajectory of system (5.10) in the right halfplane into the integral curve of equation (5.20) in the right half-plane, and change the trajectory of system (5.10) in the left half-plane into the integral
§5. EXISTENCE
97
curve of (5.21) in the left half-plane; then we apply the comparison theorem for differential equations to (5.20) and (5.21) to determine the relative positions of the points of intersection of the integral curves of these two equations with they-axis; finally we return to the (x, y)-plane, and it is easy to construct the outer boundary curve of the required annular region. First we note that for equations (5.20) and (5.21) we only have to assume f(x) and g(x) are continuous and condition (5.17) holds; then we can guarantee the existence and uniqueness of solutions of (5.20) and (5.21). This is because the function dFi(Zi) dF(x) dx f(x) dzi = liX dzi = g(x) (i = 1' 2) is continuous in the half-plane Zi > 0, so that the existence and uniqueness of solutions of (5.20) and (5.21) on the right half-plane can be guaranteed. As to any point on the y-axis except the origin, the existence and uniqueness of the solution can be guaranteed since the right side of the equation dy g(x) dx- F(x) -y has continuous partial derivatives with respect toy. LEMMA 5.1. Let F(z) be continuous, and F(O) = 0. Then the integral curve of the equation dzjdy = F(z)- y (5.22) passing through the point B(zo, F(zo)) (Zo > 0) must intersect the y-axis at two points A and A' such that YA 2': 0 and YA' ~ 0, at least one of which is a strict inequality.
Note (5.22), and observe that ldyjdzl < 1 when IYI > 1 + We know that the integral curve passing through a point Bon the isocline y = F(z) lies to the left of the line z = zo. It has negative slope above y = F(z), and positive slope below y = F(z). (Figure 5.3.) But both its ends must intersect the y-axis; let the points of intersection be A and A'. It is easy to see that A = A' is not possible. PROOF. max[o,zo]
IF(z)l.
LEMMA 5.2. Let F(z) be continuous, F(O) = 0, and F(z) < aJZ (0 < a < v's) when z > zo. Then the integral curve of (5.22) starting from any point K(O, YK) on the negative y-axis must return and cut the y-axis at a point L(O, YL) with YL 2': 0. (Similarly, if F(z) > -aJZ, then the integral curve starting from any point M(O, YM) on the positive y-axis must return and cut they-axis at a point N(O,yN) with YN ~ 0.) PROOF. If the integral curve of (5.22) starting from K does not intersect the line z = zo, then the proof is the same as Lemma 5.1. Now let the integral
98
THEORY OF LIMIT CYCLES
,
FIGURE 5.3
curve intersect z = zo at a point P, which lies below the isocline y == F(z). In the half-plane z > zo, we compare the integral curves of (5.22) and of the equation
dwfdy
= a.fW- y
(5.23)
passing through the same point P. Since (5.23) has the general integral
2w-ay.fW+y2 = Cexp{ h t a n - 1
[a~y]},
any integral curve of (5.23) has a point of intersection with the positive and also with the negative y-axis. We know that for the integral curve z = z(y) and the integral curve of (5.23)( 3 ) w = w(y) passing through the same point P, we should have z(y) < w(y) when y > yp; hence z(y) must intersect y = F(z) and then return to intersect they-axis at a point L. It is clear that YL ~ 0. REMARK. We can see from the proof of Lemma 5.2 that if in Lemma 5.1 we let F(z) > -a..(Z, 0 < a < v'B, then we must have YA > 0; if we let F(z) < a..(Z, then YA' < 0. Hence if under the condition of Lemma 5.2 we also have F(z) > -avz (or F(z) < a..(Z) when 0 < z < 6, then we can prove that YL > 0 (or YN < 0). ( 3 )1£ we let .,fiii = u, then (5.23) becomes du/dy = (au- y)/2u, when a 2 equation takes the origin as focus, and all the integral curves are spirals.
<
8; this
§5.
99
EXISTENCE
y
II
0'
FIGURE 5.4
FIGURE 5.5
PROOF OF THEOREM 5.3. Take z' in (0,8) such that F1(z') < F2(z'). Passing through B(z', F1 (z')) we construct an integral curve of the equation dyfdz = 1/Fl(z)- y,
(5.20)*
and through E(z', F2(z')) we construct an integral curve of the equation dyfdz = 1/F2(z)- y.
(5.21)*
From Lemma 5.1 and condition 1) of this theorem, we know the first curve should intersect toe y-axis at two points A and C, and the second curve should intersect the y-axis at two points G and D (Figure 5.4). Moreover, YG > YA ;:::: 0 and Yc < YD ~ 0. The figure in the (x, y)-plane corresponding to Figure 5.4 is shown in Figure 5.5. The points with the same letter in the two figures correspond to each other. Since dxfdt > 0 on the line segment A'G' on the positive y-axis and dxfdt < 0 on the line segment C' D' on the negative y-axis, we know that B' C' D' E' G' A' B' forms the inner boundary curve of the required annular region.( 4 ) Next, according to the conditions of Theorem 5.3 and the remark after Lemma 5.2 we know that the integral curve of equation (5.21)* starting from the point K on the negative y-axis must return to intersect the y-axis at L, YL > 0, and YL increases when YK decreases. Let limyK-+-oo YL = YM; then either YM = +oo, or 0 < YM < +oo.
=
( 4 )If D 0, then D' = 0. Then we use the trajectory_...£assing C' on the left half-plane and the vertical line passing E' to replace the segment 0 E'.
100
THEORY OF LIMIT CYCLES
z
N'
FIGURE 5.6
FIGURE 5.7
We first consider the case YM < +oo. In this case the integral curve of equation (5.20)* starting from M should return to intersect the negative yaxis at a point N, YN < 0; and the integral curve of (5.21)* starting from N should return to intersect the positive y-axis at a point P below M (Figure 5.6). Turning to the (x, y)-plane, we immediately see that the trajectory M'ii'P' and the line segment M' P' form the outer boundary curve of the annular region, because the trajectory when intersecting M' P' must cross it from left to right. This completes the proof of existence of stable limit cycles (Figure 5.7). When YM = +oo, the above proof is not suitable. We still need the following lemmas. LEMMA 5.3. Let Yt(z) and y2(z) be solutions of equations (5.20)* and (5.21)* respectively, and let them satisfy the same initial conditions Yt(O) = y 2 (0) = YO· For zo in condition 2) of the theorem and for any e > 0, we can take IYol sufficiently large that Yt(z) and Y2(z) are defined on the interval [0, zj, and satisfy the inequalities IYt(z)- Yol < e and IY2(z)- Yo I < e when 0 ~ z ~ Zo·
PROOF. Let m = max[o,zoJfiFt(z)l, IF2(z)j]. If IYol ;;::: m + zo/e + e, it is easy to prove Lemma 5.3. The readers can supply the details for themselves.
§5.
101
EXISTENCE
From Lemma 5.3 we can immediately obtain LEMMA 5.4. AsiYol-+ oo, 2
Yo -+ 1 (Ft(z)- Y1(z))(F2(z)- Y2(z)) uniformly in [0, zo]. LEMMA 5.5. There exists a positive number p such that Yt(zo) for IYol > p.
< Y2(zo)
PROOF. From y 1 (0) = y2 (0) and (5.20)* and (5.21)*, it is easy to establish that
y~(Y2(zo)- Yt(zo)) _
{ "0
- lo
(y2(z)- Yt(z))y~ dz {"0 (F1(z) - Y1 (z))(F2(z) - Y2(z)) + lo (Ft (z)- F2(z)) dz
~ + lor~( (Ft(z)- Y1(z))(F2(z)Y2(z)) -
)
1 (Ft(z)- F2(z)) dz
= /1 +/2 +/3. As IYol -+ +oo, we know that It -+ 0 by Lemmas 5.3 and 5.4. Moreover, since F1 (z) - F2(z) is bounded on [0, zo], I a -+ 0. Besides, according to condition 2) we know that /2 > 0. Hence, as IYol-+ +oo, the right side of the above formula approaches a positive limit; and when IYo I is sufficiently large, Y2(zo)- Yt(zo) > 0. The lemma is proved. We now prove Theorem 5.3 when YM = +oo. Take an integral curve KQRL of equation (5.21)* such that YK < -p and YL > p. Through L and K construct integral curves LST and KV of (5.20)*. From Lemma 5.5 we know that Ys < YR and Yv < YQ· From condition 2) and the comparison theorem we know that §T cannot intersect RQ and must return to meet z = zo at a point U above Q (Figure 5.8). Returning to the (x, y)-plane, we know that the section of the trajectory V'K'Q'R'L'S'T'U' and the line segment U'V' together form the outer boundary curve of the required annular region (Figure 5.9). The theorem is completely proved. REMARK 1. H for every z in (0, t5) we have
---
-av'z < F2(z)
= Ft(z) < a..[Z,
then it is easy to see that system (5.10) takes the origin as its center. This is the extension of the symmetry principle in §1. REMARK 2. It is easy to prove that Theorem 5.1 is a special example of Theorem 5.3. Hence in Theorem 5.1 we only need to assume f(x) and g(x) are continuous; then we can guarantee the existence and uniqueness of the solution. In fact, even the theorem of de Figueiredo [63], which was published
THEORY OF LIMIT CYCLES
102
" ::
FIGURE 5.8
FIGURE 5.9
a few years later than the results of Filippov, is also a special example of this theorem. Besides, from condition 1) of Theorem 5.3 we can establish a rule to determine nonexistence of closed trajectories. THEOREM
5.4. If we have F1(z) $ F2(z) for allz > 0, andF1(z) ¢ F2(z) > 0, then system (5.10) does not have a closed trajectory.
in (0, 6) for any t5
COROLLARY. If g(x) = x, F(x) = s(x) + r(x), where s(x) is an even polynomial, s(O) = 0, r(x) is an odd polynomial, and xr(x) ~ 0, then system (5.10) does not have a closed trajectory.( 5 )
m. In the following we tum to study equation (5.4). Note that we can only use general methods to transform this equation into a system of firstorder equations; we cannot use the Lienard transformation. The problem of limit cycles of this equation was first studied by Norman Levinson and Oliver K. Smith, but here we introduce the result of Dragilev [65] in which the hypotheses are weaker than the conditions of Levinson and Smith, and the method of proof has been improved, and is different from the two methods introduced in the previous subsections. THEOREM 5.5. Inequation(5.4) letf(x,v) andg(x) satisfythefollowing
conditions: 1) xg(x)
>0
( 5 )Compare
when
lxl > 0,
Exercise 13 in §3.
and G(±oo)
= +oo.
§5. EXISTENCE
103
FIGURE 5.10
2) f(O, 0) < 0. 3) There-exists xo > 0 such that f(x, v) ~ 0 when lxl ~ xo, and there exists M > 0 such that f(x, v) > -M when lxl ~ xo. 4) There exists Xt > xo such that for any decreasing positive function v(x)
1
2:1
f(x, v) dx ~ 4Mxo +a
(a> 0).
(5.24)
dvjdt =- f(x, v)v- g(x),
(5.25)
zo
Then equation (5.4) has stable limit cycles. PROOF. Consider the system of equations
dxjdt = v,
which is equivalent to (5.4), and, as in Theorem 5.1, let
From condition 1) we know that ..\ (x, v) = C represents a system of simple closed curves containing the origin; from condition 2) we know that in the vicinity of the origin we have d..\ dv dx 2 dt =vdt +g(x)dt =-v f(x,v)~O.
Hence we can take a very small C1 > 0 such that d>./dt ~ 0 everywhere on .\(x, v) =Ct. This closed curve can be taken as the inner boundary curve of the annular region (Figure 5.10).
104
THEORY OF LIMIT CYCLES
Now we study the trajectory 1 starting from a point Ao(xo, a) (a> 0) on the line x = xo. From condition 3) we know that d>.jdt ~ 0 to the right of x = x0 • However, the singular point is unique; hence 1 must intersect the x-axis and then return to intersect x = x0 at a point Aa. Since we shall take a very large later, we may as well assume 1 also meets x = x1 at two points A1 and A2. When lxl ~ xo we know from condition 3) that
dv = _ f(x, v) _ g(x) ~ M + maxlxi<xo ig(x)i < M dx v ivi
+1
'
when lvl > maxlxl~xo lg(x)l. On the other hand, from the inner boundary curve of the annular region constructed before, and the fact that dvjdtiv=O = -g(x), we know that 1 after Aa must intersect the negative v-axis and enter the left half-plane. After that, it may cross the negative x-axis from the right of x = -xo and enter the second quadrant, or it may cross x = -xo through A4. But on the left of x = -xo we have d>.jdt ~ 0; hence, later on, 1 should still intersect the negative x-axis and then turn back to intersect x = -xo again at As Using the same proof in the lower half-plane, we know 1 will cross the positive v-axis, and then intersect x = x 0 again at a point A6 • Now we shall prove that if a > 0 is taken sufficiently large, we can make A6 lie below Ao. At the same time, the section of the arc from Ao to A6 on 1 and the line segment AQ'A6 can form the outer boundary curve of the required annular region. We first prove that, for any given positive number N, we can always choose the ordinate a of Ao greater than N and so large that the arc AoA1 · · · A6 can intersect x = XI. and its part inside the strip -xo ~ x ~ X1 must lie above v = N or below v = -N. In fact, if 1 after Ao does not intersect x = x1. then we can choose instead the trajectory 1' starting from Ai (xi. a'), a' > a, to replace 1, so that when t increases 1' can still move around the origin as 1 does. It would be better if 1' can intersect x = xo as t decreases; otherwise, the negative semitrajectory of 1' must approach oo from the right of x = xo (since dxjdt > 0 and dv/dt < 0), while at the same time we can take A~ = oo, and A~ must be below A~. So the proof of the theorem is complete. Similarly, for example, if A 3 lies above v = -N, then we can take the trajectory 1" starting from A~(xo, -N- 1) to replace 1, etc. Let Vi denote the ordinate of Ai. Now we estimate v5. Use (5.24) to obtain
Vt-Vo = -
l."'' xo
1
f(x,v)dx-
[."' x0
g(x) dx V
Xl
<-
xo
f(x,v)dx
~
-4Mxo- o:.
(5.26)
§5. EXISTENCE
105
Next, along~. >.(x,v) decre~Ules Bll t incre~Ules, but G(x) hBll the same value at A1 and A2; hence (5.27) Along A--;A;, >.(x, v) still continues to decreBlle, and so !v~ G(xl); thus
+ G(x0 ) ~ !v~ +
I 3 I I 2I < 2[G(xl)- G(xo)] G(xl) - G(xo) v - v lv31 + lv2l < N ·
(5.28)
For the arc A~, we have V4
-v3 =- 1
-xo xo
f(x,v)dx-
1-xo ( ) xo
~dx v
f(x,v)dx+ { 0 g(x) dx+ ro g(x) dx. 1o V 1-xo 1-xo V Using condition 3), we can get = r
lv41-lv31 < Along
ro 1o
g~) dx +
roM dx = 2Mxo + 1-xo
G~o).
(5.29)
....G"k we have d.\jdt ~ 0; hence vs -
lv41 ~ 0.
(5.30)
Finally, for A~ and A~, similar estimates lead to
v6- Vs < 2Mxo + G(-xo)/N.
(5.31)
Adding (5.26)-(5.31), we get
v6- vo <-a+ (G(xt)
+ G(-xo))/N,
hence, we only have to take N
then we have
V6 -
2:: (1/a)[G(xt) + G( -xo)];
vo < 0. The theorem is completely proved.
IV. In the process of proving Theorem 5.5, we have already seen that if under suitable conditions we can prove there exists a trajectory 'Y whose negative semitrajectory approaches oo, and along whose positive semitrajectory its polar angle incre~Ules without bound Bll t incre~Ules, then the outer boundary curve in the Annular Region Theorem can be very eBllily constructed. This idea WBll first seen in the paper of Wu Zhuo-qun [66]. Using this idea 1Ul in Theorem 5.2 we can eliminate the conditions G(±oo) = +oo. Below we shall introduce this theorem.
THEORY OF LIMIT CYCLES
106
THEOREM 5. 6. Suppose that the following conditions are satisfied: 1) xg(x) > 0 when x 'I 0.
2) /(0,0) < 0. 3) Let F(x) = inf11 f(x,v), suppose bound, and suppose also that lim lzl-+oo
J; F(x)sgnxdx exists and has a lower
Jor(g(x)+F(x)sgnx)dx=+oo.
4) There exist C > 0 and M > 0 such that
fox(Cg(x) -F(x)sgnx)dx < M
(x > 0).
Then system (5.25) has stable limit cycles. PROOF. As in Theorem 5.5, it is not difficult to construct an inner boundary curve. Now let us construct an outer boundary curve. First we prove that if any trajectory 1 from any point (xo, vo) in the first quadrant does not intersect the x-axis as t increases, then x(t) must go to +oo. Suppose not. Since dxjdt = v > 0, 1 must remain in the region xo :5; x :5; x1 < oo, v > 0. Since 0(0, 0) is the unique singular point of system (5.25), v(t) must be unbounded; that is, lim v(t) = +oo, t-+T"
where T* is an upper bound of t which can be continued along the positive semitrajectory of 1, and may be +oo. But when T < t < T* (x(T) = x0 , v(T) = vo), we have
v(t) = v(T) -1x f(x, v) dx -1x z(T)
:5; v(T) -1x
g(x) dx
:z:(T)
V
F(x) dx;
:z:(T)
here the right side is bounded as t -+ T*, and the left side is unbounded. This is impossible. We next prove that 1 cannot remain in the first quadrant and still make x(t) -+ +oo. Since
v :5; vo-
1z
F(x) dx-
:z:o
1z zo
-g(x) dx < v0 v
-
1x
we must have lim
fx F(x) dx < +oo,
z-++oo }zo
zo
F(x) dx,
(5.32)
107
§5. EXISTENCE
for otherwise the left side of the above formula cannot always be greater than zero. Then from condition 3), we get that
fo+oo g(x) dx =
+oo,
and so by the boundedness of v the three terms in the middle of (5.32) can take negative value. This is also impossible. In short, the trajectory"' starting from any point (xo, vo) in the first quadrant must later intersect the x-axis and enter the fourth quadrant. We note that in the fourth quadrant after"' crosses v = -6, we have v ~ -6 < 0 along 1 (refer to the proof of the corresponding part in Theorem 5.5) and dxfdt < 0; that is, x remains bounded. Hence from
v-vo
~-
l:r::or: F(x)dx- l:r::r:o -g(x)d x
(x < xo)
V
we know that v has a lower bound; thus "' must intersect the negative v-axis. Since condition 3) of the theorem is independent of the sign of x, the behavior of 1 in the third quadrant is similar to its behavior in the first quadrant, and the behavior in the second quadrant is similar to its behavior in the fourth quadrant. This means "' must later go around the origin once and return to the first quadrant. We have thus proved under conditions 1), 2), and 3) that the polar angle of the point along the trajectory "' must increase without bound as t increases. Now we prove J;here exists a negative semitrajectory,(6 ) which goes to infinity by condition 4). Consider the trajectory "' passing through the point (O,vo) at t =to, where vo = -M -2/C. Let the maximum time during which "(can be continued in the negative direction beT < t ~ t 0 • Now we prove that in this interval we must have v(t) < -1/C. First, this inequality holds near t =to. If there exists t1 > T such that v(t) < -1/C when h < t ~to and v(tl) = -1/C, then in the interval h < t ~to we have
r
r
2 v(t)
(x > 0).
Lett-+ fl. We get v(tl) ~ -2/C < -1/C, which contradicts the hypothesis. Since the negative semitrajectory "' always lies below v = -1/C, and the singular point is also unique, this negative semitrajectory must be unbounded. The theorem is completely proved. COROLLARY. Under conditions 1}, 2}, and 3) of Theorem 5.6, if there exists an a> 0 such that in the interval (-oo, -a) or (a, oo), (g(x)/ F(x)) sgnx ( 6 )This
method was first seen inN. N. Krasovskil's paper 129].
THEORY OF LIMIT CYCLES
108
has an upper bound M, and F(x) cycles.
> 0, then system {5.25) has stable limit
PROOF. It is clear that if we take C
1
00
(Cg(x)- F(x)) dx
= 1/2M, then
< 0 or /_~oo (Cg(x) + F(x)) dx < 0;
thus, as in the theorem, we can prove the trajectory passing through (a, -4M) or (-a, 4M) as t increases is unbounded. EXAMPLE 2. For the van der Pol equation (p,
we have g(x) F(x)
lxl sgnx = p,(x2- 1) --+ 0
as
> 0)
lxl --+ oo,
which satisfies the requirements of the corollary. Hence there must exist a stable limit cycle. V. In the above four subsections, the reader can see that to construct an annular region, it is rather difficult to obtain the outer boundary curve, and, except for Filippov's method, the inner boundary curve of the annular region is easily obtained because we have strengthened the conditions on the singular point. However, if we would like to use the method similar to that in Theorem 3. 7 of §3 to prove the existence of limit cycles, then, contrary to the above remark, the difficult part is to obtain the inner boundary curve instead of the outer boundary curve. Of course, if the unique singular point is an elementary singular point and the real part of its characteristic root is not zero, then to distinguish its stability, and, if the equation contains a parameter, to know for what value of parameter such an elementary singular point will change its stability, becomes a very easy task. The -problem is that, when the singular point is a so-called "fine focus," to distinguish its stability requires the calculation of its "focal quantity." This task is very tedious, although in the qualitative theory of differential equations we have the familiar Poincare method. In the following we only give one example to illustrate this point; the large number of applications of this method will be discussed in the latter half of this book. EXAMPLE 3. Prove that the system of equations
rz: ,
dx 1 dt = y - ox -
3
has stable limit cycles when 0 < o « 1.
dy 1 dt = -x + 3x2y - 2y3
109
§5. EXISTENCE
PROOF. This system has 0(0, 0) as an unstable focus when 6 = 0,(1) and when 0 < 6 « 1, (0, 0) changes its stability. Hence from Theorem 3. 7 in §3, we know that in the neighborhood of (0, 0) stable limit cycles will appear.
VI. Besides the above five methods for the construction of annular regions, there is yet another method, which analyzes the properties of the infinite singular point to replace the construction of outer boundary curves. H we can prove that on the Poincare hemispherical surface( 8 ) the trajectories from the equator paBSing through the infinite singular point all run from the singular point on the equator to the finite part, and at the finite part there exists a unique negatively oriented, asymptotically stable singular point 0, then the plane with 0 deleted is the required annular region. But this method is only applicable for dynamic systems whose right sides are polynomials, since we can introduce homogeneous coordinates to study properties of singular points at the equator. Of course, sometimes there is no need to use the Poincare spherical surface. By adding a unique infinite point to the plane we make it a Gauss spherical surface if, by doing this, we have the possibility of analyzing the properties of the infinite singular point. Earlier in the fifties, Lefschetz [69] and Dong Jin-zhu [70] used the above two methods to analyze the properties of the infinite singular point of the van der Pol equation, and from this obtained the conclusions that the trajectory starting from any point in the exterior of the limit cycle takes the limit cycle as w-limit set. Recently better results were to prove that the so-called Brusselator system of differential equations dx dt
= A- (1 + B )x + x 2 y,
dy dt
= Bx- x 2 y, has a unique stable limit cycle when B > 1 + A 2 ~ 1+A2 ;
A > 0, B > 0,
(5.33)
and has an asymptotically moreover, here we have global
stable critical point (A, B/A) when B stability. This result was independently obtained at about the same time by Qin Yuan-xun and Zeng Xian-wu [71] and Zhang Di, Chen Zhi-rong, and Zhan Ken-hua [72] in China. We now give a brief sketch of the method of proof in
[71].
= x + y.
Then (5.33)
dxjdt =A- (1 + B)x + x 2 (u- x).
(5.34)
We first analyze the infinite singular point. Let u becomes
dujdt =A-x,
From this we can see that it has a unique finite singular point: (x = A, u =A+ B/A) or (x =A, y = B/A). (1)For example, see (67], Part II, Chapter 2, §1. (8)Refer to (68], Chapter 9, §5.
110
THEORY OF LIMIT CYCLES
We translate the origin to the singular point. Let x
=A+
~
and u
=
A+ B/A + 17· Then we get
~~ = -~, ~; = (B-1-A2 )~+A2 77+ B -A2A2 e+2A~17-e+e17.
(5.35)
It is easy to see when B > 1 + A2 the origin is either an unstable focus or a nodal point; when B < 1 + A2 the origin is either a stable focus or a nodal point. In order to determine under what conditions a limit cycle can exist, by Theorem 3.7, we must determine, when B = 1 + A2 , the stability of the origin. However, if we can clearly analyze the properties of the infinite singular point, and prove the equator has an effect on the outer boundary curve of the Annular Region Theorem, then we can also distinguish when limit cycles can exist. For this purpose, let x/z and yjz replace x and y in (5.33) respectively (that is, introduce the homogeneous coordinates). We obtain
dx dz x2 y 2 z dt -X dt = Az - (1 + B)xz + dy dz x2 y zdt-ydt=Bxz-z.
z'
In order to study the singular point outside the y-axis, we can let x = 1 and dt/dr = z2 in the above formulas; then we obtain
dz/dr = -yz + (1 + B)z3 - Az4 , dyjdr = Bz2 - y- Ayz3 + (1 + B)yz2
-
y2 •
(5.36)
From this we can get two singular points, (1,0,0) and (1, -1,0). For (1, 0, 0), we use the well-known method of Lyapunov to distinguish its properties. Let the right side of the second formula in (5.36) be equal to zero; we can solve y = Bz2 + · · · and substitute in the right side of the first formula to obtain dz / dr = z3 + terms of higher degrees. Hence we know that (1, 0, 0) is a saddle point.( 9 ) Since z = 0 is a trajectory, from the second equation of (5.36), we see that the two separatrices entering {1, 0, 0) are the equator, and the other two separatrices then leave from {1, 0, 0), one of which enters the upper half of the Poincare spherical surface, which is equivalent to the finite (x, y)-plane. For (1, -1, 0), we just let y = -1 + u, and from (5.36) we obtain
dz / dr = z + terms of higher degrees, du/dr = u +terms of higher degrees. (9)See the first half of [67], p. 146.
111
§5. EXISTENCE
FIGURE 5.11
Hence (1, -1, 0) is an unstable nodal point; all the trajectories except the equator start from (1, -1,0) and run to the finite plane. We can similarly prove that the singular points (-1,0,0) and (-1,1,0), symmetric with respect to the diameter of the above two singular points, have the same stability as the above two points. Similarly, for the infinite singular points (0, ±1,0) on the y-axis, we can prove they are semisaddle nodal points with index zero. Thus we can draw the phase portrait on the upper half of the Poincare spherical surface (Figure 5.11). Thus we know immediately that when B > 1 + A2 the system (5.33) has stable limit cycles. As for its uniqueness, we shall discuss the proof in the next section. In order to prove that when B ~ 1 + A2 there does not exist a limit cycle, we note if we treat B in (5.35) as a parameter, we can immediately compute p
Q
ap
aQ
oB aB
=
(B- 1- A2)~ + A2-q + B- 2A2 ~2 + 2A~-qA
~ + .!.e A
e + ~2-q -e 0
112
THEORY OF LIMIT CYCLES
when € > -A. Hence (5.35) forms a family of rotated vector fields with respect to B in the half-plane > -A. On the other hand, on the line € =-A we have
e
d€/dt = -A(B- 1- A 2 )
+ A 2 17 + AB- 2A3 - 2A2 q + A 3 + A 2 17 =A> 0;
hence closed trajectories, if they exist, can only lie entirely on the right of the line € =-A. However, we have already shown that when B > 1 + A 2 the half-plane > -A has limit cycles. As B varies, these limit cycles, following the increase in B, continuously expand; the boundary of the region covered by them should have a singular point. From Figure 5.11 we see that this singular point may be the infinite singular point (1, 0, 0) or (0, 1, 0). Finally, from the nonintersection theorem (Theorem 3.2) of §3, we know immediately that (5.35) does not have closed trajectories when B ~ 1 + A 2 • We have already introduced six main methods to prove the existence of limit cycles in this section. There are still some other methods which are not as common. For example, Seifert [47] used the theory of rotated vector fields to deduce from the existence of limit cycles for equations
e
dx/dt = v, (where ft(x,v,a) = -a(x 2
dvjdt = -g(x)- f 1 (x,v,a)v + v 2 ) + f(x,v)v), when a< 0, the existence of
limit cycles for equations (5.25); instead of the condition on the decay of the displacement (i.e., the first half of condition 3) of Theorem 5.5), he only demanded that there exist a constant J.L > 0 and a continuous function h( v) such that for all x and v
f(x, v) > J.L
fox g(x) dx- h(v).
This shows that for every fixed v, when lxl is sufficiently large, f(x,v) > 0; but it is not necessarily possible to find a lower bound xo of lxl for all v. For example, if f(x,v) = x4 /(lvl + 1) -1, g(x) = x/2, then condition 3) of Theorem 5.5 cannot hold, but Seifert's condition holds. Moreover, when g(x) in (5.10) has more than one zero, to replace the method in Theorem 5.2 for avoiding singular points other than the origin (a singular point), Sansone and Conti [73] directly studied the relative positions of two separatrices passing through a saddle point for the system having two singular points when these separatrices surround the origin. Later Yu Shu-xiang [74] and Zhou Yu-rong [75] improved their conditions. However, in short, analyzing the two separatrices passing through a saddle point to determine their relative positions, even for a polynomial differential system, is still very difficult, except for very special cases.
§5. EXISTENCE
113
There are many articles on the proof of existence of limit cycles for system (5.10) or some polynomial systems. In China alone, there have been between thirty and forty papers in recent years. In the following, we shall give a simple introduction according to the six main methods discussed in this section. The readers are advised to refer to the original papers for details. 1. The work of using broken lines to construct the outer boundary curve of the annular region was done by Liao Xiao-xin and Liang Zhao-jun [76], Zhang Di and Chen Zhi-rong [77], and Li Ji-bin [78]. In [76] the authors studied the special two-beat oscillator differential equation
x + p(ez- 2)x + x = 0,
(5.37)
and proved that the corresponding system of equations has stable limit cycles when 0 < p ~ 1. Later, in [79], the uniqueness of the limit cycle was proved. The equation under consideration in [77] is similar to (5.33), which is a system of equations concerning the theory of a trimolecular chemical reaction:
x= A -
Bx - xy2 ,
iJ = Bx + xy 2 -
y.
(5.38)
The equations studied in [78] are similar to (5.14), which is used to solve the problem of periodic solutions of the second-order nonlinear equation
X+ (aixi~ + bixi#Sx + c)x +X= 0
(5.39)
in the theory of mechanical cutting. 2. As for Filippov's method for constructing the outer boundary curve of an annular region, after the earlier work of Yu Shu-xiang [74], in recent years some research wM done by Huang Qi-chang, Shi Xi-fu and Chen Xiu-dong at Dongbei Normal University [80]-[84]. Among their work, [83] upgraded the method of Lemma 5.2 to the general concepts of characteristic trajectories, characteristic points, and characteristic intervals, and not only extended Filippov's results but also obtained a new method of constructing the Lienard equation possessing n limit cycles. 3. D. A. Neumann and L. D. Sabbagh [85], Ye Yan-qian [86], Wu Kuiguang [87], Lin Li-cong [88], and Yang Zong-bo [89] directly used the section of trajectory and its transversal to construct the outer boundary curve of an annular region. [85] extended Theorem 5.1 of Dragilev. [87] pointed out that the proof of R. M. Cooper in [90] concerning the existence of limit cycles for dxjdt = P(y),
dyjdt = Q(x, y)
under a given condition was wrong, and corrected it. [89] extended a theorem of Zhitel' zelf for the existence of limit cycles of the Lienard equation (where g(x) x), and gave sufficient conditions for existence of at least two limit cycles for this equation.
=
THEORY OF LIMIT CYCLES
114
4. Yu Shu-xiang [92], Chen Guang-qing [93], Zhou Yu-rong [94], and Wang Xian [95] used one end of the trajectory running to infinity to construct the outer boundary curve of an annular region. [92] gave more general conditions on the Lienard equation to extend the results of Theorems 5.1, 5.3, and 5.4. [93] further extended the results of [92]. [95] gave different sufficient conditions for Theorem 5.6, strengthening condition 4) but weakening condition 3) so that it was more conveniently verified. As a special example, the author of [95] obtained sufficient conditions for the existence of limit cycles for the system of hard springs dxfdt = y- F(x),
dyfdt
= -x(k + Cx 2 )
(where k > 0, C > 0, F(x) is piecewise polynomial, and F(O) = 0). [94] studied the existence of one and only one limit cycle and two and only two limit cycles for system (5.14). 5. Chen Lan-sun and Jing Zhu-jun [96]-[98] and Wang Xian 138] used the method of changing stability of the singular point and analyzing the infinite singular point to prove existence of limit cycles. [96] studied equations for the immunization reaction dxfdt = x(-a- bx + cy- dxy),
dyjdt
= y(1- X+ y),
when a, b, c, and d are positive constants. [98] studied equations for the struggle for existence dxfdt
= x(a1 + a2x + a3x 2) - xy,
dyjdt = -y + xy.
[97] studied the system for catalytic reactions dxfdt
= -x- a1x 2 + a2xy + a 4x 2y,
dyfdt
= ao + a1x2 -
a2xy- a4x2y.
[38] studied the system of soft springs dxfdt
= y- F(x),
dyfdt
= -x(k- Cx 2 ),
when k > 0, C > 0, F(x) is a quadratic or biquadratic polynomial, and F(O) = 0. REMARK. The work on the existence of solutions of a system of equations whose orbits are algebraic curves (especially closed curves), the conditions on existence of limit cycles of quadratic or cubic polynomials, and work on the existence of several limit cycles have not been mentioned here. We shall introduce these in the following sections. Exercises 1. Explain in detail the physical meaning of the conditions of Theorem 5.1. 2. Show that if the conditions of Theorem 5.1 hold, then the conditions of Theorem 5.3 also hold.
§5. EXISTENCE
115
3. From xg(x) > 0 (x i= 0) and the continuity of f(x) and g(x), prove directly that the equation
dyjdx
= g(x)/(F(x)- y)
has a unique solution (except at the point (0, 0)). 4. Prove that if f(x, v) f(x) in Theorem 5.5, then Theorem 5.5 can be deduced from Theorem 5.3 [100]. 5. If in (5.4) f(x,v) and g(x) satisfy the Lipschitz condition, /(0,0) < 0, xg(x) > 0 when xi= 0, jg(x)l + f(x,v)lvl > E > 0 when lxl > xo, and there exists an R > 0 such that f(x, v) + f(x, -v) ~ 0 when lxl +!vi~ R, then the equation has stable limit cycles (A. de Castro). 6. If in (5.10) f(x) and g(x) are odd functions, xg(x) > 0 when lxl i= 0, and there exists a 6 > 0 such that when 0 < x < 6, g(x) ~ +e:)lf(x)F(x)l, E > 0, then system (5.10) takes the origin as its center [101]. 7. For system (5.25), let
=
(!
lo(x)
= max[f(x,v),fi(x)] = minf(x,v). lvl~c5 v
Suppose the equation x + fo(x)x + g(x) = 0 satisfies condition 1) of Theorem 5.3, the equation +!I (x )x + g(x) = 0 satisfies condition 2) of Theorem 5.3, and g(x) satisfies condition 1) of Theorem 5.1. Then the system (5.25) has a unique stable limit cycle (Filippov). 8. Suppose that in system (5.10) xg(x) > 0 when x =f= 0, g(O) = 0, G( -oo) = +oo, and xF(x) > 0 when x =f= 0 and lxl is sufficiently small; and let there exist constants C > 0 and M > 0 such that
x
F(x) + C
fox g(x) dx < M
(x > 0),
and xo > 0 and K > 0 such that when x < -xo and M+1/C-F(-xo) < 1/K,
-f(x)/g(x)
~
K;
and suppose F( +oo) = -oo. Then (5.10) has an unstable limit cycle [92]. 9. The equation x+F(x)+G(x) = 0 is given, where F and G are continuous and the uniqueness of its solution is assured, yF(y) ~ 0 for IYI ~ 71t (> 0), F(y)sgny ~ C > 0 for IYI ~ 112 > 71t 1 maxjF(y)J = M > 0 for Jyj ~ Y2, and G(x) sgn x ~ M + E for lxl ~ 6 (e: > 0, 6 > 0). Then the equation has a stable limit cycle [102]. 10. If in the equation x + f(x, x)x + g(x) = 0, I and g are continuously differentiable, yf(x, y) ~ 0, xg(x) > 0 when x =F 0 and JtX! g(x) dx = +oo, then a solution of the equation which can be continued to t - +oo must be bounded, and either it is oscillating or it approaches zero (W. R. Utz).
THEORY OF LIMIT CYCLES
116
11. Prove that if in the equation x+ f(x)x + g(x) = 0 the function F(x) = J; f(z) dz satisfies g(x)F(x) < 0 for x ::/= 0, then this equation does not have a closed trajectory. 12. Apply the method of analyzing the infinite singular point to prove that for the equations
dxfdt = -x + a1xy + a2x 2y, dyfdt = 1- a1xy- a2x 2y, if a2 ~ (a1 + a2) 2, then the equations have a stable limit cycle, which lies in the first quadrant [97]. 13. Prove that if
a1 > 0,
a3 < 0,
a1 + a2 + a3 > 0,
a2 + 2a3 > 0,
then the system of equations
dxfdt
= x(a1 + a2x + a3x2) - xy,
dy/dt = -y + xy
must have a stable limit cycle which lies in the first quadrant. 14. In system (5.10) let F(x) and g(x) satisfy the following conditions: 1) There exist 82 < 0 < 81 such that f(x) < 0 when 82 < x < 81, but if x > 81 or x <~.then f(x) > 0 and F(+oo) = +oo. 2) There exists >.1 < 82 such that g(x) > 0 when x < >. 1, but if >.1 < x < 0 or x > 0, then xg(x) > 0, g(>.1) = g(O) = 0, and g'(>.!) < 0. 3) There exist C > 0 and M > 0 such that J;(Cg(x)- f(x)) dx < M (x > 0). 4) F(>. 1) ~ -(M + 1/C +c), c > 0. Then system (5.10) has a stable limit cycle [75]. 15. Suppose in the equation x + rp(:i:) + .,P(x)71(x) = 0 that 1/J(y) > 0, JtXJ(dy/1/J(y)) = ±oo, x71(x) > 0 when x ::/= 0, and rp, 1/J, and 11 are continuous, and the existence and uniqueness of the solution of the initial value problem are guaranteed. Prove the following assertions: a) If yrp(y) ~ 0, but rp(y) ¢. 0 when 0 < y < 1, then the corresponding system of equations does not have a closed trajectory. b) If rp(O) = 0, rp'(O) < 0, 71(±oo) = ±oo, and there exist constants M > 0 and k' < k such that
rp(y) IO(Y)
~
~
t/J(Y)[11(k- x)- 71( -x)] when y > M, lxl < +oo,
.,P(y)[71(k'- x) - 71( -x)] when y < -M, lxl < +oo,
then the system of equations has at least one limit cycle [103]. 16. Use the above exercise to prove that the equation
x + ax(1- 1x2 ) + (1 + .Bixl)x = 0 has a periodic solution.
(a< 0, ')',/3 > 0)
§6. Uniqueness of a Limit Cycle
The question of uniqueness of a limit cycle has not been studied as much as its existence. There are probably two reasons: first, the problem of uniqueness is more difficult than the problem of existence, and it is not as easy to obtain good results; second, the importance of the problem of uniqueness in the qualitative theory of differential equations has been ignored. In the past twenty-some years, owing to the tremendous amount of research on the equations whose right sides are quadratic polynomials carried out by Chinese and Russian mathematicians, people have gradually begun to pay considerable attention to the uniqueness problem, and produced many excellent results. In the following, we shall introduce some rather important work in this area, dividing it into several subsections according to the method used. I. Method of .point transformation. In the theory of nonlinear oscillations we often encounter piecewise linear equations,( 1 ) and often apply the method of point transformation to prove the existence of a limit cycle-with the limit cycle corresponding to a fixed point of the point transformation. We saw this in §1. If in the course of proving its existence we can prove that the fixed point is unique, then the uniqueness of the limit cycle is thereby proved. In the following we shall give an example which is much more complicated than Example 3, §3, to illustrate this method.(2)
EXAMPLE 1. We consider the natural oscillations of a valve generator with the resonant network in the grid circuit or in the anode circuit (Figure 6.1). If we neglect the anode conductance, the grid currents and the capacitances in the vacuum tube, we can write, according to Kirchhoff's law, the equation ( 1 )Readers can see from Example 1, below, that this in fact is an approximate method, used to avoid the possibility of failing to obtain a general solution of the nonlinear equation.
(2)Taken from Chapter 8 of Andronov, Vitt, and Khalkin (104]. 117
THEORY OF LIMIT CYCLES
118
E.
-11Q
FIGURE 6.1
FIGURE
6.2
of oscillations as d2 u
LC dt' 2
du
·
+ [RC- Mf(u)] dt' + u = 0,
(6.1)
where f(u) is the slope of the characteristic curve ia = ia(u). Now we take ia(u) approximately to be a piecewise linear function (Figure 6.2)(3 )
. {0S(u + uo)
ta =
when u ~ -uo, when u > -uo,
(6.2)
and introduce the dimensionless variables
x
= ufuo,
t
= wot',
wo =
1/v'LC.
(6.3)
Equation (6.1) can now be rewritten as
x + 2h1± + x = 0 x - 2h 2± + x = 0
when x < -1, when x > -1,
(6.4)
where h1 = ~RCwo and h2 = (wo/2)[MS- RC]. Changing into a system of equations, we obtain dx
dt =y,
when x < -1, - x + 2h2y when x > -1.
dy = { - x- 2h1y
dt
(6.5)
The line x = -1 divides the (x,y)-plane into two regions, (I): x < -1 and (II): x > -1. The unique singular point (0, 0) of system (6.5) lies in the region ( 3 )Note that if instead we take ia(u) = ro + r1u + r2u 2 + r3u 3 , and move the origin of the coordinate system in Figure 6.2 to the point of inflection of the characteristic curve, then the right side of the above expression has only terms of first and third degree, and (6.1) then becomes the van der Pol equation.
119
§6. UNIQUENESS
(II), and from h 1 > 0 we know that (0, 0) is a stable focus or a nodal point of the trajectory in (I). If h2 ~ 1, then (0, 0) is an unstable nodal point for the trajectory in (II), and there are two integral lines (with positive slope and negative slope respectively) which leave the origin and run to infinity through the first and fourth quadrant respectively; hence it is impossible to have limit cycles. Similarly, we know that when h2 < 0 it is not possible to have limit cycles. If h2 = 0, then in the vicinity of (0, 0) there is a system of closed trajectories, which cannot be realized in physics. Hence we may as well assume 0 < h2 < 1, and that at the same time (0, 0) is an unstable focus for the trajectory in (II). We now assume also that 0 < h1 < 1; then the trajectories in (II) are spirals. We shall study the trajectory starting from a point ( -1, -s) in the lower half of the line x = -1 (Figure 6.3). As t increases, it should intersect the upper half of x = -1 at a point ( -1, s'). It is easy to see the parametric equations of this trajectory are x = -e -h 1t [ cos w1t y
= e-
h 1t [
-SCOSW1
s + hl sm . w1t.] , +---;:;;-
t+
1 +Wth1s SlDWtt . ] ,
where w1 = y"l-h'f. Suppose the time of motion at ( -1, -s) is t the time at (-1, s') is t1 = Tt/Wt. Then we have _ 1=
s + h1 sm . Tt ] - e- h 1T1I w1 [cos r 1 + ---;:;;-
(6.6)
= 0 and
,
. Tt ] . s' = e- 1T! 1w1 [ -s cos Tt + 1 +Wtsh1 sm
(6.7)
h
From this we can solve for s and s':
s=
e"'~1 T1
- cos Tt - 11 sin Tt
J1 + "Y~ sin Tt = htfwt = htfy"l--if.
s'=
e-"'11 T1 - cos Tt
+ 11 sin Tt
----~==~~~-----
Jl + "Y~ sin Tt
(6.8)
where "Yt Now we want to study the properties of the curve {6.8) on the (s, s')-plane; here we take Tt as a parameter. Thus, we introduce an auxiliary function rp(r,"'f) = 1- e"~T(cosr- 1sinr).
It is easy to prove it possesses the following properties: 1. rp( -r, -1) = rp(r, "Y) and rp(O, "Y) = 0. 2. orpj[Jr = (1 + "Y2)e"'fT sin T.
(6.9)
THEORY OF LIMIT CYCLES
120 y
FIGURE 6.3
FIGURE 6.4
3. When "Y > 0, rp(r, "Y) has a zero r 0 ("Y) with respect to r, which lies between 1r and 2n. When r < r 0 ("Y), rp(r, "Y) > 0 (Figure 6.4). Using rp(r,"Y), we can rewrite (6.8) as 8
= e"'fl'~'trp(rl, -"Yd
(6.10)
~sinr1'
Thus we can also get
rp( r1, 1d
ds dr1
yf1 + 1~
sin 2
d8' r1'
dr1
rp( r1. -11)
= ~sin 2 r1 '
(6.11)
According to (6.10), (6.11) and the properties of rp(r,"Y) we can see that as r1 goes from 0 to 1r, 8 and 8 1 all increase monotonically from 0 and approach
+oo, and it is not difficult to draw the graph of the curve (6.10) in (81, 8)-plane (Figure 6.5). Its slope at the origin is 1, and when r1 increases from 0 to 7r, the slope of the curve 8 = f (81 ) monotonically increases and finally takes the line (6.12) 8 = e"'l111'8 1 +a as its asymptote, where
_ 11m . [8-e"'ft11' s' ]+ e"'~ 1 11') < 0. a- 211(1 ~ '1'--+11'-0
v 1 + "Yl
The curve 8 = f(s') thus determines the corresponding relation between the points (-1,-s) and (-1,8 1 ).
121
§6. UNIQUENESS
11
,. ~0+------------------~
••
0
FIGURE 6.5
FIGURE 6.6
As above, we can discuss the corresponding relation between the point (-1, 8 1 ) and the point ( -1, -81) determined by the trajectory in region (II). For convenience, suppose the motion at the timet= -r2/w2 < 0 is at ( -1, 8 1 ), and at t = 0 it arrives at ( -1, -81). In order to obtain the formulas for 81 and 8 1 similar to (6.10), we only have to note that the equation of a trajectory for system (6.5) in (II) can be obtained from (6.6) by changing h1 to -h2 and Wt to w2 = J[=~. Thus, if we again change 8 to 81 and Tt to -r2 in (6.8), we can obtain the parametric representation formulas for a and 8 1 :
where 12 = h2/w2. The above formulas can be rewritten as 8, = _ e-...,~r~fP(T2, 12).
J1+i~ sinr2
(6.13)
According to the properties of fP(T, 1), if we only let T2 vary in (1r, r~), we can get all 8 1 in (0, oo) (here T~ is the smallest positive root of ~P( T2, 12) = 0, 7r < T~ < 21r). It is easy to draw the graph of the curve 81 = g(8 1 ) indicated by (6.13) (Figure 6.6). When T2 decreases from T~ to 1r, 8 1 increases from 0 to +oo and 8 1 increases from the fixed value 8~ > 0 to +oo. When 8 1 > 0, the curve 8 1 = g( 8 1 ) has positive slope, which increases as 8 1 increases, and
THEORY OF LIMIT CYCLES
122 ,, S1
,.
FIGURE 6.7
finally approaches the limit 81
e"~ 2 7r.
= e
The equation of the asymptote is
-y21r
8
'+ 212( 1 + e"~ 2 7r) y' 1 + 1 ~ .
(6.14)
Putting (6.10) and (6.13) together, we can obtain the point transformation carrying the lower half of x = -1 into itself and (-1, -8) into ( -1, -81). Obtaining the fixed point of this point transformation is the same as superposing Figure 6.5 on Figure 6.6 to obtain the point of intersection of the two curves 8 = /(8 1 ) and 81 = g(81 ) (Figure 6.7). In the following we divide this into two different cases to discuss: Case 1. 0 < h2 < h1 < 1.(4 ) In this case the slope of the line (6.12) is greater than the slope of the line (6.14); hence the curves 8 = /(8 1) and s1 = g(8 1 ) have an odd number of points of intersection. We now prove that they have only a unique point of intersection. Suppose (s', 8 = 81) is an arbitrary point of intersection of these two curves, which corresponds to the parameters "f1 and "f2. Then from (6.10) and (6.13) we obtain
(6.15)
( 4 )This
condition is the same as 2RC
> MS > RC
and RC
< 2/wo.
123
§6. UNIQUENESS
and
ds11
ds
s=i
=
(ds1 Ids) ds' ds' s=i
=
Using {6.15), we can compute {6.16) On the other hand, since s1 > s when s' is sufficiently small, at the first point of intersection of these two curves we must have dsjds'
> dstfds' > 0.
In fact, using (6.15) it is easy to compute that at (s', 8 1 = s), we have ds - dst = e2h2r2+"11rt) 0. ds' ds' rp(r2, -12) Hence we must have 0
< dstfdsip < 1.
{6.17)
If these two curves have a second point of intersection Q, then at Q we should have ds1 > ds ds' - ds'
0 > ' which implies that
~:
L
2::: 1.
{6.18)
But this is impossible because we can see from {6.16) that as 8 increases ft increases whereas f2 decreases, and so we would have dstfdslq < dstfdslp, which, together with {6.17), indicates that (6.18) cannot hold. This proves the uniqueness of the fixed point; that is, system {6.5) has a unique limit cycle. From (6.17) we can see (according to Theorem 2.1 in §2) that it should be a stable cycle. Case 2. 0 < h1 :::::; h2 < 1. In this case if the two curves s = f(s') and s1 = g(s') intersect, then at the first point of intersection we would have dstfds = e2h2r2-"''t1't) :::::; 1, but this is impossible since f2 > 1r >ft. and 12 2::: 11 when h2 2::: h1. We can prove that when h1 2::: 1 > h2 > 0, system (6.5) also has a unique stable limit cycle, while (0, 0) is a stable nodal point for the trajectory in region (II). Hence {6.6) cannot be used again. The proof is similar to the first case, hence it is omitted here. For general nonlinear equations, since there is no way to give a clear analytic expression for the point transformation, we cannot prove the uniqueness of the limit cycle in the same way as we have done before. However, according
THEORY OF LIMIT CYCLES
124
• FIGURE 6.8 to the idea of the point transformation, we can use the following method to prove the uniqueness. (a) We first use the qualitative method to prove that any trajectory starting from any point A on a half-ray l passing through the singular point 0 must return and meet l again at B (Figure 6.8). (b) Next we prove that IDAI: lOBI (or the difference fOAI-1 0 B I) increases (decreases) monotonically as IDAI increases. Then IOAI : lOBI can equal 1 at most once (I'OAI-IOBI can at most equal 0 once); this shows that the limit cycle, if it exists, must be unique. The authors of [63] and [64] used this method to study the uniqueness of a limit cycle, and obtained better results. II. The Poincare method. This method was given in Poincare's classical paper Integral curves defined by differential equations [1]. We first prove LEMMA. Let the differential equation in polar coordinates
dp/ dw = cp(p, w)
(6.19)
be given, where the origin 0 is a singular point. Let 1/J(p) be any single-valued continuous function of p. If equation (6.19) has two closed trajectories r1 and r2. then there exists a point between rl and r2 such that one of the following five relations holds: d'ljJ
-=oo, dp
cp(p,w) = oo, ocp -=oo,
ap
~"" dp2 = oo.
(6.20)
PROOF. If one of r 1 and r 2 does not contain the other, or if they do and 0 lies in the exterior of these two trajectories or between them, or if they do and 0 is in the interior of the inner trajectory but there exists a half-ray through 0 which has mor.e than one point of intersection with one of them, then for
§6. UNIQUENESS
125
all these cases it is easy to prove between rl and r2 there must exist a point such that cp(p, w) = oo-for this we should use the well-known formula dp
tanT = p/ dw' where T represents the angle between the radius vector passing through a point and the trajectory of (6.19) passing through this point. Now suppose that 0 c rl c r2 and any half-ray w = Wo starting from 0 intersects r i at only one point (Wo' p~i)) (i = 1' 2). Study the function
~(wo) = 1/J(p~2)) - 1/J(p~t>). It is easy to see that this is a continuous periodic function of wa; hence there exist an w1 at which ~ takes its maximum value. At w = Wt we have d~(w)/dwlw=w 1 = 0;
that is,
dp) - (d"' dp) ( df/J dp . dw (wi,P~2)(wi)) dp • dw (wl,p~l)(wi))
(6.21)
We now fix w11 and consider (df/l/dp)(dp/dw) as a function of p; the interval of variation of pis [p&1>(wt),p&2 >(wt)]. From (6.21) and Rolle's theorem we know there exists p*, p&1>(wt) < p* < p~2 >(wt), such that at the point (wt,p*) at least one of the following three equalities holds: df/J dp dp dw
=.00
'
!_
ap
(d"'
dp) dp dw
=
0,
!_ ap
(d"'
dp) dp dw
= oo.
Substituting dp/dw = cp(p,w) in the above formulas, we get (6.20). The lemma is completely proved.THEOREM 6.1. If for equation (6.19) we can find a function 1/J(p) such that in some simply connected region G none of the five relations in (6.20) hold, then there exists at most one closed traiectory in G, and it must be a limit cycle.
This is obvious from the lemma. COROLLARY. If 1/J(p,w) is a quotient of two continuous functions, and the denominator in the region G is not equal to zero, and along any half-ray w = wo, cp(p,w) is a monotonically increasing (decreasing) function of p, then there exists at most one closed traiectory in G. PROOF. By hypothesis there is no point in G such that !P(p,w) = oo. Hence from the proof of the lemma we can see that if there exist two closed trajectories r 1 and r 2t then they must contain the origin in their interior, and
THEORY OF LIMIT CYCLES
126
any half-ray w = wo can only meet ri at one point. Since cp(p,w) is monotone with respect to p, we get immediately
0 = rf.. cp(p1(6), 0) dO > ftl (<)
J.
fr~
cp(p2(6), 0) dO= 0.
This is a contradiction. Hence there cannot ~xist two closed trajectories. EXAMPLE 2. Consider the system of equations
dxjdt
= -y + xF(x, y),
dyjdt
=x+
yF(x, y),
(6.22)
where
F(x, y) = a[(x- xo) 2
+ (y- Yo) 2 -
a f 0, k > 0.
k],
Then when k > x~ + y~, system (6.22) has a unique limit cycle, which is stable (unstable) when a< 0 (> 0); if k ~ x~ + y~, then (6.22) does not have a closed trajectory [105].( 5 ) PROOF. When k > x~ + y~, the circle F(x, y) = 0 contains the origin in its interior. Let V(x, y) = !(x2 + y 2 ). Then we can calculate
dV dx +ydt dy =ax ( 2 +y2)[( x-xo )2 + (y-yo )2 -k.] dt=xdt Take r > 0 so small that the circle Cr : x 2 + y 2 = r 2 lies entirely in the interior of the circle F(x, y) = 0; thus dvjdt and a have different signs on Cr. Similarly, takeR > 0 so large that the circle CR: x 2 + y2 = R 2 contains the circle F(x, y) = 0 in its interior; thus dvjdt and a have the same sign on CR. Hence there must exist a limit cycle between Cr and CR, which is stable (unstable) when a < 0 (> 0). To prove its uniqueness, we can transform (6.22) into polar coordinates, and obtain (6.23) dpjdw = pF(pcosw,psinw) = cp(p,w), where F(pcosw,psinw) = a[p2 Now take 1/J(p)
= 1/p.
d.,Pjdp-:; oo,
-
2p(xocosw+yosinw) +
pg- k],
pg = x& +Y&·
Then it is easy to see that between Cr and CR we have
o
d2.,Pjdp2
-:;
oo.
Finally,
{) ( d.,P) = ~ [ak- p~ + 2p(xocosw + yosinw)- p2] ap cp dp ap P = a (
p~P~ k -
1) "# 0.
( 5 )This example is an extension of Example 5 in §I; but (6.22), after being transformed into polar coordinates, cannot have a general integral.
127
§6. UNIQUENESS
Since p~ < k, from Theorem 6.1 we know that there exists a unique limit cycle. To prove that when k ~ xg + y~ there does not exist a closed trajectory, we only have to note that the curve of symmetry of the integral curve of (6.23) with respect to the line x0 cos w+y0 sin w = 0 satisfies the differential equation dpjd~N = -o:p[p2 + 2p(xo cosw +Yo sinw)
+ p~- k].
From this we can get (note that p~ ~ k)
d -L d-) -= 2o:p[p2 + pg - k] > o; (L p=p
hence equation (6.23) cannot have a closed trajectory. Before we introduce other methods, let us mention the history of research on the un\queness of the limit cycle for the Lienard equation
x+ f(x)x + g(x) =
(6.24)
0.
Equation (6.24) can be changed to the system of equations dxjdt = v,
(6.25)
dvjdt = - f(x)v- g(x),
by general transformation, and can also be changed to the system dxjdt = -y- F(x),
(6.26)
dyjdt = g(x)
by a Lienard transformation. This equation is very important, not only because (6.24) itself has obvious physical meaning,_ but also because there are many systems of equations, not belonging to (6.26), for which in order to prove the uniqueness of limit cycles we have to go through a coordinate transformation to change the system to the type of (6.26). The reader will see this fact clearly in the second half of this book. The first proof of the uniqueness of a limit cycle of (6.24) was due to Lienard [9]. His result was as follows:
=
THEOREM 6.2. Let g(x) x, let f(x) be an even function, and let there exist a b > 0 such that f(x) < 0 when lxl < b, f(x) > 0 when lxl > 6, and
{±oo lo
f(x) dx
= ±oo.
(6.27}
Then system (6.26} has a unique (stable) limit cycle.
Levinson and Smith (64] extended this theorem to the case where g(x) is a nonlinear continuous function, but they assumed that xg(x)
> 0 when x =F 0,
{±oo
Jo
g(x) dx =
+oo,
(6.28)
128
THEORY OF LIMIT CYCLES
and they did not require f(x) to be an even function. G. Sansone [106] still assumed g(x) x, omitting the condition that f(x) is an even function (thus its zeros need not be symmetric with respect to the origin), and changing the hypothesis to the assumption that F(x) has zeros symmetric with respect to the origin, and then proved uniqueness. Since f(x) represents the damping coefficient of an oscillating system, the conditions in the above theorems all show that when the displacement is small the damping is negative, whereas when the displacement is rather large the damping is positive, and on both sides of x = 0 the damping is equal to zero exactly once. Based on direct observations in physics, H. Serbin [107] attempted to prove: If we only require that f(x) < 0 when D-1 < x < 61, that f(x) > 0 when x > 61 or x < 6-1. and that xg(x) > 0 when x ::/:- 0, and also that formula (6.28) holds, then the system (6.26) has a unique limit cycle. However, shortly afterwards, Duff and Levinson [108] gave an example to x and f(x) is a polynomial, Serbin's result is show that, even when g(x) wrong. Their example is as follows: EXAMPLE 3. Consider the system of equations
=
=
= -x- eyf(x), (6.29) and take f(x) = A3x6 -A2x4 +A1x2-Ao-Cx, where Ao = 36/Tr, A1 = 192/Tr, A2 = 112/Tr, A3 = 64/5Tr, and C > 0 is sufficiently large. According to (4.18) in §4, we can calculate that on all the closed trajectories x2 + y 2 = h 2 of dxjdt
= y,
dyjdt
system (6.29)e=O which can generate limit cycles of (6.20), h must satisfy the equation
{27r Jo
h 2 sin2 O[A3h6 cos6 0- A2h4 cos 4 0
+ A1 h2 cos2 0- Ao- ChcosO] dO= 0; that is, c)(h)
= -h8 + 14h6 -
= -h2 (h2 Hence we must have h c)'(1) = -48
= 1, 2, 3.
< 0,
c)'(2)
49h4
1)(h2
-
+ 36h2
4)(h 2
-
9)
= 0.
Moreover, since
= 80 > 0,
c)'(3) = -2160
< 0,
when e > 0 is sufficiently small, the system (6.29) indeed has three limit cycles r 1 (stable), r 2 (unstable), and f3 (stable). When e-+ 0, they take circles of radius 1, 2, and 3 respectively as their limit positions. On the other hand, if we take C > 0 sufficiently large, it is easy to prove that f (x) = 0 has only a negative root and a positive root, satisfying the requirements of Serbin.
§6. UNIQUENESS
129
From this we can see that just limiting the zeros and sign of f(x) is not enough to guarantee the uniqueness of the limit cycle; in fact a change of sign of f' (x) and f" (x) can also affect the number of limit cycles. Zhang Zhi-fen [109] gave an example to illustrate this: if g(x) x, f'(x) > 0 when x >a> 0, and f'(x) < 0 when x
=
m. Method of comparison. There is another more effective method, the method of comparison, for proving the uniqueness of limit cycles, first created by Lienard and then further developed by Sansone and Zhang Zhifen. Liooard and Sansone compared the values of the differential of a positive definite function ~(x,v) integrated along two limit cycles r1 ::> r2, whereas Zhang compared the values of the divergence integrated along the two cycles. We first introduce Sansone's theorem.
=
THEOREM 6.3. Suppose in equation (6.24) that g(x) x and f(x) is continuous; let there exist D-1 < 0 < 81 such that f(x) < 0 when D-1 < x < Dt, and f(x) > 0 when x < D-1 or x > 81. Moreover, suppose that F(±oo) = ±oo, and there exists ~ > 0 such that F(~) = F( -~) = 0. Then system (6.26) has a unique limit cycle, which is stable. PROOF. The existence of a limit cycle is deduced from Theorem 5.1 in §5. We prove its uniqueness. Let ~ (x, y) = ~ ( x 2 + y 2 ). Then along a system of equations (equivalent to equation (6.24)) dxfdt = y- F(x),
we have
d~
d~dt
dy = dt dy
dyfdt = -g(x),
(6.26)'
1
= [x(y- F(x))- xy]_x = F(x),
or d~ = F(x) dy. We first prove that the abscissa of the extreme right point of the limit cycle r of (6.26)' must be greater than ~. and the abscissa of the extreme left point must be less than -~. H this is not true, we first assume r lies completely inside the region lxl ~ ~. Hence, moving along r in the counterclockwise direction on the right of they-axis, we have F(x) ~ 0, dy > 0; and on the left of they-axis we have F(x) ~ 0, dy < 0. Hence we always have d~ ~ 0. Thus <.ford~ < 0. However, moving around any closed
130
THEORY OF LIMIT CYCLES
1t=F(s)
FIGURE
6.9
curve once, ..X(x, y) should return to its original value; hence this inequality cannot hold. Next, suppose the extreme right point of r is on the right of x = ~. but the extreme left point is on the right of x = -~. Then r intersects the negative x-axis at a point Con the right of x = -~. and intersects x =~at a point D below the x-axis, IWI > ~. From the previous discussion we know that when moving from C in the counterclockwise direction along ron D we have d.\ < 0; hence lOCI > IWI > ~. which contradicts lOCI < ~. Similarly, we can prove it is not possible for the extreme right point of r to be on the left of x = ~. and the extreme left point of r on the left of x = -~. Suppose now there are two limit cycles f 1 :::> f 2 , and they all intersect x = ±~ as in Figure 6.9. We shall compare fr 1 d.\ and fr~ d.\. First we have
f
jH~'H~'
d.\=
f
jH~'H';
f
F(x) dy >
F(x) dy =
jH1AH-t
f
d.\,
(6.30)
jH1AH-t
because F(x) > 0 and dy > 0 on the right of x =~.and for the same y the abscissa of the point on Hf H:{ is greater than the abscissa of the point on H 1 AH4, and F(x) is an increasing function of x. Similarly we can prove that (6.31) Next, along H4DH3 and
H~D' H~
we can write
d.\= -xF(x) dx; y- F(x)
131
§6. UNIQUENESS
thus it is easy to see that
f
d>.
=f
jH'D'H' 4 3
)H'D'H' 4 3
-xF(x) dx y-F(x)
(6.32}
> {
-xF(x) dx = { d>.. JH4DH3 Y- F(x) JH.DH3 This is because along these two arcs we have dx < 0 and xF( x) < 0, and for the same x the ordinate of the point on H~D' H~ is greater than the ordinate of the point on H4DH3. Similarly, we can prove that
f
jH~B'H~
d>. >
f
jH~BH1
Finally, for the remaining four arcs on
f ~>~ h~~
f ~>~ ~~
d>..
(6.33}
r 1 we have
f ~>~ )~~
f
jH~'H~
d>. > 0. (6.34)
Adding all the inequalities (6.30)-(6.34), we get
rl
lrl
d>. >
rl
lr2
d>.,
but >.(x, y), after moving around any closed curve once, should return to its original value; hence r 1 and r2 cannot simultaneously exist. The proof is completed. (6 ) EXAMPLE 4. Prove that the equations dy -=-xy (6.35) ~~ = y 2 - (x + 1)[(x- 1) 2 + JJ], dt have exactly two (stable) limit cycles when 0 < JJ < 1 [110]. PROOF. System (6.35) has three singular points: A( -1, 0), Bt (0, ~}, and B2(0, -~. Since the vector field is symmetric with respect to the x-axis, it suffices to prove there exists a unique (stable) limit cycle about the x-axis. Note that the x-axis is a trajectory, and hence it cannot meet the limit cycle. First it is easy to see that the limit cycle r must be on the right of the line x = -1, because on this line dx/dt = y2 > 0; and the other two singular points are also on the right of x = -1. Now apply the transformation u = x, z = x 2 + y2 - (1 + JJ}, which maps the point on the upper (or lower) half of the (x, y)-plane y ~ 0 (~ 0) one-to-one to the interior of the parabola )/: z = u 2 -(1+JJ) in the (u,z)-plane (Figure 6.10). System (6.35) is changed to du/dt = z- F,..(u), dz/dt = -g,..(u), ( 6 )From the proof we can see that if we can use other methods to prove the limit cycle lllust intersect z = ±.6., then g(z) can be a nonlinear function if it satisfies condition (6.28).
132
THEORY OF LIMIT CYCLES
II
·--1 FIGURE 6.10
where F~(u) = u3 - (1- JL)u and g~(u) = 2u(u + 1)[(u- 1) 2 + JL]. Note that here F~(u) satisfies all the conditions to be satisfied by F(x) in Theorem 6.3 and g~(u) on the right of the line u = -1 (the image of x = -1) satisfies the condition
~a= g~(u) du =
oo.
Hence, exactly as in Theorem 6.3 (replacing JL with g~(u), and ~(z, u) = z 2 +u 2 with z 2 + G~(u)), we can prove its existence and uniqueness.(1) Next we introduce Zhang Zhi-fen's method. In order to prove system (6.26) has a unique stable (unstable) limit cycle, she carried out three steps: 1. Show that the unique singular point of the system is a singular point of positively oriented remote type. 2. Show that if r 2 :::> r 1 are two cycles, then the value of the divergence integrated once along r2 (in the direction of increasing t) is less than the value of the divergence integrated once along r 1· 3. Show that there does not exist a semistable cycle. Her theorem is not only suitable for system (6.26), but also suitable for more general equations
dxfdt = -
dyjdt
= g(x).
(6.36)
§6. UNIQUENESS
133
Later, L. A. Cherkas and L. I. Zhilevich [112] relaxed her conditions, and improved her method of proof.( 8 ) For simplicity and conciseness, we shall prove the theorem of Cherkas and Zhilevich. THEOREM 6.4. Suppose that for system (6.36) the following conditions hold: 1. xg(x) > 0 when x =F 0, and ytp(y) > 0 when y =F 0. 2. tp(y) is monotonically increasing, and f(O) < 0 (> 0). 3. There exist real numbers a and /3 such that the function h(x) = f(x)
+ g(x)[a + {JF(x)]
(6.37)
has simple zeros x1 < 0 < x2, and h(x) ~ 0 (~ 0) in [x1,x2]. 4. Outside [x1,x2] the function h(x)fg(x) does not decrease (does not increase). 5. All the cycles contain the interval [x1, x2] on the x-axis. Then system (6.36) cannot have more than one limit cycle; if one exists, it must be a stable cycle (unstable cycle).
PROOF. Here we only consider the case f(O) < 0. First assume f(x), g(x), and tp(y) are continuously differentiable functions. It is easy to see that (6.36) has a unique singular point 0(0, 0), which is unstable. Now suppose there exist two cycles r 2 ::J r 1 ::J 0, and r 1 is the one closer to 0. Then
1 g(x) dt = 0, 1 g(x)tp(y) dt = 0, !r, !r, ·1 g(x)[tp(y) + F(x)] dt = 0 (i = 1, 2). !r, This is because every integrated function can be changed to an integral of a single variable x or y. Thus from (6.37) we can see that the integral of the divergence - f(x) of (6.36) satisfies the equality
(i=1,2). We now prove that h2 < h1. From Figure 6.11 and Green's formula, we know that [
1E1E
ft(x) dt-
f
1DA
ft(x) dt =
1 JE1EADE1
=
Ifsl d~
ft((x)) dy g X
( ; ) dx dy
~ 0,
(6.38)
( 8 }In her original proof in the second step the method of comparison is identical to Theorem 6.3 without using Green's formula.
THEORY OF LIMIT CYCLES
134
•
FIGURE 6.11
{ j AtBt
ft(x)dt- {
h(x)dt=-J
jAB
!"ls{
ft(x)dx IP(Y) + F(x} ft(x)IP'(y) [~P(Y) + F(x)J2 dxdy ~ 0'
JA 1 B 1 BAA 1
=-
3
(6.39}
where 81 and 82 are the two regions bounded by the above two closed paths respectively. Similarly we can prove that
{
ft(x)dt~O,
h(x)dt- {
jKKt
{
lBc
ft(x)dt- {
lctDt
leD
(6.40}
ft(x)dt~O.
Moreover, we have the obvious inequalities
f
jBtK
ft(x)dt~O,
{ jKtCt
ft(x)dt~O,
f }EAt
ft(x)dt~O,
f
ft(x)dt
~ 0.
(6.41)
jDtEt
Combining (6.38)-(6.41}, we get h2 < ht. However, we already know that 0 is unstable; hence f 1 is internally stable. According to Theorem 2.3 in §2, we know that h1 ~ 0, and hence h2 < 0. If we can prove rl is not a semistable cycle, but a stable cycle, then r2 cannot exist because when rl is externally stable, r2 must be internally unstable, and so we must have h2 ~ 0, which contradicts h2 < 0.
§6. UNIQUENESS
135
Now suppose f 1 is a semistable cycle. Suppose {3 2: 0 {the proof is similar when /3 < 0). Construct a system of equations depending on the parameter 1 dxjdt = -cp(y)- F(x), dyjdt = g(x), {6.42) where
when x ~ x2, when x > x2. 2 Note that (6.42) and (6.36) are identical in the half-plane x ~ x2, and (6.42) forms a family of rotated vector fields with respect to 1 in the halfplane x > x 2 ; hence (6.42) forms a family of generalized rotated vector fields in the whole plane. Thus for sufficiently small 1 > 0 b < 0), (6.42) has an unstable (stable) cycle I\ on the outside of r 1 , and has a stable (unstable) cycle fi inside f1. On the other hand, the function F(x) and its derivative f(x) + "f(X- x2)g(x) still satisfy all the conditions of the theorem; hence, as before, h~ > h1 {here h~ and h1 are the values of the divergence of {6.42) integrated around r~ and I\ once respectively). This is impossible. Finally, if f(x), g(x), and cp(y) are only continuous, then we can handle this by Erugin's method at the end of §1. REMARK 1. If /3 = 0 and ft(x)fg(x) does not decrease when x increases in (-oo, 0) and (0, +oo ), we can obtain Zhang Zhi-fen's uniqueness theorem. REMARK 2. If the conditions of the theorem only hold in the region (a < x < b, -oo < y < oo}, then the conclusion also holds in this region. REMARK 3. If there is no hypothesis on the stability of the singular point, then the conclusion on the uniqueness cannot be obtained, since it is possible for h2 < 0 < hl that r 1 is unstable and r 2 is stable. EXAMPLE 5. Prove the uniqueness of the limit cycle of system (5.33) in §5 assuming B > 1 + A 2 • PROOF. Applying to system (5.35) the change of variables
_ F(x)
r,' = 'T'J,
{ F(x)
= F(x) +
e' =
!,
ds
J: "t(~- x2 )g(~) de
= A(1 + ~') 2 dt (for~' > -1,
i.e. x > 0),
we get d~' fds
= 'f/ 1 -
F(~'),
d'f/ 1 fds
= -g(e'),
where
F(e')
= ( A~'a _ B ~2A 2 e12 _ B- ~- A 2 g(e)
It is easy to see that F'(O)
e) /(1 + e')2,
= e/(1 + (} 2 •
= (1 + A2 - B)/A< 0, e'g(e') > 0 when e' :f. 0, e· G(e') = fo g(u) du---+ +oo when~'---+ -1 + 0,
THEORY OF LIMIT CYCLES
136
and
d
df
(F'(e')) g(€')
=A+
2
A€'2
B- A 2
-
1
+ A(1+e)2 >O
for €' # 0, e' > -1. Hence from Theorem 6.4 we get the uniqueness of the limit cycle. There is a paper by L.l. Zhilevich [113] which used the comparison method to prove the uniqueness of a limit cycle of system (5.25) when f(x, 0) has only one zero. IV. Method of topographic systems. This method was first used in the classical paper of Poincare, but he only considered some artificially constructed special equations. For more general equations, the difficulty of using this method is that we do not know how to obtain a suitable topographical system. Later, J. L. Massera [114] obtained a better way of constructing the topographical system to enable him to improve some others' results; moreover, his method was later adopted by Zhang Zhi-fen and Chen Xiang-yan to obtain some good results. In this subsection we present Massera's method. Earlier, in 1950, G. Sansone [115] proved a uniqueness theorem for limit cycles of the Lienard equation.
=
THEOREM 6.5. Suppose that in system (6.26) g(x) x, f(x) is continuous, f(x) < 0 when (5_1 < x < 6, and f(x) > 0 when x < 6-1 < 0 and x > 61 > 0. Moreover, for all x in ( -oo, 0) let f(x) be a nonincreasing function, and for x in (0, oo) let it be a nondecreasing function; finally, let lf(x)l < 2. Then system (6.26) has a unique limit cycle, which is stable.
Later, Massera [114] gave a simpler proof of the above theorem and eliminated the condition lf(x)l < 2. Theorem 6.5 was extended by Roberto Conti [116] in 1952 to become THEOREM 6.6. In system (6.26) suppose that xg(x) > 0 when x ::1 0 (! and g are continuow), and G(±oo) = +oo. Moreover, suppose that after the transformation z = J2G(x)sgnx the function ~(z) = F(x(z)) satisfies the following conditions: 1. 4i(z)/z is a nonincreasing function when z < 0, and a nondecreasing function when z > 0. 2. lcll(z)/zl < 2. Then system (6.26) has a unique limit cycle, which is stable.
We can also eliminate condition 2) of the theorem using Massera's method. In 1954, Antonio de Castro [117] also gave the following uniqueness theorem for a limit cycle of system (5.25).
§6. UNIQUENESS
137
THEOREM 6. 7. Suppose g(x) and f(x,v) are continuous, xg(x) > 0 when x # 0, g(x) is an increasing function of x, and f(x, v) satisfies a Lipschitz condition with respect to x and v. Let f(O, 0) < 0. When lxl > a > 0, let f(x, v) > 0, and suppose there exist N > 0, a> 0 such that for any continuous function v( s) /_: f(s, v(s)) ds
~a> 0,
provided that lv(s)i > N and lxl >a. Finally, suppose that when increase, f(x, v) does not decrease. Then the system dxfdt
= v,
dvfdt
= -g(x)- f(x,v)v
(6.43)
lxl
and
lvl
(6.44)
has a unique limit cycle. Zhang Zhi-fen proved later [109] that the conclusion of the theorem is wrong. She gave a counterexample, and proved that when g(x) = x the conclusion of this theorem is correct provided the following hypothesis is added: there does not exist a closed curve such that in some neighborhood of it f(x, v) takes constant value along a half-ray passing through the origin. Condition (6.43) can then be dropped. She also used Massera's method. Since Theorem 6.5 is a special case of Theorem 6.7, we now prove Theorem 6.7 (assume g(x) x). PROOF. Transforming (6.44) into differential equations in polar coordinates, we obtain
=
dpfdt
= -psin2 Of(pcosO, psinO),
dO/dt
= -1- sinOcosOf(pcosO, psinO).
We first prove that any closed trajectory of this system, if it exists, must be a star with respect to the origin. That is to say, any half-ray starting from the origin can only intersect the closed trajectory at one point (maybe touching it tangentially). Suppose not. Let the half-ray 0 = Oo and a closed trajectory L intersect at at least three points (Oo, Pi), i = 1, 2, 3, 0 < Pl < P2 < P3· Then the sign of d(J / dt at these three points should vary alternatively. We know this is impossible from the additional condition on f(x,v) in the theorem. Now suppose that under the similarity transformation x' = kx, y' = ky (k > 0) the image of L is Lk. Since L is a star, it follows that Lk contains L fork> 1, but L contains Lk fork< 1. The family of curves {Lk} forms the topographical system we need. Note that the line tangent to Lk at the point Pk(kx, kv) is parallel to the tangent to L at P(x, v ); hence the slope of Lk at Pk is dv' dx
= Lt
x -vf(x, v),
THEORY OF LIMIT CYCLES
138
FIGURE 6.12
but at Pk the slope of the trajectory of system (6.44) is
-~I
b X = --f(kx, kv) = --f(kx, kv).
kv
dx (6 . 44 )
v
From the conditions of the theorem, we know that X - - - f(x v) { V
'
~ -~- f(kx, kv), V X
2: - - - f(kx, kv), v
when k
< 1; (6.45)
when k > 1.
Thus, if as lxl and Ivi increase f(x, v) strictly increases, then any Lk (k =f-1) is an arc without contact (Figure 6.12). Hence L must be a stable cycle, and thus is a unique limit cycle. If as IxI and IvI increase f (x, v) does not strictly increase, then according to the previously mentioned hypotheses the family of equations
dxfdt = v,
dvfdt =-x- f(x/k,v/k)v
(6.46)
forms a family of generalized rotated vector fields with respect to the parameter k, and, along any closed curve, when k varies slightly the vector cannot keep its direction unchanged everywhere. From the theory of §3, we know that closed trajectories of (6.46)k belonging to different vector fields do not intersect. On the other hand, it is easy to see that Lk is a closed trajectory of (6.46)ki thus it is immediately seen that (6.44) does not have any other closed trajectory except L, for otherwise it would intersect or coincide with some Lk (k =f.l), which is impossible. The theorem is completely proved.
§6.
UNIQUENESS
139
FIGURE 6.13 REMARK. In Theorems 6.5 and 6.6 of Sansone and Conti, we assume that lf(x)l < 2, which makes d8fdt =J 0 in the system after it is transformed into polar coordinates, and thus guarantees that the closed trajectory should be a star. From the proof of Theorem 6. 7, we see this can also be assured from the monotone property of f( x, v) with respect to x and v. Following the method of proof of Theorem 6.7, we can prove a more general result of Chen Xiang-yan [50]. ' THEOREM 6. 8. Let the system of equations dx/dt
be given. If for any ..\
= P(x, y),
> 1 the
dyfdt = Q(x, y),
(6.47)
inequality
P(x,y)Q(..\x,..\y)-P(..\x,..\y)Q(x,y)~O
(or ~0)
(6.48)
always holds and the points where equality obtains do not fill completely the closed trajectory of system (6.4 7), then system (6.47) has at most one limit cycle-which, if it exists, must be star shaped (with respect to the origin).
PROOF. Suppose (6.47) has a cycle whose equations are x
= rp(t)
and
y = ,P(t). Suppose r~ is a curve determined by the equations x = ..\rp(t) and
y = A'I/J(t), ..\ E (O,oo). It is easy to see that system
r.\ is a closed trajectory of the (6.49)
From (6.48) and the theory in §3, we know that r and r.\ (A =J 1) do not intersect. Also, it is easy to see that the origin does not lie on r, for otherwise f.\ (..\ =J 1) would intersect r at the origin (Figure 6.13). Moreover, if r is not a star, suppose r and some ray through the origin intersect at two points P 1 and P2. Let ..\o = IVP21/IOP11· Then r.\o and r will intersect at P2, which is impossible. Hence r is a star with respect to the origin.
140
THEORY OF LIMIT CYCLES
r,
Now suppose system (6.47) has another limit cycle and suppose the points of intersection of r and with the positive x-axis are M(x 0 , 0) and M(xo, 0) respectively. Let X= xo/x0 • Then will intersect rx at M(xo, 0), which is also impossible. Hence does not exist.
r
r
r
COROLLARY. For the system
dx/dt = y- F(x),
dyjdt
= -x,
(6.50)
if, as x varies in (-oo,O) and (O,oo), F(x)/x is a nondecreasing (nonincreasing) function of lxl, and in the strip where the limit cycle exists, F(x)/x ¢. const, then system (6.50) has at most one limit cycle. When F'(x) exists, the above uniqueness condition is equivalent to
F'(x) - F(x)/x ~ 0
(or :::; 0)
for all x =F 0, and in the strip where the limit cycle exists the left side of the above formula is not identically zero. This result was obtained earlier by Luo Ding-jun using another method. Moreover, in [50] there is another uniqueness theorem, which is a corollary of the following theorem of de Figueiredo [63] using the point transformation. THEOREM 6.9. For system (6.50), if there exists 8 > 0 such that xF(x):::; 0 when lxl :::; 8, F(x) ¢. 0 in the vicinity of the origin, F(x) ~ 0 when x > 6, and F(x)/x is a nondecreasing function of lxl in ( -oo, 0) and [o, +oo), then the system has at most one limit cycle. REMARK. The conditions ((6.45) and (6.48)) for the system to form a family of rotated vector fields with parameter A of the similarity transformation are too strong; hence the scope of application of the above theorems is not very broad. In the second half of this book we shall see that the system
dx/dt
= -y +ox+ lx 2 + mxy + ny2 ,
dyjdt
=x
forms a family of generalized rotated vector fields with respect to the parameter 8, and for any 6, if the system has a limit cycle, we can prove it must be unique. However, the system does not form a family of rotated vector fields for the similarity transformation x' = kx, y' = ky. V. The Bendixson-Dulac method. By this we mean the special case of Theorem 1.12 in §1 when n = 2. That is, THEOREM 6.10. Suppose that for system {6.47) there exist an annular region G and a continuously differentiable function B(x, y) Uor simplicity, suppose P(x, y) and Q(x, y) are also continuously differentiable) such that
§6. UNIQUENESS
141
(8f8x)(BP) + (8/oy)(BQ) has a constant sign in G and is not identically zero in any subregion of G. Then (6.47) has at most one closed traiectory lying entirely in G.
We can make use of other similar theorems in §1 to obtain similar results, but the practical value of these methods is very little. For instance, for Example 5 in §1, we can take any circular annular region which contains the unit circle x 2 + y 2 = 1 as G, and take B 1. For further extension, as in Example 2 in this section, how to choose G and B(x, y) is not very clear. Since the Bendixson-Dulac method is very useful for proving the nonexistence of limit cycles in polynomial differential systems (as we shall see in the second half of this book), it seems still necessary to continue to explore and investigate the use of this method in the proof of uniqueness of limit cycles.
=
VI. The method of Levinson and Smith. In an earlier paper by Levinson and Smith [64], they first used the method of "proving that the divergence integrated along any limit cycle once must always take positive (or negative) value" to prove uniqueness. Hence it is natural to assume the equations have a -unique singular point (whose index is +1); thus every limit cycle, if it exists, should contain this singular point in its interior. Then according to the well-known fact that adjacent sides of adjacent limit cycles must possess different stability, we can deduce uniqueness of limit cycles in the above manner. However, the first theorem they obtained by this method not only had certain drawbacks, but also it is not easy to verify the conditions of the theorem; hence it. does not have much use. In 1970, because of the needs of qualitative work on quadratic differential systems (that is, systems (6.47) whose right sides P and Q are quadratic polynomials of x and y), and also adopting the advantages of Filippov's method of proving existence of limit cycles, G. S. Rychkov [118] obtained better results by using the method of Levinson and Smith to prove uniqueness of limit cycles. This method was recently improved and extended by Zeng Xian-wu [119], [120], so that it can include many known uniqueness theorems as special examples. For simplicity we shall only present the theorem from [118], but its proof is far more difficult than that of any of the other uniqueness theorems in this section. Suppose in the strip d1 < x < d 2 (d1d2 < 0) we are given the equations dxfdt = y- F(x),
dyfdt = -g(x),
or
(F(x)- y) dy = g(x) dx, (6.51) where xg(x) > 0 when x 1: 0, g(x) is continuous and F(x) is twice continuously differentiable. Introducing the transformation (5.18), (5.19) in §5, we change
THEORY OF LIMIT CYCLES
142
!I
z*
FIGURE 6.14
(6.51) into the two equations
(Fi(z) - y) dy
= dz,
0 $; z < Zoi
(i = 1, 2),
(6.52)
on the right half {z, y)-plane, where Zoi = limx-+ld,l Zi(x), Zi(x) being taken from {5.18) and (5.19). Every closed trajectory L of (6.51) corresponds to a unique pair of curves L1 and L2, which are solutions of the two equations of (6.52) respectively, starting from the same point on the positive y-axis, passing through the same point on the negative y-axis, and satisfying (along the clockwise direction)
~
h
(apax + 8Q) dt 8y
= =
rf, F'(x) dy
h f
jL1
g(x)
FHz) dy-
f F~(z) dy.
(6.53)
jLl
Let Yt(z) and y2(z) denote the equations of L 1 and L2 respectively, let Yu and Y21 denote the sections of arcs of these two curves on the upper parts of the perpendicular isoclines of the corresponding equations respectively, and let Yt2 and Y22 denote the sections of arcs on the lower parts of the perpendicular isoclines respectively (Figure 6.14). Let A and B denote the points of intersection of L 1 and L2 with the two isoclines respectively. On the curve y == Ft(z) we take a point C which has the same ordinate as B; that is,
143
§6. UNIQUENESS
yc = YB = F2(zB) = Ft(zc). Construct an integral curve y = y3(z) of equation (6.52)1 passing through C, and extend it both ways, so that it intersects the line z = z• (z• being the abscissa of the point of intersection of the two isoclines y = Ft(z) andy= F2(z)). This point of intersection will later be assumed to exist and be unique. Let (i,j = 1, 2), and let Yi(z) be the inverse function of Zi(y); that is, the inverse function of Yi = Fi(z) is Zi = Fi(y). THEOREM 6 .11. Suppose there exist unique zo and z*, 0
< zo < z• <
zot, such that the following conditions hold: 1) F~(z) < 0 when 0 < z < zo2.(9 ) 2) (zo- z)Fl(z) > 0 when z =I zo, 0 3) F~'(z) < 0 when 0 < z < z0 t-( 10 )
< z < z01 , and F1 (z*) = F2(z*).
4) ddy(Ft(y)-F2(y)) >0, i.e., (F~- Fn/F~F~
> 0 when /3 < y < Ft(z*), where /3 = 1.=1,2 lllax ( lim Fi(z)) . Z-+Zoi
Then equation (6.51) does not have more than one limit cycle; if it exists, it must be a single cycle. The proof of this theorem will be divided into nine small steps,(l 1 ) in which we have to use the inequalities ZA > z* and ZB > z*, as can be seen from Theorem 5.4 of §5, because in the strip lzl ~ z• it is impossible to have a closed trajectory. Moreover, if F(O) =I 0, then we can translate y to make F(O) = 0. ( 1) YA < YB; that is, the extreme right point of Lt is below the extreme right point of L2.
PROOF. Take a point A' on y = F2(z) such that YA' = YA, with A' on the right of A (Figure 6.15). Now we translate the integral curve y = y 12(z) of (6.52)1 and the isocline y = Ft(z) to the right to make A coincide with A' and obtain y = Y12(z-6.) andy= Ft(z-6.) (dotted line); here 6. = ZA' -ZA· At this timeD moves to D'. If Zc = Zv• and G is a point on y = Y22(z), then Yc ( 9 )This condition is equivalent ( 10 )This condition is equivalent
to f(x) < 0 for the Lit~nard equation when d 1 < x < 0. to condition 4) of Theorem 6.4, but the range of existence
is smaller. ( 11 )Please first see the remark after the proof of this theorem.
144
THEORY OF LIMIT CYCLES
FIGURE 6.15
is clearly greater than y D'; that is, G is above D'. Moreover, it is easy to see the curve O'K'H'A'lies entirely above y = F2(z); that is, F1 (z-~) ~ F2(z), where z E [~, ZA' ]. From this we see that if YA' ~ YB we can apply differential inequalities to the two equations (F2 (z)- y) dy = dz thereby obtaining
YD' ~Yo,
and
(F1(z-
~)-
y) dy = dz,
which is impossible; hence
YA'
= YA < YB·
(2) Y21 < IY22I when 0 < z < ZB· The geometrical meaning of this is obvious. PROOF. From Y21 = Y21- F2 and Y22 = Y22- F2 it is easy to see that Y21 and IY22I are solutions of
dY
dz
1 + YF~(z) y
(a)
and
dY = 1- YF~(z) (b) dz Y respectively. Note that F2(z) < 0 when 0 < z < ZB, and so at the same point (z, Y) dYI - dYI = dz (a) dz (b)
-2F~(z) > o.
Now if there exists Z3, 0 < Z3 < ZB, such that Y21{z3) ~ IY22{z3)l, then from differential inequalities we have Y21 > IY22I at {z3, ZB), and so d dz(Y21(z) -IY22(z)\) = -2F~(z)
+ (Y21(z) -IY22(z)\)(IY22(z)IY21(z))- 1 > 0;
145
§6. UNIQUENESS
that is, Y21(z) -IY22(z)l increases as z increases, for z E [z3,ZB)· Taking the limit, we get
which contradicts Y21(zB) = IY22(zB)I; hence Y21(z)
< IY22(z)l when 0 < z <
ZB·
(3) IY12I > IY22I > Y21 > Yu and Y21 - Yn > Y22 - Y12 (12 ) when 0 < z ~ z*. PROOF. At (O,z*) we have F2(z) < F1(z); hence from differential inequalities we know that Y21 > Yll· Moreover from the definitions of Y21 and Yu, we immediately get Y21 > Yu. Similarly we can prove"that IY12I > IY22I· Using (2), we get the first half of (3). Next, we note that d 1 1
-(Y21- Yn) = ( ) dz F2 z - Y21 F1 - Yn (Y21 - Yn) + (F1 - F2) =..:..:..__....;;...,::-:-'-..,..,-;....._-___:_ Y21Yu
and so we can solve
r
r
r
r
F1(u)- F2(u) { de } Y21 - Yn = ] 0 Y21(u)Yu(u) exp lu Y21(e)Yu(e) du. Similarly,
F1(u)- F2(u) { de } Y22 - Y12 '"']0 Y22(u)Y12(u) exp lu Y22(e)Y12(e) du. Note that F1(u)- F2(u)
> 0 and
[Y22(u)Y12(u)]- 1 < [Y21(u)Yu(u)]- 1; that is, we get the second half of (3). ( 4) Y{ 1 (z)
< 0.
PROOF. We have Y{ 1 = y~ 1 - Fi = -1/(yu- Ft)- Fi(z). When 0 < z < ~ 0; hence Y{ 1 < 0. When zo < z ~ ZA, let o:(z) = F1 - (Ff)- 1 . Then
zo, Fi(z)
[F1- o:(z)]- 1 = F{ > F{
+ F{'(F{)- 2 = o:'(z).
To prove Y{ 1 < 0 is the same as proving 1 I 1 Yu + IY12I. Moreover, this inequality shows the curved angular region formed by Y21 and yu is wider than that formed by Y22 and Y12·
THEORY OF LIMIT CYCLES
146
Now a' < 1/(Ft -a), by Yi 1 = 1/(F1 - Yu) and
Yn(ZA)
= F1(zA) < Ft(ZA)- [F{(zA)t 1 = a(zA);
hence from the comparison theorem we know that when 0 < z < ZA, Yu(z) must lie below a(z); that is, yu
(5) [(Yu)- 1 + 1Yt2i- 1 ]~ > 0. PROOF. If 0
s zo, then from F{ > 0 and the inequality in (3) we can
get
[(Yu)- 1 + IY12I- 1 l~ = Yi1 3 +
IYd- 3 + F{(Y1! 2 -IYd- 2) > 0,
and when zo < z < ZA, we have IY12I~ = -1/IYt21 (4), we get
+ F{ < 0.
From this and
[(Yu)- 1 + IYt2l- 1 ]~ = -(Yu)- 2Y{1 -IYt2l- 2 ·IY12I' > 0.
(6)
~ = -1z• F{ (~1 + l~2l) dz + 1z• F~ (~1 + IY~2l) dz > O. First we prove a lemma. LEMMA. Suppose a> b > c
> d and b + c >a+ d > 0. Then
1
1
1
1
a+ €t > li + c· PROOF. Since (a- d) 2
> (b- c) 2 and b + c >a+ d > 0, we have (a-d) 2 2(a+d)
(b-c) 2
> 2(b+c)"
Moreover, it is easy to see that
(b-c) 2
-=-=----'"'7
2(b+c)
=
(b+c) 2 -4bc (a+d)(b+c)-4bc > -'----------':....;.,-----..,:....____ 2(b+c) 2(b+c)
Combining these, we get (a-d) 2 (a+d) 2 -4ad a+d 2ad 2(a+d) = 2(a+d) = -2-- a+d (a+ d)(b +c)- 4bc a+ d 2bc > =---2(b+c) 2 b+c"
Hence
be b+c
ad a+d'
-->--·
§6. UNIQUENESS
147
that is,
b+c
T
1
a+d
1
1
1
= "b + -c < ---;;;r- =a+ "J·
PROOF OF (6). From (3) and the lemma we deduce that
Note that F~
1
1
Yu
+ IY12I >
1 Y21
1
+ IY22I.
< 0, and hence ci>
> foz• (F~- F{) ( IY:21 +
~~) dz.
Since (5) holds, we can apply the mean value theorem to the right side of the above formula, and we get
> [(Yu(z*))- 1 + IYt2(z•)l- 1]· [Ft(~)- F2(~)] since F1 > F2 at (0, z*).
~ 0,
ci>
(7)
1
y 31 (z")
F~ (z3(y))
dy ~
Yal(z•)
0 :::;
e: :; z*,
1Y11 (z")
Ff(z1 (y)) dy. Ytl(z•)
PROOF. Consider a fixed yo. Since in Figure 6.14 we have Yo= YB > YA and F{ ( z) < 0, it follows that C is on the upper left of A and the integral curve y3(z) is on the left of Yt(z); thus Z3(Yo) < Zt(Yo). Since F{'(z) < 0, we have F.l(z3(Yo)) > F.l(zt(Yo)); thus
l
!tat (z")
F.l(z3(y))dy >
Yal(z•)
1Y3t (z•)
F.l(zt(y))dy. Y32(z•)
But Yu(z") 1
F{(z 1 (y))dy=
Ytl(z•)
1y31(z")
F{(zt(y))dy+
1Ya2(z")
Y32(z•)
F.l(z1(y))dy
Yt2(z•)
Yll (z•)
+ /.
F{{zt(y)) dy,
Yat(z•)
where the values of the last two integrals on the right are negative. The proof of (7) is then obtained.
(8)
PROOF. The function y3(F1 (F2(z)) and Y2(z) are solutions of the equations
(F1
-
y)
dy = /3*(z) dz and (F2- y) dy = dz
THEORY OF LIMIT CYCLES
148
respectively, where
!3*(z)
= F~(z)[F~(F1(F2(z)))]- 1 .
From condition 4) of the theorem, noting that Fi (z•, zc), we conclude that /3* (z) < 1. Moreover,
y3(F1(F2(zB))) Hence when z•
~
z<
ZB,
< 0 (i = 1, 2) when z E
= Y2(zB) = F2(z.a).
we must have
(i=1,2). This can be obtained by the method of contradiction as in (2). Let z = F1(F2(~)). Then(1 3 )
· FHz) dz = FHF1(F2(~))) · (F1(F2(~)))' d~ d
-
= d~ {F1(F1(F2(~)))} d~ = F~(~) d~.
This completes the proof. (9) From (6)-(8) we can easily deduce that the right side of (6.53) is greater than zero. Hence from (7) and (8) we obtain
f. Yn
(z•)
FHz2(y))dy;:::
Y22(z•)
f.Yu
(z")
FHz1(y))dy;
Ytl(z•)
that is,
L2
f.-.. F~ dy ~ L1 f---. F~ dy.
jPBM
jQAN
(6.54)
Moreover, the first term in the middle of (6) is equal to
A
,.-.,. FFi(z)dz ( ) RQ 1 Z - Y
h
h '( ) h
_- ,.-.,. F1 z dy + ,.-.,. F'( 1 z )dy,. + N,.-.,. FFi(z)dz ( ) D 1 Z - Y RQ ND
. (6.55)
(13)This technique has an important effect on the proof of this theorem; it will be frequently used in the next section.
149
§6. UNIQUENESS
and the second term in the middle of {6) is equal to -
f_.,. F~(z) dy- f_.,.F~(z) dy; }MD }R_p
therefore {6) is equivalent to
f_,.F~(z)dy+ f__F~(z)dy< f__FHz)dy+ f__FHz)dy. }MD }R_p }R.q JND
{6.56)
Adding (6.54) and {6.56), we immediately get f
jLt
F{(x)dy> f
jL2
F~(z)dy
(clockwise direction), which shows that the right side of {6.53) is greater than zero. The theorem is completely proved. REMARK. The proof of this theorem is quite difficult. In outline, it can be divided into three steps. From {1) to {3) we study the geometrical properties and the relative positions of the arcs L 1 and L 2 of the trajectories; (4) and (5) are preliminaries to the proof of (6) since we have to use (5) when we apply the second mean value theorem to the integral of (6); finally, the three inequalities in (6)-(8) are used to compare the integrals on the corresponding sections of arcs of L1 and L 2 ; hence in order to compare the integral values on PM and QN we use the integral value of the integral curve MlP of equation (6.52) passing through C as an intermediary. This is the key point of the proof. Zeng Xian-wu [120] allowed the case when y = F2(z) is not monotone,(1 4 ) and can have any finite number of extreme points. This makes the theorem more difficult to prove. Although generally speaking it is not easy to estimate the sign of the di· vergence integrated once along a limit cycle, for some equations we can get a rather easy estimate. We use the following theorem of N. V. Medvedev [121] as an example. THEOREM 6. 12. Suppose that in the equation
x+ f(x)xk+ 1 + g(x) =
0
(6.57)
k is an even number, xg(x) > 0 when x ::1 0, and there exist Xt < 0 < x2 such that f(xl) = f(x2) = 0, f(x) < 0 when x E (x., x2), and f(x) > 0 when x < Xt and x > x 2 • Moreover, let G(xl) = G(x2) and G(±oo) = +oo. Then the system of equations dx/dt
= y,
dyfdt =- f(x)yk+l- g(x)
(6.58)
(1 4 )Note that yin (120] is equivalent to-y in Theorem 6.11; hence the drawing of the two vertical isoclines is different from Figure 6.14.
THEORY OF LIMIT CYCLES
150
cannot have more than one limit cycle. If it exists, it must be a stable single cycle. PROOF. Take
A(x, y)
= y 2 + 2G(x).
Then A(x, y)
= C represents a family
of closed curves, and
dAjdt
= -2f(x)yk+ 2;
hence, if a closed trajectory exists, it lies in the exterior of the closed curve A(x, y) = 2o = 2G(xt)-that is, along any closed trajectory, we have A(x, y)2o > 0. Moreover, from the hypotheses we know that yk (G - a) f ~ 0 always holds. Now, integrating the inequality
around the closed trajectory once (in the direction of increasing t), we obtain
{T
-1
rh
Jo ykf(x)dt= k+ 1 Tdiv(P,Q)dt>O; thus we immediately obtain what we need to prove. The method of proving uniqueness by directly estimating the integral of the divergence was used also in [16], [106], [110], [123], and [124]; it will be presented in §14. VII. The method of Andronov and Leontovich. This method is similar to Lyapunov's method for distinguishing the center and the focus, which is only applicable to a small neighborhood of a focus; in fact, it is an extension of Theorem 3. 7. We study an analytic system with parameter A, assuming its linear approximate system takes the origin as its center when A = 0; then, after an affine transformation, the system can be changed to
dxjdt
= a(A)x- b(A)y + P2(x, y, A),
dyjdt = b(A)x + a(A)y + Q2(x, y, A),
(6.59)
where P and Q are polynomials of degree at least two. The characteristic roots of the linear approximate system are a( A)± ib(A), a(O) = 0. We may as well assume that b(O) > 0, as the case b(O) < 0 can be discussed similarly. We introduce polar coordinates, eliminate dt, and expand the right side of this equation into a power series of r; thus we obtain
§6. UNIQUENESS
151
where R 1(0, >.) = a(>.)jb(>.), and ~(0, >.) is a polynomial in cos 0 and sin 0. Now we look for a solution of (6.60) in the form
T =ToUt (0, >.) + T~u2(0, >.) + T3ua(O, >.) + · · ·
= /(0, To,>.),
(6.61)
where To is the initial value of r; we can obtain the equations satisfied by the functions Uk (0, >.): du1 ( ) du2 2 (6.62) dO = u1R1 0, >. , dO = u2R1(0, >.) + u 1R2(0, >.), ... , and the initial conditions
(k = 2,3, ... ). (6.63) Ut(O, >.) = 1, Uk(O, >.) = 0 From (6.61) we see that the necessary and sufficient condition for T = f(O, To,>.) to be a periodic solution is /(2rr, ro, >.)-To= [ut(2rr, >.)- 1]To
+ u2(2rr, >.)T~ + ua(2rr, >.)T3 + · · · = 0. Cancel the factor To
# 0, and rewrite the above transcendental equation as
cp(>., ro) = vt(>.) + v2(>.)To + va(>.)T~ + · · · = 0. (6.64) In order to study whether a closed trajectory appears in the vicinity of the origin as >. varies, we have to study whether the equation cp(>., r 0 ) = 0 has a real root with respect to To. H we treat cp(>., To) = 0 as a curve in the (>., To)-plane, it is clear it must pass the origin. Moreover, from the equation satisfied by u2(0, >.)it is easy to see v2(0) = u2(2rr,O) = 0. Now suppose that • a'(O) # 0, va(O) # 0. (6.65) We wish to show that when>.# 0 and takes the suitable sign, system (6.59) has a unique limit cycle in the vicinity of the origin. Note that v1 (>.) = Ut (2rr, >.) - 1 = exp ( 1 (')
vl ""
= 2 b(>.)a'(>.)- a(>.)b'(>.)
b2(>.)
7r
We have vi(O) = 2rra'(O)/b(O)
# 0.
:~~~ 2rr) exp
1,
( 2 a(>.)) 7r b(>.) •
Thus
8cpj8>.l(o,o) = v~(O) # 0. Hence in the vicinity of the origin we can solve for >. from (6.64) as a singlevalued function of To, >. = >.(ro). Next,
d>.l = [- ocp ;acp] = _ v2(0) = dro (0,0) oro a>. (0,0) vi (0) ~>.I 2va(O) = _ b(O)v3(0) # 0 dr5 (O,O) vi (0) rra'(O) '
__
o,
THEORY OF LIMIT CYCLES
152
(a)
(b)
FIGURE 6.16 and hence A= A(ro) takes extreme value at (0, 0). In the following we discuss four different cases, depending on the signs of a'(O) and v3(0). 1. a'(O) > 0 and v3(0) < 0. Then A(ro) has a minimum at the origin (Figure 6.16(a)). Since lPAjdr~l(o,o) > 0, dAjdro increases in some neighborhood above ro = 0; hence the function inverse to A = A(ro) is also single-valued in the vicinity of the origin in the first quadrant; that is, for every sufficiently small A > 0, there is a unique ro > 0 which satisfies rp(A, ro) = 0, i.e., system (6.59) has a unique limit cycle in the vicinity of the origin. Moreover, from a'(O) > 0 and a(O) = 0 we see that, for A < 0, we have a(A) < 0 and the origin is a stable focus; when A(O) > 0 we have a(A) > 0 and the origin is an unstable focus; hence the limit cycle is stable. 2. a'(O) > 0 and v3(0) > 0. Here A(ro) has a maximum at the origin (Figure 6.16(b)); when A> 0 the origin is an unstable focus, and when A< 0 it is a stable focus; hence the limit cycle which appears when A < 0 must be unstable. 3. a'(O) < 0 and v3(0) > 0. An unstable limit cycle appears when A> 0. 4. a'(O) < 0 and v3(0) < 0. A stable cycle appears when A< 0. Here we only discuss the conditions for system (6.59) with one parameter to have a limit cycle in the vicinity of the origin. Later, in §9, we study the same problem for systems with several parameters. Exercises 1. Prove that when ht ~ 1 > h2 > 0, system (6.5) has a unique (stable) limit cycle. 2. If, in system (6.44), g(x) = x and J(x, v) = 0 represents a real ellipse in which the distance from the center to the origin is less than the distance from the origin to the ellipse, then the system has a unique limit cycle.
§6. UNIQUENESS
153
=
3. Prove that if, in system (6.26), g(x) x and F(x) is a polynomial of degree not higher than three, then either system (6.26) does not have a limit cycle or it has a unique limit cycle. 4. Study whether Theorem 6.11 can contain some previous theorems as special examples. 5. Prove that when 0 < J.L < 1 the limit cycle of system (6.35) must intersect the lines x = ±~, and when J.L ~ 0 or J.L ~ 1 the system does not have a closed trajectory. 6. Prove that the function f(x) in system (6.29) has a positive zero and a negative zero when C > 0 is sufficiently large. 7. Prove that the equation x + hx + x = k when x > 0; = 0 when x < 0 (where h > 0 and k > 0 are constants) has a unique periodic solution, which is stable. 8. Suppose the right side
dxfdt
= -y- F(x),
dyjdt =X
F(x) is continuously differentiable, F(O) = 0, and F'(x) - F(x)/x ~ 0 (or ~ 0) when x =I 0, then the system has at most one closed trajectory. Explain the geometrical meaning of these conditions. 10. Suppose f{x,y) and F(x,y) are defined in the rectangle R = {c ~ x ~ c+a,iy-.81 ~ b}, and f(x,y) < F(x,y). Suppose y(x) and Y(x) are differentiable functions in (c, c + 6), 0 < 6 < a, and satisfy equations y'(x) = f(x, y(x)) and Y'(x) = F(x, Y(x)) and the initial conditions y(c) = ,8, Y(c) = ,8. Prove that y(x) < Y(x) when c < x ~ c + 6 [125]. 11. Suppose that, in the equation x + 0, 1, the equation does not have a periodic solution [79].
§7. Existence of Several Limit Cycles H a system of equations has more than one limit cycle, then they naturally
can be distributed in many different ways. For example, three limit cycles can be distributed in any of the four ways shown in Figure 7.1. In this section we discuss the first kind of distribution, since the equations we are going to discuss have only a unique singular point and the interior of each limit cycle must contain the singular point. Other kinds of distribution will be discussed with respect to the quadratic differential systems in the second half of this book.
0
0
@
(a)
00 (c)
(b)
(d)
FIGURE 7.1
Back in 1958, M. I. Vollokov [127] pointed out that if we added suitable requirements to the geometrical figure of the function y = F(x), we could guarantee that the system of equations dxjdt = y- F(x),
dyjdt = -x
(7.1)
would have exactly n limit cycles, where n is any given positive integer. However, since the system involves too many parameters, it is rather difficult in that paper to determine that a given system (7.1) has exactly n limit cycles. In 1966, G. S. Rychkov [128] gave a condition to guarantee that the system of equations dxjdt = y- F(x), dyjdt = -g(x) (7.2) 155
156
THEORY OF LIMIT CYCLES
has at least n limit cycles. As far as the problem of limit cycles is concerned, it is much more useful to be able to answer the following question: when the system of equations is given, how can we show that there exists a limit cycle? If limit cycles exist, how many are there? In 1975, Rychkov [129] proved that when F(x) = a2x 5 + a1x 3 + aox in (7.1), the system has at most two limit cycles, and also pointed out that if F(x) = e-(x 5 -~-tx 3 + x), where E > 0 and IJ > 2.5, then (7.1) satisfies the condition of his previous paper [128], and so this system has exactly two limit cycles. This result can be said to be the earliest one on the existence of two and only two limit cycles. The idea of the proof is to study dh/da1 (the rate of change of h(L) with respect to the ordinate of a11 where h(L) is the integral value of the divergence along a spiral L leaving from, for example, a point a1 on the positive y-axis, going around the singular point (0, 0) and returning to another point a2 on the positive y-axis) and to prove that dhjda1 has a fixed sign, e.g. always positive. Since two adjacent sides of two adjacent cycles have different stability, we know at once that the system of equations has at most two cycles, the integral value of the divergence along the inner cycle is less than zero (or :50), and the integral value along the outer cycle is at least 0 (or > 0). In the proof the technique of proving step (8) in the proof of Theorem 6.11 in §6 is used. Since 1979 the Chinese mathematicians Huang Ke-cheng [130], Zhang Zhifen and Ke Qi-min [131], [132], Chen Xiu-dong [133], Huang Qi-chang and Yang Si-ren [134], and Ding Sun-hong [135] have given sufficient conditions for system (7.2) to have at most or at least n limit cycles; Zhang's result [131] solved a previous conjecture. In the following discussion we assume that F(x) E Cl, F(O) = 0, g(x) E C, and xg(x) > 0 when x '# 0. It is easy to see that (0, 0) is a unique singular g(e) de. We first point of (7.2). As before let F'(x) = f(x) and G(x) = present the result of [130].
J;
LEMMA 7 .1. Suppose there exist constants a, a', b, and b' (b' < a' < 0 < a
§7. EXISTENCE OF SEVERAL CYCLES
FIGURE
'II= F(x) at P1 and
PI·
157
7.2
Suppose yp1 =-Nand YP: = M. Let
.>t2(x, y) = !(Y- M) 2 + G(x), and let .>t,(x, y) = .>t,(A) if (x, y) are the coordinates of A. We compute ~arately the total derivatives of .>tt(x,y) and ..X2(x,y) with respect tot (along the trajectory of (7.2)):
d.>tl/dt = -g(x)[N + F(x)], d.>t2/dt = g(x)[M- F(x)].
(7.3) {7.4)
From condition 2) we know that F(x) ::; M when xp: ::; x::; 0, and F(x) ~ M when b' ::; X ::; X p:. We inspect the parts of r 1 and r 2 on the left half-plane because
{7.5)
Similarly, we can see that
158
THEORY OF LIMIT CYCLES
and (7.6) From (7.5) and (7.6) we at once get ..\2(A2) - ..\2(D2) ;::: ..\2(A1) - ..\2(D1); that is,
(YA~- M) 2 - (YD~- M) 2 ;::: (YAt- M) 2 - (YDt- M) 2.
(7.7)
Similarly, ..\1(D2)- ..\1(A2);::: ..\1(D1)- ..\1(Al); that is,
(YD~
+ N) 2 -
(YA~
+ N) 2 ~ (YDt + N) 2 -
(YAt
+ N) 2.
(7.8)
From condition 3) we know that equality cannot hold simultaneously in (7.7) and (7.8). Adding these two inequalities and simplifying, we get
But M + N ;::: 0, YA 1 - YA~ < 0, and YD 1 - YD~ > 0, which is a contradiction. The lemma is proved. Corresponding to Lemma 7.1, we obviously have LEMMA 7. 2. Suppose there exist constants b' < a' < 0 < a < b such that the following conditions hold: 1) F(x) ~ F(a) when 0 ~ x ~a, and F(x) is monotonically nondecreasing on [a, b]. 2) F(x) ;::: F(a') when a'~ x ~ 0, and F(x) is monotonically nondecreasing on [b', a']. 3) F(x) ¢. 0 when a' ~ x ~a. Then in the strip b' ~ x ~ b, system (7.2) has at most one limit cycle which can intersect both the lines x = a and x = a'.
Next we prove another lemma. LEMMA 7. 3.
Suppose there exist constants N ;::: 0, a > 0, and b' < 0 such
that 1) F(x) ;::: -N when 0 ~ x ~ a, and
2) F(b') ~ -N- y'2G[cij. Then the limit cycle of (7.2) which intersects the line x the line x =a.
= b'
must intersect
PROOF. Let the trajectory starting from A( a, -N) as t increases intersect the negative y-axis at A', and let the trajectory starting from B(b', F(b')) when
§7. EXISTENCE OF SEVERAL CYCLES
159
FIGURE 7.3
t' decreases intersect the negative y-axis at B' (Figure 7.3). From condition 1) and (7.3) we know that YA' ~ -N- J2G(a) and YB' < YB = F(b'); hence
YB' < YA', i.e., B' is below A'. From this we can see at once that the lemma holds. Similarly, we can establish the following three lemmas. LEMMA 7. 4. Suppose there exist constants M ~ 0, a
> 0, and b' < 0
such that 1) F(x) ~ M when 0 ~ x ~a, and 2) F(b') ~ M + ~J2G(a). Then the limit cycle of system (7.2) which intersects the line x intersect the line x = a.
= b'
must
LEMMA 7. 5. Suppose there exist constants N ~ 0, a' < 0, and b > 0 such that 1) F(x) ~ -N when a'~ x ~ 0, and 2) F(b) ~ -NThen the limit cycle of system (7.2) which intersects the line x = b must intersect the line x = a'.
JWM.
LEMMA 7. 6. Suppose there exist constants M ~ 0, a' < 0, and b > 0 such that 1) F(x) ~ M when a' ~ x ~ 0, and 2) F(b) ~ M + y'2G(a'). Then the limit cycle of system (7.2) which intersects the line x = b must intersect the line x = a'.
THEORY OF LIMIT CYCLES
160
From the above lemm88 it is not difficult to give sufficient conditions for system (7.2) to have at most n limit cycles. For example, THEOREM 7 .1. Suppose in system (7.2) F(x) and g(x) satisfy the follow-
ing conditions: 1) F( -x) = -F(x) and g( -x) = -g(x). 2) In the interval (0, b), f(x) has only n zeros 0 < 01 < 02 <···
(k
= 1, ... , n- 1),
where .Bk+l E (nk+t. ak+2) and F(.Bk+t) = F(ak)· Then in the strip lxl ~ b, system (7.2) has at most n cycles.
PROOF. Since F(x) and g(x) are odd functions, the closed trajectory of system (7.2) is symmetric with respect to the origin. From the conditions of the theorem we know that (-1)k+1F(x) is monotonically incre88ing when x E [.Bk,Ok+t],
(-1)kF(x) ~ (-1)kF(.8k) when x E [O,,Bk],
(7.9)
where k = 1, ... , n, .BE (nt.n2), and F(.Bt) = 0. It is e88y to see for lxl ~ .B {7.2) does not have limit cycles. From (7.9), condition 3) and Lemm88 7.1-7.4, we know that (7.2) h88 at most one limit cycle in the strip lxl ~ 02. Any limit cycle intersecting the line x = Ok must intersect the line x = -.Bk (k = 2, ... , n), and in the strip lxl ~ Ok+l there is at most one limit cycle which can intersect x = Ok. Thus we can see that in the strip lxl ~ b there exist at most n limit cycles of system (7.2). In §5 we indicated that the systems of equations
dxfdt = y + x2 sinx,
dyfdt
= -x
and
dx/dt = y + x2 sin x, dyfdt = -2x/(1 + x2) 2 have at le88t n limit cycles in the strip lxl ~ mr+ ~'11'. Now from Theorem 7.1
we also know that every system h88 at most n limit cycles in lxl ~ n'll' + !11'· Hence, every system in this strip h88 exactly n limit cycles. From Lemm88 7.1-7.6 we know that even if F(x) and g(x) are not odd functions, it is not difficult to obtain sufficient conditions for system (7.2) to have at most n limit cycles. In the following we introduce again the work of Rychkov [129] and Zhang [254]. We first prove some lemm88.
§7. EXISTENCE OF SEVERAL CYCLES
161
LEMMA 7. 7. Suppose there exist constants 0 ~ a < e < {3 such that 1) F(a) = F({3); and 2) f(x) > 0 (/(x) < 0) when X E (a, e), and f(x) < 0 (/(x) > 0) and X E (e, {3). Then along any arc 8 of the trajectory of the system
dxjdt in the strip a
~
x
~
= v,
{3, v = v(x)
dvjdt = -x- f(x)v ~ 0,
l-f(x)dt > 0
(7.10)
x E [a, ,8], we have (l-f(x)dt
PROOF. We only consider the case outside the parentheses. Let z = F(x), x E [a,,B]. Suppose F(a) = F(,8) =a and F(e) =b.. From condition 2) we
know there exists an inverse function
X= Xt(z) E [a, eJ, and Xt(a) =a, Xt(b) = e; X2(a) = {3, X2(b) =e. Thus we obtain
re r
{ f(x) dt = [ 13 f(x) dx = f(x) dx + [ 13 f(x) dx la la v(x) la v(x) Je v(x) b dz dz = a v(xt(z)) + Jb v(x2(z))
1
=
1
1
b[
a
v(xt(z))
-
1
v(x2(z))
] dz
(7.11)
< 0,
since
v(x2(z))- v(xt(z))
=
1
z2(z)
zl(z)
dv -d dx X
=
1z2(z) Zl(z)
X
--( ) dx < 0 V X
(7.12)
when z E [a, b]. The lemma is completely proved. Similarly we can prove LEMMA 7. 8. Suppose there exist constants a < e < {3 ~ 0 such that 1) F(a) = F(,B); and 2) f(x) > 0 (/(x) < 0) when x E (a, e), and f(x) < 0 (/(x) > 0) when X E (e,,B). Then along any arc 8 of any trajectory of system (7.10) in the strip a ~ t ~ ,8, v = v(x), x E [a, ,8], we have
-1
f(x)dt < 0
( -1/(x) dt >
0) .
THEORY OF LIMIT CYCLES
162
Also we prove
7.9. If the conditions of Lemma 7.7 hold, then along two arcs and s2: v = v2(x), v2(x) > v1(x) ~ 0 when x E [o:,,B], of any two trajectories of system (7.10) in the strip o: ~ x ~ ,8, we ha'!e LEMMA
= v1(x)
s1: v
f - f(x) dt > f - f(x) dt
181
182
( f - f(x) dt < f - f(x) dt).
181
(7.13)
182
PROOF. We consider the case outside the parentheses. From (7.11) we get
f - f(x) dt- f - f(x) dt
1s1
1s2
= =
lb [v1(x~(z))- v1(x~(z))] lb [v2 (x~(z))- v2 (x~(z))] lb [(v1(x~(z))- v2(x~(z)))- (v1(x:(z))- v2(x:(z)))] dz-
dz
dz,
and
d(v2(x)- v1(x)) = __ x_ + _x_ = x v2(x)- v1(x) > O, dx v2(x) v1(x) v1(x)v2(x) when x E ( o:, ,B). Again from (7.12) we get (7.13) at once. The proof is complete. Similarly, we have
-
LEMMA 7 .10. If the conditions of Lemma 7.8 hold, then along two arcs = v1(x) and s2: v = v2(x), v2(x) > v1(x) ~ 0 when x E [o:,,B], of any
s1: v
two trajectories of system (7.10) in the strip a
1 1
-f(x)dt <
1 2
-f(x)dt
(1
1
~
x
~
-f(x)dt
,8, we have
>
1 2
-f(x)dt).
We still need two more lemmas. LEMMA 7 .11. Suppose f(x) ~ 0 (f(x) ~ 0), and f(x) ¢ 0 when x [o:, ,8]. Then along two arcs s1: v = v1(x) and s2: v = v2(x), v2(x) > v1(x) 0 when x E [o:, ,8], of any two trajectories of system (7.10) in the strip o:
x
~
.B
we have
1 1
-f(x)dt >
1 2
-f(x)dt
(1
1
-f(x)dt <
1 2
-f(x)dt).
PROOF. Since
f - f(x) dt- f - f(x) dt = ff3 - f((x)) dx- [(3 - f((x)) dx
1s
1
1s
2
1a
=-
we know at once that the lemma holds.
V1 X
1:
f(x)
1a
V2 X
[v1 ~x)- v2~x)] dx,
E ~ ~
§7. EXISTENCE OF SEVERAL CYCLES
163
FIGURE 7.4
LEMMA 7. 12. Suppose when x E (a, /3] (a
> 0)
1) f(x) > 0 (f(x) < 0), and 2) f(x) is monotonically nondecreasing (nonincreasing). Then along any two arcs of traiectories 1i (i = 1, 2) of system (7.10) passing through the points (.B;, 0) (i = 1, 2) and intersecting the line x = a twice, we have l
-f(x)dt "'11
where a
> l-f(x)dt "'12
(l
-f(x)dt
"'11
-f(x)dt), "'12
< f3i < /32 ::; f3.
PROOF. We consider the case outside the parentheses. As shown in Figure
7.4, 11 get
= A~
and 12
= AB = A~ U .4;B'; U B;B.
r_ -
-f(x)dt
From condition 1) we
< 0.
jAA2UB2B
Let x = p cos (} and v = p sin (}. System (7.10) is transformed to
= -psin2 Of(pcosO), dOjdt = -1- cosOsinOf(pcosO). dp/dt
(7.14)
THEORY OF LIMIT CYCLES
164
From condition 2) we know that ..(Bi can be expressed asp= Pi(9) (i and p 1 (9) < P2(9) when 61 ~ 9 ~ 9'1. (Figure 7.4). Thus
f
-f(x)dt-
1A2B2
f
= 1, 2},
-f(x)dt
1A1B1
{61
= 19
2
-f(P2(9) cos9) d9 -1- cos9sin0/(P2(9)cos9)
{61
-f(p 1 ( 9) cos 6) d9
(7.15)
- 19 -1- cos6 sin6/(p1 (9) cos6) = [92 f(Pt (9) cos 9) -f(P2(0) cos 6) dB < 191 { ·[-1-cos9ein9f(P1(9}coe9}] [-l-coe9sin9f(P2(9)coe9}]} - 0. · 2
From (7.14) and (7.15) we at once get
1
-f(x)dt>j-l(x)dt.
"'tl
"12
Now we present Rychkov's theorem, but the proof we give is that of Zhang Zhi-fen; moreover, the conditions are somewhat weakened. THEOREM 7. 2. Suppose that when x E ( -d, d) the following conditions hold:
1} /(-x)=/(x). 2) /(x) has only positive zeros a1 and a'J., 0 < a1 < a2 0 and F(a2) < 0. 4) / (x) monotonically increases for x E [a2, d). Then system (7.12} has at most two limit cycles in the strip lxl < d.
PROOF. We consider system (7.10} in the strip lxl < d. Since f(x) is an even function, the closed trajectory of (7.10} is symmetric with respect to the origin. Let {3 be the smallest positive zero of F(x) (at < {3 < a2). It is easy to see that system (7.10) has no limit cycles in the strip lxl ~ {3. Moreover, f(x) < 0 when x E ({3, o:2); hence, according to Lemmas 7.7 and 7.8, if (7.10) has a limit cycle in the strip lxl ~ 0:2, then it must be a unique unstable cycle. On the other hand, from Lemmas 7.9-7.12 we know that if system (7.10) has two limit cycles Lt C £2, both intersecting the line x = a'J., then
,/.. - f(x) dt > rf- - f(x) dt.
~1
~2
(7.16)
From this we know that system (7.10) does not have a compound cycle and a periodic cycle. In the following we divide our discussion into two cases.
§7. EXISTENCE OF SEVERAL CYCLES
165
(I) In the strip lxl ~ a~, system (7.10) has a unique unstable limit cycle
Lt. H there exists a limit cycle of (7.10) intersecting the line x = a~, we assume the cycle closest to L1 is L~ (:::) L 1 ); then L~ must be internally stable. In the following we prove that L~ must be externally stable; otherwise L~ should be an internally stable but externally unstable semistable cycle. We examine the system of equations
dxfdt = v,
dvfdt = -x- [/(x) + a-y(x)]v,
(7.17)
where a> 0,-y(x) = 0 when lxl < a2, and -y(x) = (lxl-a2)~ when a~ :Sixl < d. From the theory of generalized rotated vector fields we know that, when a is suitably small, system (7.17) has two cycles L~1 ) C L~~> in the strip lxl < d which intersect the line x = a 2 such that
~ < > -It (x) dt?:. 0, ~
h2
2
(ll
h-2
-/t(x) dt :50,
where ft(x) = f(x) +a-y(x), which still satisfies the conditions of the theorem. This contradicts (7.16). Hence L~ must be a stable cycle. From (7.16) again, we know there is no cycle outside L~. (II) System (7.10) does not have limit cycles in the strip lxl :5 a~. Since the singular point 0 is stable, if (7.10) has cycles, then the cycle L 1 closest to 0 must be internally unstable. H L 1 is also externally unstable, then, similarly to (I), we can prove that the exterior of L1 has at most one more cycle. H Lt is externally stable, then~ 1 -f(x) dt = 0. Hthere is also a limit cycle L2 outside Lt, then L~ must be internally unstable, which would contradict (7.16). Hence system (7.10) has only a unique semistable cycle L1. Considering the above discussion, we see that (7.10) has at most two limit cycles. COROLLARY.
If F(x) = a2 x5 + a1x3 + aox, then system (7.1) has at most
two limit cycles. PROOF. We may as well assume a~ > 0. When F(x) has at most one positive zero (a multiple zero is counted as one), system (7.1) has at most only one limit cycle; if F(x) has two different positive zeros, then from Theorem 7.2 we know that system (7.1) has at most two limit cycles. By Theorem 5.2 we can prove that if F(x) = x 5 - 50x3 /3+45x, then (7.1) has exactly two limit cycles. Many persons (see [136]-[139]) have studied the special case of system (7.1)
dxfdt = y + psinx,
dyfdt = -x
(7.18)
THEORY OF LIMIT CYCLES
166
and conjectured that, for allp. ¥= 0, system (7.18) has exactly n limit cycles in the strip lxl ~ (n + 1)11". Zhang Zhi-fen [131] completely solved this problem. Her result is as follows: THEOREM
7. 3. The system of equations
dxfdt = v,
dvfdt
= -x + (p.cosx)v
(7.19)
(equivalent to (7.18)) has exactly n limit cycles in the strip lxl
~
(n + 1)1r.
It is clear that we only have to discuss the case p. > 0. For convenience of presentation, we still let
f(x) = -p.cosx,
F(x)
= foz f(s)ds = -p.sinx.
First we prove some lemmas. LEMMA 7 .13. In the half-plane v ~ 0, along the same trajectory v = v(x) of system (7.19), we have v(x) > v( -1r-x) when x ~ 0, and v( -x) > v(1r+x) when x ~ 0. PROOF. From
(7.19) we get
v(1r+x) -v(-x)
11"+z e
= 1-z
v(e) de
1.,.+z --ed e = l -zo + 1.,. + o .,. v(e) = e[v(7r +e)- v( -e)I de
r lo
v(-e)v(1r+e)
r v(e) e de- lor
- lo
11" v(7r +e) de,
X~
0.
Differentiating both sides yields
d dx[v(1r+x) -v(-x)]
=
x[v(1r+x)-v(-x)] 1r v(-x)v(1r+x) - -,v(:-1r-+-x"7)
<
x[v(1r + x)- v( -x)] v( -x)v(1r + x) '
X~
0,
thus
d:
[(v(7r+x)-v(-x))exp(-
foz v(-e~~!+e))] <0 whenx~O.
But v(1r) - v(O) < 0, and so v(1r + x) - v( -x) < 0 when x ~ 0. Similarly we can prove the other inequality. Now we construct a positive definite state function >.(x, v) = x 2 /2 + (v+F(x)) 2 /2. Differentiating along the direction field defined by (7.19) yields
§7. EXISTENCE OF SEVERAL CYCLES
167
d>..jdt = -xF(x). For simplicity, let >..(x) = >..(x,v(x)), where v = v(x) is the equation of the trajectory of (7.19). In the following we study the variation of the state function along the trajectory of (7.19). LEMMA 7.14. If the trajectory v = v(x) of system (7.19) intersects the lines x = ±m1r in the half-plane v ~ 0, then along this section of the trajectory we have m~l.
(7.20)
= A.((k + 1)1r)- A.(k1r) and Kk = >..( -k1l")- >..( -(k + 1)1r) 0 is an integer). First we prove the inequalities
PROOF. Let .dk
(k
~
.6o > 0,
Since d>..jdt
Ko > 0, 1 (-1)k+ (.6k + Kk+1) > o, k ~
(7.21)
( -1)k+ 1 (Kk
(7.23)
o, + .6k+l) > o, k ~ o.
(7.22)
> 0 when 0 < lxl < 1r, inequality (7.21) holds. Moreover,
.6k + Kk+1 =
=(-1)k
l
(k+l),.. -xF(x)
k71"
( )
V X
r[ t1r+x)_ k1r +X
}0
V
V
dx +
~-(k+ 1 ),..
-xF(x)
-(k+2)11"
V X
()
dx
(k1r+1r+x )]IF(x)idx, -k1r- 1l"- X
k~O.
From Lemma 7.13 we know that the function under the integral sign is less than zero; hence (7.22) is proved. Similarly we can prove (7.23). Now let ,6_1 = K-1 = 0. Then from (7.21)-(7.23) we have m-1 ( -1)m+l[>..(m7r)- >..( -m1r)] = ( -1)m+ 1
L (.6k + .6k)
k=O [(m-1)/2)
= {-1)m+l
L
(.6m-2k-2
+ 'Km-2k-d
k=O [(m-1)/2)
+ (-1)m+l
L
('Km-2k-2
+ .6m-2k-d
k=O
>0. Lemma 7.14 is completely proved. Now let f[a,/3) f(x(t)) dt denote the integral of f(x) along a section of the trajectory of (7.19), where [a, .8] is an interval of variation of x. Let x(tt) =a and x(t 2 ) == ,8. Then the interval of variation oft is [t 1 , t 2 ].
168
THEORY OF LIMIT CYCLES LEMMA
7.15. In the half-plane v
(-1)m- 1
~
0
r
f(x(t)) dt > 0,
(7.24)
1[-m'lr,m'lr]
where m is a positive integer. PROOF. Let
d1c =
i
f(x(t)) dt =
(k1r,(lc+1)1r]
= (-1)/c
1
(k+1)1r
v( x)
k1r
r v(k1rf(x)+ x) dx
f(x)
-
dx
lo
= (-1)/c r/2 v(krr + x)- v(krr + 11"- x) if(x)l dx, )0 v(krr + x)v(krr + 1r- x)
dk
=r
f(x(t)) dt
= ~-k'lr
1(-(lc+1)1r,-lc'lr]
= ( -1)/c
r v(-krrf(x) dx x)
-(k+l)'lr
k
~ 0,
(7.25)
f(x) dx v(x)
)0
= (-1)/c r/2 v(-krr- x)- v(-krr -'II"+ x) if(x)i dx, k ~ 0. (7.26) lo v( -krr- x)v( -k1r- 1r + x) First we prove the inequalities
+ d~c) > 0, (-1)/c+l((llc+l + d~c) > 0,
(7.27) (7.28)
(-1)/c+l(dlc+1
It is easy to see that
d~c 1 +die= (-1)/c+l +
r 12 v(k1r + + x) -v(k1r + 2 x) if(x)l dx v(krr + + x)v(krr + 2rr- x) 11"-
11"
lo
1r
- ( -1)/c+l r/2 v( -k'll"- x) - v( -krr- 11" + x) lf(x)l dx.
lo
v( -krr- x)v( -krr- 1r + x)
(7.29)
From Lemma 7.13 we have
v(krr + 1r + x) < v( -krr- x), 0 ~ x ~ rr/2, k ~ 0, v(krr + 2rr- x) < v(-krr -'II"+ x), 0 ~ x ~ rr/2, k ~ 0.
(7.30)
§7. EXISTENCE OF SEVERAL CYCLES
169
From (7.19) and Lemma 7.13 we have
v(k1r + 1r + x)- v(k1r + 27r- x) = fk1r+21r-x ( (x) + f(x)) dx
=
Jk1r+1r+x
1 1
k1r+1r-X 7r + X
V X
dx k1r+x v(1r + x) k1r+11'-X X > --dx k1r+x v( -x)
= v(-k1r- x)- v(-k1r -1r + x) > 0, 0 ~X~ 1rj2,
k;::: 0.
(7.31)
From (7.29)-(7.31) we obtain (7.27). Similarly we can prove (7.28). From Lemmas 7.7 and 7.8 we can get
do = f
f(x(t)) dt > 0,
(7.32)
][0,7r)
do = Let d-1
= iL1 = 0.
f(x(t)) dt > 0.
f
(7.33)
j[-1r,O)
From (7.27)-(7.33) we get m-1
f(x(t)) dt = ( -1)m-l
(-1)m-l {
I: (dk + dk) k=O
J[-m1r,m7r)
[(m-1)/2)
I:
= ( -1)m-l
(dm-2k-l + dm-2k-2)
k=O [(m-1}/2) + (-1)m-l
I:
(dm-2k-l +dm-2k-2)
k=O
> 0,
m;::: 1.
Lemma 7.15 is completely proved. Since the closed trajectory of system (7.19) is symmetric with respect to the origin and (-1)m-l f(x);::: 0 when m1r ~ x ~(2m+ 1)7r/2, from Lemma 7.15 we have LEMMA 7 .16. If the closed trajectory Lm of system (7.19) intersects the interval [m1r, (2m+ 1)7r/2] in the positive x-axis, then
( -1)m-l J.
h ...
f(x(t)) dt > 0,
m;:::
1.
THEORY OF LIMIT CYCLES
170
Next we prove LEMMA 7.17. If system (7.19) has two closed trajectories Lt C L2, and
= ±(2m+ 1}rr/2 in the half-plane v ~ 0,
they both intersect the lines x
f
(-1)m
(f(x2(t))- /(xt(t))] dt > 0,
m
then
> 0.
1(-(2m+1)11'/2,(2m+l)11'/2]
PROOF. First, from Lemmas 7.9 and 7.10 we get
(-1)kDk = (-1)k
f
[f(x2(t))- f(xt(t))]dt
1((2k+ 1)11' /2,(2k+5)11' /2]
> 0,
k ~ 0,
(-1)k Dk = ( -1)k
(7.34)
f
[f(x2(t))- /(xt(t))] dt
1(-(2k+5)11'/2,-(2k+l)11'/2]
> 0,
k ~ 0,
f
(7.35)
[f(x2(t))- f(xt(t))] dt
< 0.
(7.36)
1(-11',11']
> 0 when
When m is odd, by (7.34)-(7.36) we have f(x) and by Lemma 7.11 we have
f
1r
~ lxl
< 3rr/2,
[j(x2(t))- /(xt(t))] dt
1[-(2m+1)11'/2,(2m+1)11'/2]
f
=
+f
1[-311'/2,-x] (m-1)/2
+
E
+f
1(-11',11']
(D2k-1
[f(x 2 (t))- f(x 1(t))] dt
1[11',311'/2]
+ D2k-d < 0.
k=1
When m is even, by (7.34) and (7.35) we have f(x) and by Lemma 7.11 we have
f
< 0 when 0 ~ lxl < rr/2,
[j(x2(t))- f(x1(t))] dt
1(-(2m+1)11'/2,(2m+1)11'/2] ·
=f
m/2-1
[f(x2(t))- f(xt(t))] dt +
1(-11'/2,11'/2]
E
(D2k
+ D2k) > 0.
k=O
Lemma 7.17 is completely proved. From Lemmas 7.12 and 7.17 we get
*
LEMMA 7.18. If system (7.19) has two closed trajectories L1 C L2 and both intersect the interval [(2m+ 1)rr /2, (m + 1)rr] in the positive x-axis, then
{-l)m
[.fr:
3
j(x2(t))dt-
1
f(x1(t))dt]
> 0,
m;:::
1.
§7. EXISTENCE
OF SEVERAL CYCLES
171
PROOF OF THEOREM 7.3. (I) We show that system (7.19) has at least n limit cycles in the strip lxl ~ (n + 1)11'. Suppose the trajectory starting from a point Pm(-m1r,O) of the negative x-axis, after passing through the half-plane v ~ 0, intersects the positive xaxis at Qm. From Lemma 7.14 we know that when m > 0 is odd we must have XQm > m1r; when m > 0 is even, XQm < m1r. Since the directional field is symmetric, we know that when the trajectory from the point Pm(m1r,O), after passing through the half-plane v ~ 0, intersects the negative x-axis at a point 'lJm, we must have XQm = -XQm· Let PmQ:n and PmlJ,;: represent the segments of the trajectories passing through the points Pm,Qm and Pm,Qm respectively; let QmPm and QmPm represent the line segments joining Qm, Pm and 7Jm, P m respectively. Set
r m = -p;;:Q;, U QmPm U PmQm U QmPm. Then in r 1, r 2 , ••• , r m, ... , r n+ 1 every pair of adjacent simple closed curves form the inner and outer boundary curves of a Poincare annular region, and there exists at least one closed trajectory between them. Hence in the strip lxl ~ (n + 1)11', there exist at least n limit cycles. (II) We show that there exist exactly n limit cycles of system (7.19) in the strip lxl ~ (n + 1)11'. We divide this into two cases. 1. There exists a closed trajectory Lm which intersects the interval [m1r, (2m+ 1)11'/2] in the x-axis. By Lemma 7.16 we know that when m is odd (even), Lm is stable (unstable) and there are no more trajectories intersecting (m1r, (2m+1)11'/2]. Now we prove there is no closed trajectory which intersects the interval ((2m+1)11'/2, (m+1)11']. Suppose the contrary. Then the trajectory L~ closest to Lm and containing it must be internally unstable (stable). Next we prove L~ cannot be semistable. In fact, suppose L~ is semistable. Consider the system of equations dxfdt
where fa(x) =
-JLCOSX
= v,
dvfdt = -x- !a(x)v,
(7.19*)
+ O')'m(x), a> 0, and when when
lxl ~(2m+ 1)11'/2, Ixi > (2m+ 1)11'/2.
System (7.19)* forms a family of generalized rotated vector fields with respect to a, and when a is very small, there exist closed trajectories L!.!> C L~) of (7.19)* which intersect ((2m+ 1)11'/2, (m + 1)11'] in the x-axis, and, moreover, (-1)mrf. h~l
fa(x)dt~O.
172
THEORY OF LIMIT CYCLES
This contradicts Lemmas 7.12 and 7.17. Hence L~ must be an unstable (stable) cycle. From the proof of (I), at the same time there exists at least one one-sided, or even both-sided stable (unstable) closed trajectory outside L~ which intersects the interval ((2m+ 1}n/2, (m + 1)1r]. This contradicts Lemma 7.18. Hence there cannot be another closed trajectory which intersects
((2m+ 1)11"/2, (m + 1)1r]. 2. There is no closed trajectory which intersects [m1r, (2m+1)?r/2] on the xaxis. From the proof of (I), we know there exists at least one closed trajectory which intersects ((2m+ 1)11"/2, (m + 1)1r]. Suppose Lm is the one closest to the origin; when m is odd (even), it must be internally stable (unstable). Lm cannot be semistable, for otherwise by the proof of (I) there would exist at least one one-sided or even both-sided stable (unstable) closed trajectory outside Lm which intersects ((2m+ 1)11"/2, (m+ 1)1r]; this contradicts Lemma 7.18. Hence, Lm must be a stable (unstable) closed trajectory. Then, similarly to case 1 we can prove there is no other closed trajectory except Lm which intersects the interval ((2m+ 1)11"/2, (m + 1)11"]. From the above discussion, we know that when n is odd (even), system (7.19) has a unique stable (unstable) closed trajectory which intersects [m1r, (m + 1)1r] in the positive x-axis. Also, in the strip lxl ~ 1r, system (7.19) does not have a closed trajectory; hence in the strip lxl ~ (n + 1)11" there are exactly n limit cycles, and stable and unstable limit cycles are arranged alternately. Theorem 7.3 is completely proved. After proving this beautiful theorem, it is fitting for us to introduce the history of this problem. We know that the van der Pol equation
x + J.tf(x)
+ x = o,
(7.37)
where f(x) = -x+x3 /3,(1) has a unique limit cycle with respect to all J.t =f:. 0. Similarly, we can prove that if f(x) is a polynomial of degree 2n + 1 which contains only terms of odd degrees, then for sufficiently small IJ.tl equation {7.37) has at least n limit cycles. Eckweiler [136] first noticed that if we take f(x) = sinx, then in order to obtain a solution from the family of solutions z:l + z2 = A 2 for p = 0 that will produce a periodic solution for J.t =/:- 0, the amplitude A must satisfy the equation( 2 )
Jor1f sin~sin(Asin ~) d~ = 21rJ1 (A) = 0, =
(7.38)
( 1 )Differentiating (7.37) with respect to t, and letting :i y, we obtain the well-known form. Equation (7.37) is sometimes called the Rayleigh equation; it first appeared in Rayleigh's book Theory of aound (1894), and was much earlier than van der Pol. (2 )See (4.18) in §4.
§7. EXISTENCE OF SEVERAL CYCLES
173
where J1(A) is a first-order Bessel function. It is easy to calculate the derivative of the above formula, and get
{211:
lo
sin2 tp cos(A sin tp) dtp
= 21r[J0 (A) -
J2 (A)].
(7.39)
By the properties of Bessel functions,( 3 ) Jt(A) has an infinite number of positive zeros, and the large zeros obey the asymptotic law
An= n1r + 11'/4 + 0(1/n).
(7.40)
Under this condition, the right side of (7.39) becomes
4( -1Y'.../211'/An + 0(1/n).
(7.41)
Hence we see that this value is different from zero provided that n is sufficiently large. Thus [136] affirmed that the equation
x +~sin±+ x = 0
(7.42)
has an infinite number of limit cycles when 1~1 « 1. Later, Hochstadt and Stephan [138) pointed out that this affirmation is not rigorous because it may not be possible to find an interval for the same ~ such that the value of~ in this interval and the infinite number of limit cycles corresponding to the zeros of J1(A) can coexist. The authors of [138) applied more asymptotic properties of J 1 (z) and the method of constructing the Poincare annular region, first proving that the asymptotic equation
d1 / d8 = J.t sin 8 sin("Y sin 8) of the equation (7.42) in polar coordinates d""f _ ~sinOsin(""(sinO) dO- 1 + (~h)cos8sin("Ysin8)
(7.43)
has an infinite number of limit cycles when 1~1 > 0 is sufficiently small, and then showing that this conclusion also holds for (7.43) provided 1~1 is taken slightly smaller. Later, R. N. D'heedene [139) improved the method of constructing the annular region in [138] to obtain the above conclusion when 0 < IJ.tl < 2, but for IJ.tl ~ 2 he only proved that outside a sufficiently large neighborhood of the origin there exist an infinite number of limit cycles, and he conjectured that Theorem 7.3 should also hold. Zhang Zhi-fen in [131) completely proved his conjecture. REMARK 1. For the van der Pol equation
dxjdt = y, (3)Refer to [140].
dyjdt = -x + JJ(Y- y 3 /3),
(7.44)
174
THEORY OF LIMIT CYCLES
FIGURE 7.5
if we let x = -J.Lv, then we get
dy dv
= J.L
2y
- y 3 /3 + v
,;;.____;;__"""y--'
(7.45)
and we can prove( 4 ) that as J.L ---+ +oo the limit position of this periodic solution of (7.45) is a discontinuous periodic solution, as shown by the bold line in Figure 7.5. Applying to (7.42) the transformation x = J.LV, or
dx/dt = y,
dy/dt
= -x- J.Lsiny,
we get
dy/dv = -J.L2 (v+siny)jy.
(7.46)
We conjecture that as J.L---+ +oo we can obtain some results on (7.46) similar to those for (7.45). REMARK 2. Exercise 5 of §6 of the first edition of this book (1965) (Exercise 3 in this edition) has shown that when F(x) in (7.1) is a cubic polynomial of x, the system has at most one limit cycle. This result was also obtained independently by A. Lins, W. de Melo and C. C. Pugh [141]. They also conjectured that if F(x) is a polynomial of degree 2n + 1 or 2n + 2, then (7.1) has at most n limit cycles. This conjecture has not yet been confirmed or disproved. We think it is correct.
( 4 )See
Appendix IV in (137).
§8. Structural Stability of Differential Systems Briefly speaking, a property possessed by a differential system is called stable if this property remains after a slight change of the system itself. For example, in §4 we have already seen that if an ordinary differential system (P(no),Q(no)) possesses a single limit cycle r 0 , then in the family ofrotated vector fields with a as parameter, for all a in a sufficiently small neighborhood of n 0 , the system (P(a), Q(n)) also has a unique single cycle in the vicinity of fo; hence the property of "possessing a single limit cycle" with respect to the rotation of vector fields is stable. However, if (P(nt), Q(nt)) possesses a semistable cycle f 1 , then for a close to n 1 on one side, (P(a), Q(n)) does not have a limit cycle in the vicinity of r 1 , but for a close to nt on the other side, (P(a), Q(n)) has at least two limit cycles in the vicinity of ft: this illustrates that the property of ''possessing a semistable limit cycle" is unstable with respect to the rotation of vector fields. In dynamical systems in the plane, the study of the existence and the number of limit cycles is an important part of the global qualitative theory; hence it must involve the problem of whether the global qualitative structure is stable. This is the structural stability to be discussed in this section. The importance of structural stability in practical applications is very obvious, since mathematical models abstracted from practical problems (which are in general represented by algebraic, ordinary or partial differential equations) have gone through processes of approximations and simplifications. To make sure the properties obtained from the study of mathematical systems can correctly reflect the state of the practical problems, We must demand that these models be structurally stable. In this section we first introduce a rigorous definition of structural stability of plane ordinary differential systems, and give topological characteristics of phase-portraits of structurally stable systems; then we discuss the problem of structural stability of polynomial systems. This section is preparatory for a discussion on the problems of limit cycles and global structure for quadratic differential systems in the following sections. 175
THEORY OF LIMIT CYCLES
176
Let the plane system
dxjdt = P(x, y),
dyjdt
= Q(x, y)
(I)
be defined in a region B of the (x, y)-plane, let P, Q E C 1 (i.e., continuously differentiable), and let the boundary of B be a simple closed curve having no points of contact with respect to {I). For definiteness, we may as well assume the trajectory of (I) crossing the boundary of B does so from outside to inside. All the systems whose P and Q satisfy the above conditions form a set X. Now in X we introduce a metric p as follows: Let another system in X be
dx dt = P(x, y) + p(x, y),
dy dt = Q(x, y)
+ q(x, y).
(II)
Then we define the distance between (I) and (II) in X to be
p(I,II)
+
= m;x (IPI lql +~~~~+I ~:I+ ~~!I+ ~~:1).
We can prove that, under such a definition of the metric 1 X becomes a Banach space. DEFINITION 8.1. If there exists a 8 > 0 such that there is a topological map T from B into itself which can carry trajectories of {I) into trajectories of (II) provided that p(I, II) < 8, then we call system (I) a structurally stable system or coarse system in B, and system (II) a permissible perturbed system of (I); and p and q are called perturbations. The concept of coarse systems was first introduced in 1937 by A. A. Andronov and L. S. Pontryagin [142]. They assumed P and Q were analytic functions, and for the map T, besides the requirements of Definition 8.1, they added the following restriction: for any c > 0, we can make d(M, T(M)) < c(l) for any point M in B provided 8 > 0 is sufficiently small; here d( , ) represents the distance in R 2 • They have pointed out that the necessary and sufficient conditions for system (I) to be structurally stable are 1. It has only a finite number of elementary singular points, and the characteristic roots of its corresponding linear approximate system do not have zero real parts. 2. It has only a finite number of closed trajectories, which are all single limit cycles. 3. It does not have a trajectory from a saddle point to another saddle point. (1 )We shall call any map satisfying this conditione-homeomorphic.
(III)
§8. STRUCTURAL STABILITY
177
The details of their proof, however, have never been published. In 1952, H. F. De Baggis (143) weakened the requirement of analyticity of P and Q to P, Q E C 1 , and gave a detailed proof for the necessary and sufficient conditions. In 1959, M. M. Peixoto [144] further proved that Definition 8.1 and Andronov's original definition are equivalent, and generalized to n-dimensional syste1ns. In 1962, Peixoto [145) gave and proved necessary and sufficient conditions for structural stability of differential dynamical syste1ns on compact two-dimensional manifolds. His results will be mentioned at the end of this section. Since Peixoto has proved the equivalence of the two definitions of structural stability, we may as well add the requirement d(M, T(M)) < e in Definition 8.1 in order to simplify our proofs. Because we only have to consider perturbations p and q whose absolute values are very small, in order to prove the necessity of (ill) we can assume that the singular point of (I) under the map T becomes the singular point of (II) in its vicinity; the same hypothesis is also applied to the limit cycle. These hypotheses greatly shorten the proof of the following theorem. THEOREM 8. 1. The conditions (III) are necessary conditions for system
(I) to be structurally stable. The proof will be divided into the following four lemmas, in all of which we assume that (I) is structurally stable. LEMMA 8. 1. System (I) can only have a finite number of singular points and a finite numbe1> of closed trajectories. PROOF. From the Weierstrass approximation theorem, in any neighborhood of (I) there always exists a polynomial system (II) (that is, the right sides of the system are polynomials) such that the two polynomials of (II) do not have a common factor; hence the number of singular points of (II) can only be finite. Also, from Dulac's well-known theorem [146], [147], we know that (II) can only have a finite number of closed trajectories; hence (I) can only have a finite number of singular points and a finite number of closed trajectories. LEMMA 8. 2. System (I) can only have elementary singular points, and the characteristic roots of its corresponding linear approximate system do not have zero real parts. PROOF. Take any singular point, which we may assume to be (0, 0). If it is not an elementary singular point, then a(P,Q)
I -o
l a(x, y) (0,0) -
,
178
THEORY OF LIMIT CYCLES
which shows that the two curves P = 0 and Q = 0 have a point of intersection at (0, 0) which is not simple.(2) Thus we can make a slight perturbation of one of them such that its singular point changes from one to more than one. This contradicts the hypothesis that (I) is structurally stable. We now prove that the characteristic roots of its linear approximate system at (0, 0) do not have zero real parts. Expand (I) into dx dt =ax+ by+ P1(x,y),
dy dt =ex+ dy
+ Q 1(x,y),
Jx
2 + y 2 ). Suppose its characteristic root has zero where P 1 and Q1 are o( real part, i.e., a+d = 0; thus system (I) takes the origin as its center or focus. Now we add a perturbed term o:x, 0 < lo:l « 1, on the right side of dxfdt; then the perturbed sytem still takes the origin as its singular point and the real part of its characteristic root is -o:. If (0, 0) is a center of (I) but is a· focus with respect to the perturbed system, it is clear that the center and the neighborhood of the focus cannot be €-homeomorphic, which contradicts the hypothesis that (I) is structurally stable. If (0, 0) is a stable (unstable) focus of (I), then we use Theorem 3.7 in §3 and take o: < 0 (> 0), which can make the origin become an unstable (stable) focus of the perturbed system and can generate new limit cycles in its vicinity; hence these two systems are not €-homeomorphic. In short, the real part of the characteristic root of the linear approximate system of (0, 0) cannot be zero. This lemma is completely proved.
LEMMA 8. 3.
There is no trajectory connecting two saddle points of sys-
tem (I). PROOF. To get a contradiction, assume there is a trajectory of (I) which
connects two saddle points A and B (which may coincide). Construct a family of rotated vector fields dxfdt = P- o:Q,
dyfdt = Q
+ o:P;
(IV)
when o: < 0, the vector field rotates in the clockwise direction and the arrows in Figure 8.1 indicate the directions of the trajectories of (IV). It is easy to see that (IV) still has A and B as its saddle points, but at the same time the separatrix entering B does not come from A, and the separatrix starting from A does not enter B; that is, (I) and (IV) are not €-homeomorphic, which contradicts the hypothesis that (I) is structurally stable. This completes the proof. (2)This implies that these two curves touch tangentially at (0,0), or at least one curve has (0, 0) as a multiple point.
§8. STRUCTURAL STABILITY
FIGURE
179
8.1
LEMMA 8. 4. Closed traJectories of system (I) are single limit cycles. PROOF.
Let "ft, ... ,
"'N be all the limit cycles of (I), and set
h("tt) =
i( 'll
aP ax
+ aQ) ay
dt = 0.
Now we shall construct a function cp 1(x, y) E C 1 such that 'Pt(x, y) = 0 on i= 0 outside an q-neighborhood N.,., of "ft. where 17 > 0 is small enough that N.,., does not meet with any "fi (i :j: 1). For this, we only have to take "fl, acptfax ~ 0 on "ft. and 'Pl
• 2 1rn(x, y) + ( '1f"2 sm n x, y ) exp [217
'Pt(x,y) =
t
y)] an 2 1rn(x, 211
when (x, y) EN.,.,, 1f"
when (x, y) ¢.
2
N.,.,,
"'l·
where n(x,y) represents the distance of a point (x,y) of N.,., from From 1 n(x, y) E C it is easy to prove that 'Pt(x, y) indeed satisfies the above conditions. Similarly, for any "fi we can define 'Pi(x, y). Now we study the system of equations dy ~ acpl (V) - = P(x y) + c'P1'P2 · · · 'PNdt = Q(x,y). dt ' ax' It is easy to see that (V) and (I) both take "ft. ... , as closed trajectories, and when c > 0 is sufficiently small, they are permissible perturbed systems. Using the fact that h("tt) = 0 and the properties of 'Pl (x, y) we can compute that, for system (V),
"'N
hy("tt) =
i
ccp2 · · · 'PN (
"'ll
a;:l)
2
dt.
"'ll
Since acptfax ~ 0 and is continuous on and 'Pi(x, y) > 0 on the right side of the above formula is not equal to 0, and has the same sign as c. Now assume that 11 is an externally stable cycle of system (I). From the theory of rotated vector fields in §3, we know that in the outer neighborhood of there exists a curve r such that all the trajectories of (I) intersecting
"'t
THEORY OF LIMIT CYCLES
180
r
move in the exterior-to-interior direction. Take E > 0 small enough so that r is still a curve without contact with respect to (V), and (V) does not have singular points between "Y1 and r. On the other hand, since E > 0, "11 has become an unstable cycle of (V); hence in the outer neighborhood of "11 there appear closed trajectories of (V) different from 11 ,, .. , "fN, which is impossible. The lemma is completely proved. Combining Lemmas 8.1-8.4, we know immediately that conditions (III) are necessary for system (I) to be structurally stable. REMARK. Lemma 8.4 also tells us that systems possessing a multiple cycle can, after a slight perturbation, make the multiple cycle split into at least two cycles. This important fact has not been mentioned in §§2 and 4. Using similar methods, Ding Tong-ren (148] proved that if a multiple cycle has even (odd) multiplicity, then the system can, after any arbitrarily small perturbation, split the multiple cycle into an even (odd) number of single cycles (not less than 2). THEOREM 8. 2. Conditions (III) are sufficient for system (I) to be structurally stable.
For this we first introduce the concept of singular trajectory and regular trajectory. DEFINITION 8.2. If the positive direction and the negative direction of a trajectory are orbitally stable, or the positive direction of this trajectory is orbitally stable and the negative direction runs out of the region B, then the trajectory is called a regular traj'ectory. If a trajectory has at least one direction in B which is orbitally unstable, then it is called a singular traj'ectory. It is easy to see that a center or central focus is a regular trajectory, but any other singular point is a singular trajectory since it always has one direction which is orbitally unstable. Periodic cycles and compound limit cycles are regular trajectories, but other closed trajectories are singular trajectories. A separatrix entering or leaving a saddle point must be a singular trajectory. If both ends of a trajectory run to a saddle point or a focus or approach a limit cycle, or if its positive direction does so but its negative direction runs out of the region B, then this trajectory must be regular, since all the trajectories in its vicinity possess the same properties. Thus, as far as a structurally stable system is concerned, all its singular points, limit cycles and separatrices are singular trajectories, and there are no other singular trajectories. We now turn to the proof of Theorem 8.2. Since (I) satisfies conditions (III), there exist only a finite number of singular trajectories (including singular points) in the region B, which divide B into a finite number of subregions, called normal regions; and the interior of each normal region is completely
§8. STRUCTURAL STABILITY
181
filled with regular trajectories. There are two kinds of normal regions: one kind takes the boundary of B (or part of it) as part of its boundary, and another kind does not contain any boundary point of B on its boundary. DEFINITION 8.3. Stable nodal points, foci, and limit cycles are called sinks, and unstable nodal points and foci and the boundary of B are all called sources. It is easy to see the boundary of every normal region has at least one source and one sink; for otherwise the regular trajectory in this normal region would have nowhere to go as t-+ ±oo.( 3 ) LEMMA 8. 5. The boundary of every normal region has one and only one source and one sink.
Suppose that on the boundary of the normal region G there exist two sinks M1 and M2. We shall obtain a contradiction. Use a smooth arc C lying entirely in G to connect Mt and M2. Since Mt is stable, the regular trajectories passing through points on C close to M1 all enter M1 (as t -+ +oo); the totality of all the points on C entering M 1 form a corresponding open set 81. Similarly, the totality of all the points on Centering M2 form a corresponding open set 82. The sum of 82 and S 1 cannot be equal to C, since there must exist a point N on C such that the trajectory passing through N as t -+ +oo enters a third sink M3. We may as well assume N is a boundary point of the open set 8 1 ; but the totality of all the points on C entering M3 also form an open set, and so the trajectory passing through the points in the vicinity of N in 81 will enter M3. This contradicts the definition of N. The lemma is completely proved. From this lemma it is easy to see that normal regions of structurally stable systems can only have the following three types: 1. There are one source and one sink on the boundary (Figure 8.2(a)-(d)). 2. There are one source, one sink, two saddle points, and four separatrices (Figure 8.3). 3. There are one source, one sink, and three separatrices (Figure 8.4(a), (b)).(4) (In Figures 8.3 and 8.4, S=source and H=sink.) PROOF.
LEMMA 8. 6. If system (I) satisfies conditions (III), then for sufficiently
small 8 a system (II) which satisfies p(I, II) < 8 also satisfies conditions (III) and has the same number and same type of singular trajectories as (I). ( 3 )Since we have assumed that conditions (III) hold, the regular trajectory in the normal region cannot be a closed trajectory. ( 4 )The source in Figures 8.3 and 8.4 can be a limit cycle or boundary of B; the sink lnay also be a limit cycle. For simplicity, here they are each represented by one point.
182
THEORY OF LIMIT CYCLES
(a)
(b)
(c)
(d) FIGURE 8.2 H
s FIGURE 8.3 PROOF. For a singular point Oi we can construct a very small simple closed curve ci which contains oi but does not contain any other singular points; the index of Ci is the index of Oi. When 6 > 0 is sufficiently small, the vector fields determined by (II) and (I) on Ci do not take opposite directions. Hence the indices of Ci are the same in these two systems. Moreover, since we
§8. STRUCTURAL STABILITY
(a)
183
(b) FIGURE 8.4
o
only have to take the radius of Ci and very small, for any singular point of (II) in the interior of Ci, the real part of the characteristic roots of its linear approximate system is not zero, and has the same sign as the characteristic root of (I) at Oi; hence (II) has one and only one elementary singular point in the interior of Ci, whose index is the same as that of Oi. However, we should note that when Oi is a (critical) nodal point of (I), there may appear in the interior of Ci a focus of (II). After we remove the interiors of Ci from the region B, IP(x, y)l + IQ(x, y)l has a positive lower bound in the remaining region; hence, when is sufficiently small, the quantity
o
IP(x, y) + p(x, Y)l + IQ(x, y) + q(x, y)l also has a positive lower bound in that region. Thus we have proved that (II) and (I) have the same number of elementary singular points possessing the same index in B. For every limit cycle ri of (I), we can construct a small neighborhood ~ of r i such that its inner and outer boundary curves are curves without contact. When o > 0 is sufficiently small, the boundary curves of R; with respect to (II) are still curves without contact possessing the same properties. Moreover, in R; there are no singular points of (II); hence (II) has at least one limit cycler; in every R;. Consider that H(s) ds does not change sign (for fJ > 0 sufficiently small), and the adjacent limit cycles possess different stability. We know (II) has only one limit cycle r; in every R;. Moreover, when fJ > 0 is sufficiently small, in the vicinity of an elementary singular point of (I) there are no limit cycles of (II), and in the other parts of B also no limit cycles of (II) appear; hence (I) and (II) have the same number and the same type of limit cycles. Finally, since a solution is continuous with respect to its initial conditions and its parameter, we can easily see that if (I) has a separatrix "Yl from a
J
THEORY OF LIMIT CYCLES
184
B
D*0
A FIGURE 8.5
s
FIGURE 8.6
source A 1 to a saddle point B and separatrices "Y2 and "Y3 from the saddle point B to the sinks A2 and Aa respectively, and the singular points of (II) in the neighborhood of Ai and B are Ai and B* respectively, then (II) also has a separatrix "Yi from Ai to the saddle point B*, and separatrices "Y2 and "Y3 from the saddle point B* to A2 and Aj respectively. Except for these separatrices, which correspond one-to-one to the separatrices of (I), (II) cannot have any other separatrices. The lemma is completely proved. According to Lemma 8.6, there is a one-to-one correspondence between the finite number of normal regions of (I) and (II), and corresponding normal regions have the same type. In the following we construct a homeomorphic transformation which maps a normal region G1 of (I) to the corresponding normal region G 2 of (II) in such a way that their trajectories correspond to each other, and then combine the point transformations of different normal regions to form a homeomorphic transformation of the region B into itself. Now we use the above-mentioned normal region of type 2 to illustrate the construction of this transformation; the construction for regions of types 1 and 3 is about the same.
-------
. B on the boundary~ Df the Suppose first that the source A and the smk normal region G 1 of type 2 are not limit cycles, and let the saddle points in G 1 be Do and D1 (Figure 8.5). Let b~} (0:::;; .X:::;; 17) represent the totality of trajectories in G 1 , where "Yo and "Y., represent the separatrices from A to Do and D 1 respectively, which have extensions "Yb and "Y~ (also separatrices) to B respectively. From a point M 0 on "YO to a point M., on "Y., we construct an arc without contact Co. in this normal region, and let the point of intersection of
§8. STRUCTURAL STABILITY
185
"t>. and Co. be M>., and denote by A the arc length from Mo toM>. along Ca. We now define a homeomorphic transformation ~ from the normal region G 1 (excluding the points A and B) to a curvilinear rectangle 61 in the (s, A)plane such that the image of a trajectory 1>.• in G1 is a line segment A= A* in 6 1 . Suppose we measure an arc length s• from M>.· on 1>.• to a point M~: (negative towards A, and positive towards B), and set the image of M~: to be (s*, A*). No matter whether A and Bare foci or nodal points, the arc length along 1>. towards A orB always approaches a finite limit; hence the left and right ends of 61 are bounded. Suppose that under this transformation ~ the image of Do is a point D0 on A = 0, and the image of D 1 is a point Di on the line A = TJ. AB before, we can define a homeomorphic transformation "Ill on the normal region G2 (except the two points, source A and sink B) of type 2 of (II) to a curvilinear rectangle 6 2 in the (s, X)-plane. Suppose that under this transformation the images of the saddle points Do and D1 in G2 are the point D~ on the line X= 0 and the point D~ on the line X= Tj respectively. Now it is not difficult to establish a homeomorphic transformation 8 between 61 and 62 such that A = 0, D0, A = TJ, and Di correspond to X= 0, D~, X= Tj, and D~, and the line segment parallel to the s-axis in 61 corresponds to the line segment parallel to the s-axis in 6 2 • Combining the three homeomorphic transformations ~. 8, and w- 1 , and setting the images of A and B to be A and B respectively, we get a homeomorphic mapping between the normal regions G1 and G2. When A is a limit cycle, if we follow the above method of construction, then the left sides of 61 and 62 will become unbounded, and so it is not obvious how to define the transformation between points of A and the points of the corresponding limit cycle A in order for the map between G1 and G2 to be homeomorphic. Now suppose R is a very small open neighborhood of A, whose boundary r (two simple closed curves) is a curve without contact with respect to (I) and (II) (the directions of trajectories crossing it are the same), and the interior of R has a unique limit cycle A of (II). We would like to establish a homeomorphic transformation from R into itself such that the image of A is A, and the trajectories of (I) become the trajectories of (II). On A we measure from a fixed point to a point whose arc length is a, and construct a normal line T/ai the points of intersection of trajectories of (I) with T/a are arranged in order. Let i = ±1, ±2, ... represent the points of intersection of TJa with every trajectory 1b (exterior of A) or 1b' (interior of A) in this order. Here the first point of intersection is on r, the positive integer i represents the point of intersection of 1b and TJa, and the negative
THEORY OF LIMIT CYCLES
186
"'a·
integer -i represents the point of intersection of 1b' and Thus in R U r all the points except A can be represented by (a,b,i) and (a,b',-i), where i is a positive integer. Suppose we have defined a homeomorphic transformation T 1 of B \ R( 5 ) into itself, which changes trajectories of (I) into trajectories of (II), and T1 on the boundary r of R determines the transformation
(a, b, 1) -+ (
(a,b,i)-+ (n,,P(b),l), (a,b',-i)-+ (n,w(b'),-z), where n =
!{1 + 1/i)
1. represents the ith point of intersection of the line segment of TlA outside A
-z
with the trajectory 1,p(b) of (II), and represents the ith point of intersection of the line segment of inside A with the trajectory 1w(b') of (II). First we can see T2 is a one-to-one transformation. Otherwise, suppose a2 > a1 are such that the points (at. b, i) and (a2, b, i) correspond to the same point (n, ,P(b), 1.); then (1 + 1/i)[
"'a
which contradicts the fact that both
\ R represents the remainder set of R in B.
§8. STRUCTURAL STABILITY
187
The above theorems were extended by M. M. Peixoto [145] in 1962 to the two-dimensional compact differential manifold M 2 • He proved THEOREM 8. 3. The necessary and sufficient conditions for differentiable systems defined on M 2 to be structurally stable are conditions (III) and the follwing condition: 4. The a- and w-limit set of every trajectory can only be a singular point or a closed trajectory.
A more important contribution of Peixoto, however, is another theorem in the same paper.
8. 4. In the Banach space B of differentiable systems on M 2 , the set E of all structurally stable systems is open and dense in B.( 6 ) THEOREM
People who are interested in the theory of limit cycles naturally will consider the following problem: if we limit the functions P and Q on the right sides of the system (I) to a narrower class of functions (e.g. polynomials), will conditions (III) still be necessary and sufficient for structural stability? It is easy to see that if we study polynomial systems(1) on the closed Poincare hemispherical surface 0 (that is, the hemispherical surface with equator), then the equator E often becomes a trajectory connecting two saddle points, and so number 3 in the conditions (III) cannot hold. Hence, the characteristics of structurally stable polynomial systems in 0 are as follows: 1. There are only a finite number of singular points in the interior of 0 and on E, and they are all hyperbolic singular points (that is, elementary singular points of the linear approximate system whose characteristic roots do not possess zero real parts). 2. In the interior of 0 there are only a finite number of closed trajectories; E can also be a closed trajectory when it does not contain a singular point. These closed trajectories are single limit cycles. 3. There is no trajectory connecting two saddle points, except, perhaps, E.
(VI)
Next, in the Banach space of polynomial systems (assuming the degrees of P and Q do not exceed some fixed natural number n), is the set Xs of all the ( 6 )In 1978, Carlos Gutierrez [149] pointed out that M should be assumed to be orientable. (T)The reader will see later that it is essential to proceed in this way. Conversely, in 0 we can only study polynomial systems, since only to such a system can we apply the spherical projection and homogeneous coordinates to map it into n.
THEORY OF LIMIT CYCLES
188
structurally stable systems possessing properties (VI) open and dense? For this problem, we can prove THEOREM 8. 5. In the above notation, Xs is an open and dense subset of X. PROOF. The fact that Xs is open can be obtained at once from the structural stability. We now prove that it is dense. Thus, we introduce some intermediate sets between Xs and X: Xoo = {(I) E X: If (I) has a singular point on E, it can only be a hyperbolic singular point; if E is a closed trajectory of (I), then it must be a single limit cycle}. XH ={(I) E X 00 : (I) only has hyperbolic singular points in 0\ E}. Xc ={(I) E XH: (I) does not have a nonequatorial trajectory connecting two saddle points}. From the definitions, it is easy to see that Xs C Xc C XH C Xoo C X. Hence we have to prove that Xoo is dense in X (see [151]), XH is dense in X 00 , and Xc is dense in XH (see [152]). These are rather easy to prove, and we leave them as exercises. In the following we shall only prove that Xs is dense in Xc. On this point, the proof in [152] is questionable. Take any point in Xc, i.e., a polynomial system, which possesses the properties of (VI) except that it may have a multiple limit cycle r in the finite plane. Now we make a slight perturbation of this system (but this perturbed system should still be a polynomial system of degree ~ n) and eliminate this property; then we obtain a point in Xs. Take the perturbed system to be (P- aQ, Q + aP), where 0 < lal «: 1. When r is a periodic cycle or a multiple cycle of even multiplicity, from §3 we know that when lal > 0 and a takes suitable sign, (P- aQ, Q + aP) will not have a limit cycle in the vicinity off; then our goal has been achieved. However, if r is a multiple cycle of odd multiplicity, then the proof of Lemma 8.4 is of no use, since its perturbed system (V) may fail to be a polynomial system. Hence we have to use other methods to prove this lemma. Let the parametric equations of r be X = cp(t) andy = '1/;(t), 0 ~ t ~ T. In a neighborhood of r we can construct a successor function which can be expanded as a power series in no and a:
,y,( ) k ~ no, a = akon 0 =
+ ao1a + ak+t,on 0k+l + auano + ao2a 2
+ terms of higher degrees (ako + ot(1))n~ + (ao1 + o2(1))o,
§8. STRUCTURAL STABILITY
189
where k is an odd number greater than 1, 01(1) and o2 (1) represent the terms which go to zero as no, a - 0, and ako =F 0 since we have assumed r is a k-multiple cycle. From §4 it is easy to prove that
....
~ ,..(0)! ""'(0) exp ( f (P. + Q,) dt) loT exp (- fo (Px + Qy) dt) !! ds. X
8
I'
Since the family of vector fields (P- aQ, Q + aP) forms a family of rotated vector fields with respect to a, aoI aa ~ 0; therefore aOl "' 0. Hence, if we only take lal and lnol suitably small, the number of roots of 1/J(no, a) = 0 can be completely determined from the equation akon~ + ao1 a = 0. It is easy to see that this equation has a unique simple root no, which corresponds to the unique single cycle of (P - aQ, Q + aP). This shows that in the vicinity of the vector field (P, Q) there always exists a vector field (P- aQ, Q + aP), which has a single cycle in the vicinity of r. Hence Xs is dense in Xc. The theorem is completely proved. For the study of the global properties of differential dynamical systems on differentiable manifolds, after the above-mentioned paper of Peixoto [145], the schools led by Stephen Smale in the United States, D. V. Anosov and V. I. Amol'd in the Soviet Union, Liao Shan-tao in China have done a lot of important work, which has become an important part of the new branch of mathematics known as "global analysis". In recent years, people have used some of these more abstract results to study bifurcation theory and the qualitative theory of concrete ordinary differential equations, which deserve serious consideration. Interested readers can refer to [153]-[158].
§9. Work of M. Frommer and N. N. Bautin Starting now we proceed to study problems of limit cycles and the global structure of trajectories for polynomial systems
dxjdt
= Pn(X, y),
dyjdt
= Qn(X, y).
(9.1)
However, when n is a general positive integer, only fragmentary work has been done, except for a long paper of Henri Dulac in 1923 [4]. Dulac's paper was translated into Russian and published in book form in 1980 [147], and a Chinese translation directly from the French paper by S. W. Ye will appear soon; hence its material will not be presented here, although we shall mention a little of this work either in the text or in the exercises of subsequent sections. Hence from this section on we shall mainly discuss the problems of limit cycles and the global structure of trajectories of quadratic polynomial differential systems (which will hereafter be called quadratic systems): •
dxfdt =
2
L
2
dyfdt =
aikxiyk,
i+k=O
L
bikXiyk
(9.2)
i+k=O
We can say that the study of limit cycles of system (9.2) is the most interesting and inspiring part in the theory of limit cycles. The reason is very simple: the system of first degree (i.e., the linear system) always has a general integral; hence it does not have a limit cycle, and detailed discussions of the qualitative state of its trajectories have been presented in textbooks on ordinary differential equations. Hence (9.2) is the simplest nonlinear system. Generally speaking, except for some special cases, there does not exist a general solution of (9.2); even a first integral cannot be obtained. However, the horizontal and vertical isoclines of (9.2) are all quadratic curves; hence their limit cycles and trajectories can possess some rather special properties, and it is possible to study these curves more deeply; moreover, from this we can derive many general problems worthy of consideration, and solving these problems will undoubtedly promote the development of qualitative theory in 191
192
THEORY OF LIMIT CYCLES
the plane. On the other hand, from the viewpoint of practical applications, there are a large number of problems in engineering, technology, and the natural sciences whose mathematical models can be transformed into quadratic and cubic systems. The readers have already seen some of these in §§5 and 6. Hence the qualitative study of (9.2) has important practical significance. For the quadratic system, the earliest research was to find necessary and sufficient conditions for the coefficients of the system to satisfy in order for it to have a center. Both Dulac [5] and M. Frommer [6] studied this problem; in [6] not only were necessary and sufficient conditions obtained, but also the figures of the corresponding trajectories were drawn. Finally, in [159], the figures of the trajectories in the projective plane for all the cases involving the appearance of a center were drawn. In this section, we shall narrate this problem mainly following [6], but we also adopt the method in [5] of obtaining the first integral. Dividing the two equations in (9.2) and eliminating dt, we rewrite (9.2) as dy x + ax 2 + (2b + a)xy + cy 2 (9.3) dx = - y + bx2 + (2c + {J)xy + dy2 ' which is permissible for the problem of studying its center. Since (9.2) takes the origin (0, 0) as its singular point, if its right sides do not have linear terms, or even if they have linear terms but the characteristic roots of the linear parts are not pure imaginary, then (0, 0) cannot be a center; when the characteristic roots of the linear parts of (9.2) are a pair of pure imaginary roots, then we can make (9.3) possess the above form through a linear transformation of x andy. I. If a = {J = 0, then (9.3) can be integrated to obtain a family of cubic algebraic curves:
~(x 2 + y2 ) + ix3 + bx2y + cxy2 + ~y 3 = const.
(9.4)
For the figures of the family of curves (9.4), we can classify them according to the number of their asymptotes and the number of singular points of (9.3). For example, in Figures 9.1-9.4 the families of curves all possess only a real asymptote; Figure 9.1 has one real singular point other than the origin but Figures 9.2 and 9.3 have three different real singular points other than the origin, and equation (9.3) at the same time has two centers. By continuous deformation we can obtain Figure 9.4 from Figure 9.2, where Figure 9.4 has a double singular point and two single singular points. In Figure 9.5 there is a triple singular point other than the origin, which can be seen from the equation dy dx
=
x+xy
(9.5)
§9. WORK OF FROMMER AND BAUTIN
FIGURE 9.1
FIGURE 9.3
FIGURE 9.2
FIGURE 9.4
where (0, -1) is a triple point of intersection of y + ~x 2 + y 2 = 0 and x(1 + y) = 0. Figures 9.6-9.8 all have three real asymptotes, and the boundaries of the families of closed curves surrounding the center have three saddle points, two Saddle points and one saddle point respectively, but there are three saddle Points in each figure. Let the saddle point on the lower-left corner of Figure 9.8 go to infinity; then two asymptotic directions coincide, and the line at infinity is a double asymptotic direction, on which two elementary singular Points of index +1 and the saddle point which goes to infinity coincide to become a singular point of higher order and index +1; then we obtain Figure 9.9.
193
194
THEORY OF LIMIT CYCLES
FIGURE 9.5
FIGURE 9.6
FIGURE 9.7
FIGURE 9.8
FIGURE 9.9
FIGURE 9.10
§9. WORK OF FROMMER AND BAUTIN
195
Figure 9.10 is the case when three asymptotic directions coincide, and this case can be realized from dyjdx
= -xj(y- y 2).
(9.6)
II. Ia: I+I.BI =/:- 0. According to the well-known method of computing the first focal quantity of the singular point (0, 0) of (9.3) (see [67], Part II, Chapter 2), we get (9.7) D1 = [a:( a+ c)- ,B(b +d)].
i
Hence, in order to have a center, we must have a(a+ c)= ,B(b +d).
(9.8)
In the following, we discuss three cases. Il1. a+ c = b + d = 0. It is easy to prove that this property is invariant under the rotation of axes. We rotate the coordinate axes by an angle
a= acos3
(9.9)
hence we can always select
=
x +a'xy Y + b'x2 + c'xy- b'y2.
(9.10)
From this we can see that this equation has another singular point (0, 1/b') besides (0,0), and the integral line y = -1/a'. H there are other singular points, they should lie on this line. Now we apply the transformation ~ x=--
71
1 + b'77'
y = 1 + b'77'
(9.11)
in order to move the singular point (0, -1/b') to infinity. Thus (9.10) becomes d77
de
e[1 +(a'+ 2b')77 + b'(a' + b')77 2 ] =77[1 + b'(a' + b')~ 2 ]
c'e-
(9.12)
We can obtain a general integral of (9.12) by separation of variables, and, generally speaking, it has four integral lines, one of which is 1+b'77 = 0. Hence (9.10), generally speaking, has three integral lines, but when c'2
+ 4b'(a' + b') < 0
there is only one. Of course, the line at infinity can be considered as the fourth integral line of (9.10).
196
THEORY OF LIMIT CYCLES
FIGURE 9.12
FIGURE 9.11
Integrating (9.12), we obtain a general integral in the form of a power series
Returning to the (x, y) coordinates, we get !{x2 + y2 ) - ![c'x3
-
3b'x2 y +(a'- b')y3 ] + · · · = const.
(9.13)
This power series has a nonzero radius of convergence; hence (0, 0) is a center because the locus of (9.13) in the vicinity of (0, 0) is a closed curve. Of course, the general integral of (9.12), and then (9.10), can be represented by a finite combination of power functions, inverse trigonometric functions and exponential functions. Hence it is possible to determine the figure for the global structure of the integral curves. The family of integral curves can be classified according to the number of integral lines, and whether there are two or three integral lines which coincide. Figure 9.6 (which can appear as the figure of both case I and case ITt) represents the case when the three integral lines are distinct;(!) Figure 9.11 represents the case when there exist two integral lines which coincide;(2} Figure 9.12 represents the case when three integral lines mutually coincide; and Figure 9.13 represents the case when all the integral lines coincide at infinity (i.e., a'= b' = c' = 0) and the family of integral curves is a family of concentric circles. H there is only one real integral line at infinity (that is, the previously mentioned case c'2 + 4b'(a' + b') < 0}, then we have Figure 9.14. (1 )All the figures in this section can be considered as projective figures, in which every line can be treated as a line at infinity. (2)The original figure of Frommer is wrong; we follow A. N. Berlinski1 [160] for its correction.
§9. WORK OF FROMMER AND BAUTIN
197
FIGURE 9.14
FIGURE 9.13
lh. a+ c = 0, b + d # 0. Here we must have {3 = 0. Now if a= c = 0, then (9.3) can be rewritten as x + o:xy = - y + f3x2 + "YY2 .
dy dx
(9.14)
The plane vector field determined from this equation is symmetric with respect to the y-axis. This case can also be further divided into several different cases, but these will be discussed in detail in the following case III. Now suppose a= -c # 0. Then we can use a similarity transformation to make a = -c = 1. Thus (9.3) becomes
dy dx =
x + x 2 + (2b + o:)xy- y 2 y + bx2 - 2xy + dy2
(9 .15)
The second focal quantity of the focus of (0, 0) can be computed as
(9.16) since b + d # 0. To make D2 = 0, we must have o: = 0 oro:= -5(b + d)1r. The case o: = 0 (we already know that {3 = 0) has previously been discussed. Hence we assume o: = -5(b +d). Then (9.15) becomes
dy dx =
x
+ x2 -
(3b + 5d)xy- y 2 y + bx2 - 2xy + dy2
(9 .17)
Next we compute the third focal quantity of the focus as
D3
= (b + d) 2(bd + 2d2 + 1).
(9.18)
Since b + d # 0, the necessary condition for existence of its center is
bd + 2~ + 1 = 0,
or
b = - 2d2d+ 1 .
(9.19)
THEORY OF LIMIT CYCLES
198
Here we can assume d :f:. 0, since when d = 0, to make Da = 0 we must have b = 0, but we have already assumed b + d :f:. 0. In the following we use Dulac's method [5] to prove that when D 1 = D 2 = D 3 = 0, (0, 0) is a center. Rewrite (9.17) as [x + x 2
-
(3b + 5d)xy- y2 ] dx + [y + bx 2
2xy + d1J2J dy = 0,
-
(9.20)
and then apply the transformation 1( I I) X= 2i X - y '
y
1
= 2(xl + yl),
(9.21)
to obtain the equation
b+ d b + d 12] d [12y + (b +3d 1) X12 + -4-X y X I
1 1
- 4 - - 2i
+ [!2 X
I _
b + d l2 b + d 1 1 2 X + 4 Xy +
I
-2-y
(
b + 3d 4
.!..) y12] dy - 0•
+ 2i
1 _
Again we apply the transformation I 2 X =-b+dX1,
to obtain
2
2i [Y1 - b +b3d ++ d X1 - X1Y1
+ [x1 + 2x 2 1 -
X1Y1-
+ 2y2] 1
dx1
2]
b + 3d - 2i b + d Y1 dy1 = 0,
or simply [Yl where
+ 2y~ -
X1Y1 +ax~] dx1
+ [x1 + 2x~ -
b+ 3d+ 2i a=- b+d
X1Y1 +by~] dy1 = 0,
b- b+3d- 2i -
-(b+ d) .
(9.22) (9.23)
Making use of condition (9.19), we can easily prove that a and b satisfy the condition ab = 1. (9.24) Moreover, it is easy to prove that equation (9.22) under condition (9.24) has an integrating factor [/(xl, yt)]- 512 = (1 + 2xl
+ 2yl +ax~+ 2X1Y1 + byn- 5/ 2 ,
and the general integral (9.25)
§9. WORK OF FROMMER AND BAUTIN
199
where
F(x1, yt) = 1 + 3(xl
+ yt)
+~[(a+ 1)x~ +(a+ b + 2)x1y1
+ (b + 1)y~] +! [a(a + 1)x~ + 3(a + 1)x~y1 + 3(b + 1)x1y~ + b(b + 1)yf]. Finally, using the transformation X1
= -b +2 -d( -zx. -
y ),
b+d(. Yl = - zx-y ) 2
to return to the original variables x andy, we see at once that (9.17) has a general integral
[tf + 2d(d2 + 1)y +(if+ 1)(x- dy) 2 ] 3 = C[d2 + 3d(d2 + 1)y + 3d2 (d 2 + 1)y2 - 3d(d2 + 1)xy + (d 2 + 1)(dy- x) 3 ] 2 •
(9.26)
Hence the origin is a center, and the family of integral curves is a family of algebraic c1,1rves.( 3 ) Ila. a+ c # 0, b + d = 0. This case is the same as Ih. The condition for getting a center is
b + d =a= .B + 5(a +c)= ac + 2a2 + d2 = 0. III. The case of a symmetric vector field (9.14). In this case the equation has the following special integral curves: 1) the line at infinity; 2) the line y = -1/a; and 3) the quadratic curve x 2 = Ay2 + 2By + C, where
"Y-a-.8 , a+ .B (a+ .B)( a+ 2.8) C-"Y+a+.B - .B(a +.B)( a+ 2.8)"
"Y A=---,
B=
The singular points are (0, 0) and P(O, -1h) on the axis of symmetry (yaxis), and possible points of intersection of the integral lines and the quadratic integral curve. The figures of the family of integral curves are determined mainly from the following three quantities: i) {a- "Y)h: when this is greater {less) than 0, Pis a center {saddle point); when it is equal to 0 or oo, P is a singular point of higher order; ( 3 )The
global phase-portrait of this case can be found in [159].
THEORY OF LIMIT CYCLES
200
FIGURE 9.15
FIGURE 9.16
FIGURE 9.17
FIGURE 9.18
FIGURE 9.19
ii) (o:- "t)/{3: when this is greater (less) than 0, y = -1/o: has (does not have) real singular points; when it is zero, two real singular points coincide, i.e., the integral line and the quadratic integral curve touch each other tangentially; iii) "1/(o: + [3), which decides the type of the quadratic curve: when it is greater (less) than zero, it is an ellipse (hyperbola); if it equals oo, it is a double integral line y = -1/ o:. The figures of the family of integral curves are Figures 9.12, 9.7, 9.3, and 9.15-9.19.
§9.
WORK OF FROMMER AND BAUTIN
201
IV. a+ c # 0, b + d # 0, but (9.8) still holds. In this case we can rotate an angle tp so that a+c 1 tantp = - - - = - b+d k' and require that k satisfies a' = ak 3
-
(3b + a)k 2 + (3c + {J)k- d = 0.
Thus it is easy to see that the equation after rotation of axes has a' = c' = {31 = 0; that is, it takes the origin as its center. Summarizing the above results, we get THEOREM 9 .1. Equation (9.3) has a center if and only if one of the following conditions holds: 1) a= (3 = 0. 2) a+ c = b + d = 0. 3) a=c=f3=0 (orb=d=a=O). (9.27) 2 4 4) a+ c = {3 =a+ 5(b +d) = bd + 2~ + a = 0, but b + d # 0( ) (orb+ d =a= (3 + 5(a +c)= ac + 2a2 + ~ = 0, but a+ c # 0). 5) a/(3 = (b +d)/( a+ c)= k, ak 3 - (3b + a)k 2 + (3c + (3)k- d = 0. REMARK. By Theorem 9.1 and the above several figures we know that a quadratic system can have at most two centers; when it has one center, although it may have a focus (Figure 9.14), it cannot have a limit cycle in the vicinity of tl:re focus.( 5 ) V. T. Borukhov [161] gave a concrete example to show that for a quadric polynomial system a center and a limit cyc'le can coexist. His equations are dyjdt =ax- x 3 ,
(9.28)
where a = 2A + A 2 and 0 < A < 5 · w- 5 • He proved that (9.28) takes 0(0, 0) as its center, and there exists limit cycles in the vicinity of each of the two foci (±ya, a 2 j(a-A)). V. M. Dolov [162] obtained the result that for a cubic polynomial system, a center and a limit cycle can also coexist. His equations are dxjdt = y- y3 , (9.29) where IJ. < 0. He proved that this system takes (0, 0) as its center, but the foci (y'=JL,±l) change their stability when IJ. varies and passes through -1; hence their neighborhoods can also have limit cycles (see Theorem 3.7 in §3). ( 4 )Note ( 5 )The
that before we derived condition (9.19) we changed a to 1. rigorous proof is given in [159].
THEORY OF LIMIT CYCLES
202
The conditions for existence of a center for quadratic systems, besides (9.27), can also be given in the form introduced by N. N. Bautin. For this we write the quadratic system in a form different from (9.3):
dx/dt = A1X- y- A3X 2 + (2>.2 + As)xy + A6y 2, dy/dt = x + >.1y + A2X 2 + (2A3 + A4)xy- >.2y 2.
(9.30)
Here 0(0, 0) is a coarse focus when A1 =I 0, but in the second equation the coefficient of x2 and the coefficient of y2 only differ by a sign. From this it appears that (9.30) is more special than (9.3). In fact, (9.2), after a suitable coordinate transformation, can always be changed to (9.30) provided that (9.31) that is, 0(0, 0) is a focus or a center of (9.2). For this we can first use the transformation
1 x = - -bb [(a10- a1)17 + b1~], 1 10
1 y = --TJ bt
(9.32)
to change it to
d17jdt1 = bt~ + a111 + B2oe + Bu~TJ + Bo217 2, d~jdt1 = a1~- b111 + A2oe + Au~TJ + Ao211 2,
(9.33)
where a1 ± ib1 (b1 =I 0) are characteristic roots of the linear approximate system of (9.2). Then, after a suitable rotation of axes which leaves a1 and b1 unchanged and makes the sum of the new B2o and Bo2 equal to zero, on dividing both sides of (9.33) by b1 and making t = b1t1. we can rewrite (9.33) as (9.30). For (9.30), we can easily deduce from Theorem 9.1 THEOREM 9. 2. System (9.30) takes the origin as its center if and only if at least one of the following four conditions holds: 1) A1 = A4 =As = 0. 2) A1 = A3 - A6 = 0. 3) A1 = A2 = As = 0. 4) At = As = >.4 + 5A3 - 5A6 = A3A5 - 2A~ - A~ = 0. An important contribution of Bautin [21] is THEOREM 9. 3. When 0(0, 0) is a center of (9.30), after a slight variation of its coefficients there exist at most three limit cycles in the vicinity of 0; for a center of type 4) it is possible to generate three limit cycles.
§9. W
203
For the proof, we introduce the polar coordinates x = p cos cp, y and get from (9.30) dpjdcp = p{A1 + p[->.a cos3 cp + (3>.2 + >.s) cos 2 cpsin cp
= p sin cp,
+ (2>.a + >.4 + >.6) cos cp sin 2 cp- >.2 sin3 cp]} · {1 + p[>.2 cos3 cp + (3>.3 + >.4) cos2 cp sin cp - (3>.2 + >.s) cos cp sin 2 cp- >. 6 sin3 cp]} -l = p[>.t + pA(cp)]/[1 + pB(cp)]
(9.34)
Rt = >.1, R2 = A(cp)- >.tB(cp), Ra = -A(cp)B(cp) + >. 1 B 2 (cp), ... , Rk = (-1)kA(cp)B(cp)k-2 + (-1)k-1>.tB(cp)k-1,
(9.35)
where
} ....
For all the >.i in the vicinity of a fixed point>.; in the >.-space (for example, all the >.i satisfying ll>.i- >.;II < e) and for all cp, there always exists a "f(e, >.i) such that when IPI < "((e, >.i), the series on the right side of (9.34) converges. It is well known that the solution p(cp) of (9.34) satisfying the initial condition p(O) = Po can be expanded as
p = PoVt (cp, >.i) + p~v2( cp, >.i) + pgva( cp, >.i) + · · ·, (9.36). where vk(cp, >.k) satisfies the conditions v 1(0, >.i) = 1 and vk(O, >.i) = 0 when k 2:: 2. Substituting (9.36) into (9.34) and comparing the coefficients of like powers of p, we can determine the equations of all the Vk: dvtfdcp = VtRt. dv2jdcp = v2R1 + vf R2, } dvafdcp = vaRt + 2v1v2R2 + vrRa, ....
(9.37)
For all sufficiently small p and all >.i satisfying ll>.i- >.ill < e, some segment of the line cp = 0 is a segment without contact. Letting cp = 211" in (9.36), we can obtain a successor function P(Po, 2rr) = povt(211", >.i) + p~v2(21r, >.i) (9.38) in a sufficiently small line segment 0 < p < p1 , cp = 0, within the radius of convergence of the power series. LEMMA 9.1. vk(211", >.i) is an integral function of all >.i and is a homogeneous polynomial of degree k- 1 for >.2, ... , >.6 when >.1 = 0. PROOF. From (9.37) and the theorem for dependence of a solution on the parameters, we know that vk(21r, >.i) is an integral function of all >.i. In
204
THEORY OF LIMIT CYCLES
particular, if >.1 = 0, then Rt = 0, and Rk is a homogeneous polynomial of degree k- 1 of >. 2, ... , >.6. Since at the same time every equation of (9.37) is separable, vk(21r, >.i) is also a homogeneous polynomial ?f degree k- 1 of ).2,····).6· LEMMA
9.2.
Vt(2rr, Ai) = e 2 ,..~ 1 , v2(2rr, Ai) = >.10~ 1 ) 1 v3(2rr1 Ai) = v3 + >.10~ 1 ), v4(2rr1 Ai) = v39i3>+ >.tOi1 >, vs(2rr, >.i) =tis+ v3(}~3 > + >.10~ 1 ) 1 v5(2rr1 >.i) = tls9~5 ) + v30~3 ) + >. 1 9~ 1 ), \ ) - +-V5 (J(S) +-V3 0(3) + .1\1 \ 0(1) V7 (2 1r 1 Ai = V7 7 7 7 1 vk(2rr1>.i) = v19r> + vsOks) + v30k3>+ >.t0k1}
(9.39)
(k
> 7),
where
= -i>.s(>.3- >.6)1 } tis = ~ >.2>.4(>.3 - >.6)(>.4 + 5>.3 - 5>.6), 117 = -~~rr>.2>.4(>.3- >.6) 2(>.3>.6- 2>.~- >.~),
V3
and
oii>
(9.40)
is an integral function of >.i.
Vt(2rr,>.i) can be obtained at once from (9.33). According to Lemma 9.1 1 we can write PROOF. The formula for
vk(2rr, >.i) = vi0 > + >.t0k1)
(k > 1),
where vi0 ) is a homogeneous polynomial of degree k - 1 of all the Ai (i == 2, ... , 6), without containing >.1. From the fourth condition for existence of a center in Theorem 9.2, we know that vi0 ) = 0 when >.s = A4 + 5>.3 - 5>.6 == >.3>.6- 2>.~ - >.~ = 0. Hence
vk(2rr, Ai) = (>.3>.6- 2>.~- >.~)()~'
(9.41)
+ (>.4 + 5>.3 - 5>.6)9~ + .As9A: + .At9k1),
but when >. 1 = .As = >.2 = 0 and .At = >. 5 = A4 = 0, a center also appears; that is, the vk(2rr, Ai) (k -#1) should all be equal to zero, and so
We then use the second condition for the existence of a center, >.1 0, and get -
-(7)
Ok = (>.3 - .A5)0k ,
-.
(5)
Ok = (>.3 - .Aa)Ok ,
= A3 -
9A: = (>.3 - .Aa)Ok(3) .
>. 6 ==
§9. WORK OF FROMMER AND BAUTIN
205
Substituting all this in (9.41) yields vk(27r, .xi) = (..\3..\s- 2.x~ - .x~)..\2..\4(..\3 - .Xs)ofl
+ (..\3- ..\s)(..\4 + 5,\3- 5..\s)..\2,\40k5 ) + ..\s(..\3 - ..\s)Ok3l + ..\10k1l (k :/: 1). Now suppose k = 2. Then from Lemma 9.1 we know that ofl 0~3 ) = 0 on the right side of (9.42); hence
(9.42)
= 0~5 ) =
V2(27r, Ai) = ,\10~ 1 ). When k compute
= 3,
as before, we have ofl ::::: 0~5 )
:::::
0, and from (9.37) we can
hence
+ ..\ 1 (}~1 ) = v3 + ,\ 1 0~ 1 ). When k = 4, as before, we know that Oi7 l :::::: oi5 l = 0, and hence v4(21r, Ai) = ..\s(,\3- ..\s)Oi3l + >.10i1l :::: v30i3l + >. 10i1l, v3(21r, Ai) = ( -71"/4)..\s(..\3- ..\s)
where the coefficient of 1!3 can be denoted by 0i3). We then compute vs(27r, Ai)· From Lemma IU we have note that
ofl = 0.
Next we
= R1vs + R2(2v2v3 + 2v1~1 ) + R3(3v1v~ + 3v~v3) + 1R 4 v~v 2 + R 5 v~. then R1 = 0 and v1 0; thus
dvsfdtp If we let ..\1 = 0,
=
dv 5 jdtp = 2R2v2v3 + 2R2v4 + 3R3v~ + 3~~'!13 + 4R4v2 + R5 .
(9.43)
Note that here we have dv2jdtp
= R2,
dv3jdtp
=
dv4jdtp = R2(v~ + 2v3) + 3~~p3 + R 4; hence the right side of (9.43) can be written a&
R3 v3
+ 2R4v2 + Rs + 4v~ ~v2 + 2v3 dv3 '1-tp
dtp
2d(v2v4) _ dtp
3d(v3v~) dtp
.
THEORY OF LIMIT CYCLES
206
Integrate both sides of (9.43) from 0 to 271", and take .As :;::; 0. Since under the condition .A1 =.As = 0 we have
it follows that vs(27r, .Ai) =
1 2
1r
(R3v3
+ 2R4v2 + Rs) dcp.
Using this formula to compute vs(27r, .Ai), we get (when .At = .As = 0) vs(27r, .Ai) = (7r/24).A2.A4(>.3- >.6)(>.4
+ 5.A3- 5.A6)·
Hence for >.1 :f. 0 and .As :f. 0 we have vs(27r, .Ai) = (7r/24)(.A2.A4(.A3- .A5)(.A4
+ 5>.3- 5.A6)
+ .As(>.3 - >.6)8~3 ) + .A18P)
-- -Vs +-V3 (}(3) \ ()(1) s + "1 S · For k = 6, note that by Lyapunov's classical theory the first nonzero vk(21r, Ai) (k > 1) must have odd subscript, and when .At = .As = >.4 + 5).3 - 5>.6 = 0 we have V2 = ... = Vs = 0; hence ofl = 0 by (9.42). Thus v6(21r, .Ai) = vs8~s)
+ v3 8~3 ) + >. 1 0~ 1 ).
Fork= 7, from (9.42) we have V7(21r, Ai) = .A2>.4(A3 - >.6)(>.3>.6 - 2>.~+-V3 ()(3) + 1\1 \ ()(1) + -Vs (J(S) 7 7 7 • As before, we let >.1 = >.s = >.4 we can determine that
+ 5>. 3 -
.A~)()fl
5.A6 = 0 to compute v7(21r, .Ai)· Thus
(}7(7) = -(25/32)11" ( >.3- >.6)· Hence '\ ) =V7 V7 (2 11",1\i
+-Vs (J(S) +-V3 ()(3) + V1 ()(1) 7 7 7 •
Finally, we prove the representation formula for vk(27r, .Ai) for k > 7. By (9.42) it suffices to show that ofl contains a factor >. 3 - >. 6 . We know that efl is a homogeneous form of degree k- 6 of >. 2, ... , >. 6, and, making use of the indeterminate nature of the forms of efl, ois), and (Ji3), we can always move the terms involving >.4 and >.s of efl to oi5) and oi3), so that ei7 ) does not contain >.4 and >.5. In this way we can prove that, under the conditions
§9. WORK OF FROMMER AND BAUTIN
or)
At =As = A4 + 5A3 - 5A6 = 0, contains a factor J.t = A3 same time, since A4 = -5(A3 - A6), we have
207 -
A5.( 6) At the
and
(9.44)
=
Hence this problem turns into the problem of proving that 1/J(po, 0) 0 on some line segment of rp = 0. The proof of this fact is very difficult, and the method used by Bautin deserves our careful attention. Consider the system of equations
dx/dt
= -H~(x,y) + J.tp(x,y),
dyjdt
= H~(x, y) + JJ-Q(X, y),
{9.45)
where
When h > 0 is very small, H(x, y) = h represents a family of closed curves near the origin which is first integral of
dx / dt = -y - A5X 2 + 2A2XY + A6Y 2, dyjdt =X+ A2X 2 + 2A6XY- A2y 2
(9.47)
0
(9.47) can be obtained from system (9.30) by letting At = As = A3 - A6 = ..\4 = 0. p(x, y) and q(x, y) are polynomials in x and y. If we take p = -x 2 and q = -3xy, we immediately obtain (9.30) with At = As = A4 + 5A3 - 5A6 = 0. Now in the region of convergence of the series (9.36), we can take a curve Cho from the family H(x, y) = h, and move the coordinate system to the curve Cho and use the curvilinear coordinates (8, h). Here his a parameter corresponding to the curve of the family H(x, y) = h, and 8 represents the periodic coordinate on ch such that d8/dt = 1 on cho; but on ch we have dsjdt = Th 0 /Th, where Th is a period of Ch, Tho is a period of Cho and 8 is measured from the x-axis. From this we can see that any ch in the vicinity of Cho has a fixed period Tho, and the transformation of coordinates
x=ft(s,h), is analytic. When h
(9.48)
= ho, we get the equations of Cho, denoted as x = /t(s, ho) = rp(s),
-(7)
y = h(s,h)
Y = h(s,ho)
= 1/J(s).
(9.49)
(6)1£ (}k contains a factor A4, then when A4 + 5A3 - 5A6 = 0 we can produce a factor >. 3 - A6 from A4, although A3 - A6 originally is not a factor of iifl .
THEORY OF LIMIT CYCLES
208
Using the relation formulas H~fL~
H(ft(s, h), h(s, h))= h, H~f~h
+ H~f~a =
0,
+ H~f~h = 1,
we obtain the equation (9.50) in the new coordinate system. In this equation we let h = ho
+ 6, and obtain
d6 /ds = R(6, s, J.L);
(9.51)
the right side of this equation is analytic in 6 and J.L when 161 and IJ.LI are very small, and is also an analytic periodic function of s. Expand the solution 6 = ,5(s) of (9.51) in a power series in J.L and its initial value 60 = 6(0), and denote it as
6 = Cw(s)8o + Co1(s)J.L + C2o(s)85
(9.52)
+ Cu(s)8oJ.L + Co2(s)J.L2 + ···.
In this formula we lets= T (= Th 0 ), and note that Cw(r) = 1 and Cko(r) = 0 for k 2:: 2, and we obtain at once the successor function on the x-axis:
8(r) = 6o + Co1(r)J.L + Cu(r)8oJ.L + C 0 2(r)j£2 + · · ·. LEMMA 9. 3. 1/J(po, 0)
(9.53)
=0 is equivalent to Co2 (r) 2=0, which in turn shows
that no matter how Cko is chosen, the coefficient of J.L in the expansion (9.53) is always equal to zero.
PROOF. Let p0 denote the abscissa of the point of intersection of Cho and
cp = 0. Expanding the right side of (9.44) in powers of Po- p0, we obtain
p- Po= J.L 2[1/J(p0,o) + .,p~o (p0,O)(po- p0) + J.L'I/J~(p0 , o) + .. ·].
=
(9.54)
Since p0 and p0 are arbitrary, .,P(po, 0) 0 is clearly equivalent to .,P(p0,0) Now we prove that the latter identity is equivalent to Co2(r) 0. From the equation H(p, 0) = ho + 8 we get !P2 + -!A2P3 = ho
=
=0.
+ 8.
(9.55)
hO·
(9 · 56)
When 8 = 0 and p = p0, we have 1 •2 2Po
+ gi\2Po 1\ •3 -
Subtracting (9.56) from (9.55), we get {j
= .\32 (p- Po)3 + ~(1 + 2.\2p'Q)(p- p'Q)2 + (p- p'Q)(p'Q + A2Po2).
(9.57)
§9. WORK OF FROMMER AND BAUTIN
209
Let p = po, and get
t5o = >.32 (Po- p(i) 3 +
~ (1 + 2>.2p())(Po- p(i) 2 +(Po- p(;)(p(; + >.2Po 2).
(9.58)
Now subtracting (9.58) from (9.57), we get
t5-t5o= ~2 (p-po) 3
+ [~(1+2>.2Po)+>.2(Po-Po)] (P-Po) 2 (9.59)
+ [(p(j + >.2p() 2) + (1 + 2>.2p())(Po- Po) + >.2(Po - Po) 2](p- Po).
Substituting (9.54) in the right side of (9.59), we get a power series expansion of t5 - t5o in terms of po - p0 and J.t. But this expansion can also be obtained by substituting (9.58) into the right side of (9.53). Comparing the coefficient of J.t2 in these two power series expansions, we get
(p0+ >.2Po2)1/J(p(),O) = Co2(r). From this we at once get the proof of the lemma. In the following we shall prove that Co2 (T) 0. Expanding the right side of (9.51) in power series of t5 and J.t, and noting that
=
R(O,s,O) = R6(0, s,O) = RZ~(O,s,O) = · · · = 0, we obtain
~! =R~(O,s,O)J.t+2 [nz~'(O,s,O)t5J.t+~R~2(0,s,O)JL 2 ]
+ .. ·.
Substituting (9.52) into both sides of the above formula, and comparing the coefficients of like powers of t50 and J.t, we obtain
Cw = 0,
Cot = R~ (0, s, 0),
Cu = 2CwRZ~' (0, s, 0),
C2o = 0,
Co2 = 2CotRZ~' (0, s, 0) + R~2 (0, s, 0),
... '
(9.60)
where Cii represents the derivative of Cii with respect to t. It is clear that has the initial conditions
cij
Cw(O) = 1,
Cij(O) = 0 when (i,j) =P (1,0).
(9.61)
From (9.50) it is easy to compute R~(O, s, 0)
= ~P- tj:;q,
R~2 (0, s, 0) = 2(~p- tj:;q)(f~hP- f~hq) lo=O'
RZ/'(0, s, 0)
=
! (!~
. . P-
f~ . . q) IJ'=O=O
= [J;hsP- f~1hsq
+ f~..,(p~J~h + P~f~h) - f~ ... (q~f~h + q~f~h)lo=O·
(9.62)
THEORY OF LIMIT CYCLES
210
From (9.60)-(9.62) we get
C10(s)
= 1,
Co1(r) =loT (-Jp- cpq) ds,
Cu(r) = loT 2RZil ds =
2[J~hP- f~hq](i + 2 loT[- J~h(P~f~s + P~f~s) + f~h(q~f~s + q~f~s) + f~s(p~f~h + P~f~h)- f~s(q~f~h + q~f~h)]8=0 ds
= 2 loT
U~sf~h- f~hf~s)8=0(P~ + q~) ds, (9.63)
But
' !'lh (!2s
-
!'2h !'ls )8=0 =
(!'lh dy , dx) dt - f2h dt Jl=c5=0
= (H~f~h + H~f~h)8=o and so
Cu(r) = 2 loT (p~
=1,
+ q~) ds.
(9.64)
Finally,
Co2(r) = 2 loT (los (-Jp- cpq) ds)
[f~1hsP- f~'hsq + f~s(P~f~h + P~f~h) - f~s(q~f~h
+ q~f~h)]8=0 ds
+ 2 loT (-Jp- cpq)(f~hP- f~hq)8=o ds = 2 loT
(foB (-Jp- cpq) dS) · [:s
U~hP- f~hq) + U~sf~h- !~hf~sHP~ +. q~)L=o ds
+ 2 loT (-Jp- cpq)(/~hp- ~~hQ)8=0 ds = 2 loT [los (p-J - cpq) ds]
(p~ + q~) ds
+ (f~h(r)p- f~h(r)q)Col(r).
(9.65)
Now we pick p and q so that system {9.45) still takes the origin as its center; then all the coefficients of the successor function should be zero. From this we can obtain some identities, and using these we can prove Co2 (r) 0. 1. Take p 0 and q = cp ·1/J = x · y. Then system (9.45) corresponds to (9.30) under the condition -" 1 = A3 - -"6 = 0; hence the origin is a center.
=
=
§9. WORK OF FROMMER AND BAUTIN
211
Then from (9.63)-(9.65) we get
C01 (r)
=-loT xyxds = 0,
C 11 (r) = 2
C02 (r)
=-loT (x loa xyxdt)
ds = 0.
loT xds = 0, (9.66)
=
2. Take p = xy = rp · '1/J and q 0. System (9.45) still corresponds to (9.30) under the condition At = A3- A6 = 0. As before,
Cu(r)
= 2 loT yds = 0.
(9.67)
3. Take p = -x2 = -rp2 and q = 2xy = 2rp'I/J. Then system (9.45) corresponds to (9.30) when At =As = A4 = 0. Hence the origin is a center. From this we obtain Cot(r)
=-loT (x iJ + 2xyx) ds = 0. 2
Using the first formula of (9.66), we get
loT x
2
y ds =
0.
(9.68)
4. Take p = -x 2 and q = -3xy. We get the case A1 = As = A4 +5A3 -5A6 = 0. But now the origin is not necessarily a center; hence it is not certain whether Co2 (T) 0. Now we use formulas (9.66)-(9.68) which we have already obtained, and the obvious identities
=
loT iJ ds = loT x ds = loT x x ds = loT xx ds = loT yiJ ds =loT (xy + xiJ) ds = 0 2
(9.69)
=
to prove C 0 2(r} 0. First we note that by (9.66) and (9.68) we know the right side of (9.65}:
Cot(r)
=loT (-x iJ + 3xyx) ds = 0. 2
lienee from (9.65} we get
Co2(r) = -5
loT [x lo 8(3xyx- x 2y)dt] ds.
Dsing the third formula of (9.61), we see that
~Co2(r) =loT (x los x2 ydt)
ds.
(9.70)
THEORY OF LIMIT CYCLES
212
But
loa x2iJ dt = [x y] 0- 2 loa xyx dt, 2
and so the right side of (9. 70) becomes
loT x 3 yds-x 2(0)y(O) loT xds-2 loT (x los xyxdt)
ds.
By (9.66) the second and third terms of the above expression are equal to zero; hence it suffices to prove that
loT x 3 y ds = 0. We use (9.47) to compute
(9.71)
J; x 2yds and J; x 2x ds, and obtain
loT x 2(x + >.2x2 + 2>.6xy- >.2y 2) ds = 0,
(9.72)
loT x 2 ( -y- >.6x2 + 2>.2xy + A6Y 2 ) ds = 0.
(9.73)
Multiplying (9.72) by >.6, (9.73) by >.2 and adding, we get
2(>.~ + >.~) loT x 3 y ds + >.6 loT x3 ds -
>. 2
loT x 2y ds = 0.
(9.74)
Moreover, using (9.66) and (9.67), we also have
loT xds = ->.6 loT x 2 ds + 2>.2 loT xy ds + >.6 loT y2 ds = 0,
(9.75)
loT iJ ds = >.2 loT x2 ds + 2>.6 loT xy ds- >.2 loT y2 ds = 0.
(9.76)
Multiplying (9.75) by >.2, (9.76) by >.6 and adding, we get
2(>.~+>.~) loT xyds=O. We may as well assume >.~ + >.~ Co2(r) = 0 clearly holds. Hence
=f. 0,
for otherwise the origin is a center and
loT xyds = 0, and, substituting in the right side of (9.75), we get
->.6
loT (x 2 -
y 2) ds = 0.
(9.77)
213
§9. WORK OF FROMMER AND BAUTIN
Moreover, we also have
loT xx ds = - ).61T x 3 ds + 2).2 loT x 2y ds + ).6loT xy 2ds = 0,
r yyds = ~r x .yds + ~r xy ds- lor 2
).2
~
2).6
2
).2
y 3 ds = 0,
faT (xy + xy) ds = -).61T x 2y ds + 2). 21T xy 2ds + ).61T y 3 ds + ..X21T x3 ds + 2).6 loT x 2y ds Eliminating
J; xy 2(
2
ds and
).2
loT xy 2 ds = 0.
J; y 3 ds (using (9.77)), we get
).~ + ,\~) [,\61ar x3 ds -
.\2
loT x 2y ds] = 0.
Using [ ] = 0 in the above formula, we can obtain (9.71) immediately from (9.74). Lemma (9.2) is completely proved. Now we prove Theorem 9.3 stated earlier in this section. By Lemma 9.2 we can write p- Po= Po[211'.\1(1 + A1
+ Pot/Jl(po, Ai))
+ V"g(l + Pot/Jg(po, .\i))P~ + Us(l + PotPs(po, ).i))P~
+ V"7(l + PotP7(Po, Ai))pg), or simply {9.78) where all the 1/Jj are power series of po,(1) and their coefficients are integral functions of all the ).i· These power series converge when I -Xi -..Xi II < e and Po < -y(e, ..x:). Now suppose system (9.30) corresponding to -Xi takes the origin as its center. We prove we can always choose eo and t5o such that when I -Xi- . Xi II < co the equation p - Po = 0 cannot have more than three positive roots in a <5o-neighborhood of the origin; this also shows that the system (9.30) cannot have more than three limit cycles in a !50 -neighborhood of the origin. (7)By virtue of this formula, if Al f. 0, then (0, 0) is a coarse focus, which is equivalent to the divergence being nonzero at (0, 0). If Al = 0 but li3 f. 0, then (0, 0) is a fine focus of first order; if Al = li3 = 0 but tis "! 0, then (0, 0) is a fine focus of second order; if >.1 ""'ii3 = ti 0 = 0 but i/7 f. 0, then (0, 0) is a fine focus of third order. Hence the focus of a QUadratic differential system is at most of third order. This definition will often be used in the sections of §11.
THEORY OF LIMIT CYCLES
214
0
First, we can find c:1 < c: and a1 such that when I!Ai - >.iII < c:1 and < p0 ~ a1 we have 1/Jj 2:: ~ (j = 1, 3, 5, 7). We rewrite (9. 78) as (9.79)
when Po is sufficiently small, every 1/Jj /1/Ji can be expanded into a power series in Po and >.t. and
= 1 + >.1 cp~i) + Pol/J~i) = 1/Jj* (j = 3, 5, 7). Then we take c:2 and a2 such that for I >.i- >.iII < c:2 and 0 < Po ~ 62 we have 1/Jj /1/Ji
1/Jj* 2:: ~, and it is easy to see that when 0 < Po ~ 62, the positive roots of the equation p - Po = 0 are positive roots of 1/Jo = 21r >.1 + 1i31/J3* P~ + 1is1/J5* P~ + 'ih1/J1* pg = 0. Let
1fi =~~~/Po = 2V3(1 + -\1cp~3 ) + Pol/J~~)) + 41is(1 + >.1cp~5 ) + pol/JW)pg + 61:17(1 + -\1 cp~7) + Pol/JW)p~.
(9·80 )
We know that for 0 < Po ~ 62 the number of positive zeros of the function 1/Jo can be at most one greater than the number of positive zeros of tfi. Now we note that the right side of (9.80) and the terms in the square brackets on the right side of (9. 79) have the same form, and their coefficients also possess similar properties; the degree of Po in the latter (the functions in the parentheses are taken as coefficients) is lower than that of the former by two. Hence for 1fi we apply the previous method two more times, and deduce that there must exist a 63 > 0 such that when 0 < po ~ 03 the number of positive zeros of the function tij can be at most two greater than the number of positive zeros of the function =;;J
=
48v7(1 + >.1cp&7) + Pol/J~il).
1fi it is clear that there exist C:4 and 64 such that for all Ai satisfying I!Ai --'iII < C:4 and 0 < Po ~ a4, =;;J does not have zeros; thus for all these Ai
But for
and Po the function 1/Jo has at most three positive zeros. Similarly we can prove that if system (9.30) corresponding to -'i does not take the origin as its center (that is, not all the functions vk(21r, >.:) are equal to zero), then when the first term not equal to zero in the parenthesis on the right side of (9. 78) is >.i, U3, Us or V7 respectively, we can find c:0 and 6o such that for all >.i satisfying ll>.i- >.;II <eo and 0
215
§9. WORK OF FROMMER AND BAUTIN
equation (9.20) does not have any, or has at most one, two, or three, limit cycles in the Do-neighborhood of the origin respectively. Next we prove the second part of the theorem. Suppose the system corresponding to .At takes the origin as its center, and .Ai
= .A5 = ).: + 5).j -
5).6
= ).j.A6 -
2.A6 2
-
.A2 2
= 0,
but .A4 ::/:- 0, .A2 ::/:- 0 and .Aj - .A{i ::/:- 0; that is, the origin is a center of the fourth type in Lemma 9.1. Now we vary .A2 slightly to make ).2 = 0, and make .Aj.A6 - 2.A6 2
-
).~
::/:-
so that here v7 ::/:- 0 (we may as well assume v7
0,
> 0), and
vs = va = 1It = 0. We know that for any e and 8, 0 < e <eo, 0 < 8 < 8o, when 0 < p < 8 and II.Ai- .At II < e, the condition 1/Jj 2:: always holds. Now we vary .A4 to become ). 4 in such a way that va becomes negative, but we still have 1J7 > 0; and we also require lvsl to be so small that for all PoE {8d1),8d2 )) C (8/2,8), the inequality p - Po > 0 still holds. On the other hand, when IPo I is very small, the right side of {9. 78) and v5 have the same sign; hence in (0, 8) the equation p - Po = 0 has at least one positive root. Suppose p = p' is the smallest root among them. Now we vary .A5 to become ). 5 in such a way that va > 0 but Ivai is so small that for all Po E (8p>, 8i 2 )) C (8d 1 ), Od 2 )) the inequality p- Po > 0 still holds, and for all po E (8~ 1 ), 8~ 2 )) C (0, p') the inequality p- Po < 0 holds. Since, for IPol sufficiently small, p- Po will have the same positive sign as va, we know that p- Po has at least two roots in (0, 8); suppose p = p" is the smallest root among them. Finally, vary .Ai to become ). 1 < 0, so that 2 >), the inequality p- Po > 0 holds, 1 ), 1. for all PoE (8~ 1 ), 8~ 2 )) C 2 1 2. for all PoE (oJ•>,oJ >) c (o~ >,oJ 2 >), the inequality p-p0 < 0 holds, and 3. for all PoE (8~ 1 >,8~ 2 )) c (O,p''), the inequality p-Po > 0 holds. Moreover, we also know that when IPol is sufficiently small, p- Po has the same negative sign as .At; hence now in (0, 8) the equation p - Po = 0 has at least three roots. For the other types of centers, it is not certain whether we can prove they can generate three limit cycles; the method of discussion is the same as before, and is therefore omitted. Similarly, we can also prove that the focus of (9. 78) which makes the first nonzero coefficient on its right side be v7 can generate three limit cycles, the focus with the first nonzero coefficient v5 can generate two limit cycles, and the focus with the first nonzero coefficient v3 can generate one limit cycle. The proof is omitted.
!
(oi oi
THEORY OF LIMIT CYCLES
216
Summarizing the above, we can see that the difficulty in the proof of Lemma
9.2 lies in proving the last formula of (9.39); that is, proving that 1/J(po, 0) in (9.44) is identically zero in some line segment of tp = 0.(8 ) Bautin's method can also be divided into the following three steps: 1. Seek a family of closed curves H(x, y) = h, and with this introduce curvilinear coordinates (c5,s) to replace the original polar coordinates (p,tp), and write the successor function (9.53) of system (9.45) in this new coordinate system. 2. Expand c5 (T) - c5o in a power series in I" and po - Po by two different methods. Hence we can obtain the proof of equivalence between 1/J(po, 0) = 0 and Co2(r) 0. 3. Select some special p(x, y) and q(x, y) in (9.45) such that the origin is still its center, and, at the same time, for the corresponding (9.53) c5(r)-c5o = 0 should hold. From this we can obtain the identities (9.66)-(9.68) depending only on the family of curves H(x, y) = h, and then use these identities and some obvious identities (9.69) to prove that the successor function of system (9.45) corresponding to system (9.30) with At = As = A4 + 5A3 - 5A6 = 0 under the new coordinate system has Co2 (T) 0 in its expansion. Necessary and sufficient conditions for the existence of a center for quadratic systems, other than those given in Theorems 9.1 and 9.2 in this section, will later be provided in other forms for convenience of application. Similar problems for cubic systems which do not contain terms of second degree have been solved by N. A. Sakharnikov [163].
=
=
Exercises 1. Prove that (9.25) is a general integral of (9.22). 2. Move [ ] on the right side of (9.26) to the left and then expand it near the origin to prove that the origin is a center. 3. Prove in detail that vs(27r, >.i) = (11"/24)>.2>.4(>.3- >.6)(>.4 + 5>.3- 5>.6)· 4. Prove the quadratic curve x2 = Ay 2 + 2By + C is indeed an integral curve of equation (9.14). 5. Without using the transformation (9.11), prove that (9.10) has an integral factor 1 1+a'y
--[(a'+ b')b'x2 - (1 + b'y) 2 + c'x(1 + b'y)]- 1 • 6. Prove that (9.14) has an integrating factor (1 + o.y) 2f3/o.-l. ( 8 )Note that in order to prove Theorem 9.3, practically we only need the representation formula for vk(27r, .>.;) given just before (9.44).
§9. WORK OF FROMMER AND BAUTIN
217
7. Prove in detail that there exist limit cycles in the exterior of each singular point (±JC.i, a 2 f(a- A)) (use the Annular Region Theorem). 8. Prove in detail that system (9.29) has limit cycles in every neighborhood of (J=j:t,±l) for some J.L < 0. 9. Prove that the system of equations dxjdt = y,
takes (0, 0) as its center, and for some value of o: there exists a limit cycle in every neighborhood of (±2, 0) (see [162]).
§10. Global Structural Analysis of Some Quadratic Systems without Limit Cycles
For a given plane polynomial system, if we know the number of its singular points on the finite plane and the equator, the topological properties of each singular point, the existence or nonexistence, number, and relative positions of closed trajectories, and the directions of separatrices passing through the singular points, then the global structure of this system can be determined. Since the fifties, people have studied the global analysis of quadratic systems without llm1t cycles, and have drawn their global phase-portraits. In this section we present some of the work in this area. In §13, the reader will see that if we add one term to a system without limit cycles so that limit cycles can be generated, then the number of limit cycles and their relative positions are closely related to the global structure of the original system without limit cycles. I. Global topological classification of homogeneous quadratic systems. The system 2 2 { dxjdt = aux + a12XY + a22Y , dyjdt = bux 2 + b12XY + b22Y 2
{10.1)
is called a homogeneous quadratic system. Clearly it does not have a limit cycle. When the right sides of (10.1) do not have a common factor, {0, 0) is its unique singular point. L. S. Lyagina [164] studied the structure of {10.1) near the origin, showed that there were sixteen possible cases, and gave methods for classification. Lawrence Markus [165] studied the global topological classification of {10.1), and, according to whether it had a straight line filled with singular points and the number of straight line solutions, obtained a standard form under the linear transformation. It is worth mentioning that this paper uses the methods of nonassociative algebra. Recently, N. I. Vulpe 219
220
THEORY OF LIMIT CYCLES
and K. S. Sibirskil [166] studied the global topological classification and geometrical classification of systems (10.1) whose right sides have and do not have a common factor, and pointed out some criteria on their coefficients for classification. In the present section we introduce all possible structures for systems (10.1) whose right sides do not have a common factor. For the case with a common factor the problem is much simpler, and is left to the reader as exercises.(!) We first do some preparatory work, which is always given in general textbooks on qualitative theory. The exceptional direction 9 = 9o of (10.1) is determined from the equation
G(9) = - a22 sin3 9 + (b22- a12) sin2 9cos9 + (b12- au) sin9cos2 9 + bu cos3 9 = 0.
(10.2)
It is easy to see that equation (10.2) has at least a pair of real roots 9 = 90 and 9 = 9o + rr in [0, 2rr). We may as well assume 9o = 0; thus bu = 0. Replacing {10.1), we only have to discuss the system
dx I dt = aux 2 + a12xy + a22Y 2, dy I dt = bl2XY + b22Y 2. Here we must have au common factor. Now (10.2) becomes
(10.3)
=I 0; otherwise the right sides of (10.3) would have a
G(9) = sin9[-a22 sin2 9+(b22-a12) sin9cos9+(b12-au) cos 2 9] = 0. (10.4) The discriminant inside the brackets in the above formula is
(10.5) If we introduce polar coordinates x = rcos9, y = rsin9, then {10.3) b&comes
d9 G(9) r dr = H(9)'
where
H(9) =au cos3 9 + a12 cos2 9sin9
+ (b12 + a22) cos 9 sin2 9 + b22 sin3 9. Integrating (10.6), we get r = r1 exp
---1
(i: ~~:~
d9)
(10.6)
(10.7)
(10.8)
( )Recently, Nikola Samardzija (see [299]) pointed out that (10.1) also has characteristic values and characteristic vectors which are used to study the stability of the singular point
(0,0).
§10. QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
221
In order to study the infinite singular point, we change (10.3) into homogeneous coordinates, again letting x = 1 and dtjdr = z; we get
+ a12Y + a22y 2), -a22Y3 + (b22- a12)y 2 + (b12- au)y.
dz/dr = -z(au dyjdr =
(10.9)
The Yi of the infinite singular point (1, Yi, 0) is a root of the equation (10.10) whose discriminant is the same as (10.5). The product of the two roots A1 and A2 of the characteristic equation at the infinite singular point (1, Yi, 0) is A1A2
=-
(au+ a12Yi + a22Yl)
x [-3a22Yl
+ 2(b22 -
a12)Yi
+ b12 -au].
(10.11)
Now we begin our discussion of (10.3). 1. It has a pair of exceptional directions. Here again we can divide into two cases. (i) () = 0 and () = 1r are the unique pair of single exceptional directions; that is, suppose
a < 0,
and hence a22(bt2- au)
< 0.
(10.12)
Now y = 0 is a unique integral line. In order to study the state of the integral curves in the sector neighborhood of the exceptional direction () = 0, let the point (rt, Ot) in (10.8) lie in a sector neighborhood of()= 0, and expand G(O) and H(O) into power series of 0; then we get
H(O) G(O)
=
au [1 + .. ·]. (b12 - au)O
From (10.8) we know that if au(bt2 - au) > 0, then r -+ 0 as () -+ 0. In this case () = 0 is a so-called ray of nodal type, as in Figure 10.1. If a 11 (b 12 - au) < 0, then r-+ 0 as()-+ 0, and()= 0 is a ray of isolated type, as in Figure 10.2. On the other hand, from (10.10) we know that in this case there is a unique singular point (1, 0, 0) at infinity. Again from (10.11) we know that if au(bt2- au)> 0, then (1,0,0) is a saddle point; if au(bt2- au)< 0, then (1, 0, 0) is a nodal point. The conclusion for () = 1r is the same. Although the above discussion is local, we note that (10.3) is a homogeneous system, does not have a limit cycle, has only the above-mentioned singular
222 THEORY OF LIMIT CYCLES
~e-o 0 FIGURE 10.2
FIGURE 10.1
point, and y = 0 is the unique integral line. Hence we get THEoREM 10.1. For system (10.3), suppose the right sides of the two equations do not have a common factor, and suppose 6 < 0. Then when au(b 12 - au)> 0 its global phase-portrait is shown in Figure 10.5(a), below, and for a 11 (b12- au) < 0 its global portrait is shown in Figure 10.5(b).
(ii) (} == 0 and (} = 71" is a pair of triple exceptional directions; that is, suppose (10.13) Thus we must have a22 =f. 0, for otherwise the right sides of the two equations of (10.3) would have a common factor. Now y = 0 is still a unique integral line, and H(O) au · G(O) = - a2203 {1 + .. ·]. From (10.8) we know that if aua22 > 0, then r ~ oo as(}~ 0, and(}= 0 is a ray of isolated type as in Figure 10.2; if au a22 < 0, then r ~ 0 as (} ~ 0, and (} = 0 is a ray of nodal type as in Figure 10.1. Under the above assumptions, system (10.9) becomes dz/dr = -z(au
+ a12Y + a22Y 2),
dy/dr = -a22Y 3,
whose unique infinite singular point (1, 0, 0) is of higher order. From the theory of higher order singular points, we know that if aua22 > 0, then (1, 0, 0) is a nodal point, and if aua22 < 0, then (1, 0, 0) is a saddle point. The conclusion for (} = 71" is still the same. Thus we have THEOREM 10. 2. For the system ( 10.3) whose right sides do not have a common factor, suppose b22- a12 = b12- au = 0. Then when aua22 > 0 its global phase-portrait is as shown in Figure 10.5(b), and when aua22 < 0, its global phase-portrait is as shown in Figure 10.5(a). 2. If (10.3) has only two pairs of exceptional directions, then among them at least one pair is double. We may as well assume (} = 0 and (} = 71" are double exceptional directions, while (} = 71" /2 and (} = 371" /2 are single exceptional
§10.
QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
223
J--/0 0 FIGURE 10.3
FIGURE 10.4
directions; that is, suppose a22
= 0,
= 0, x = 0
b12- au
hence b22 ~ a12·
(10.14)
Thus there are two integral lines, and y = 0. In the following we shall investigate the state of integral lines in the sector neighborhood of the exceptional directions 0 = 0 (0 = 1r) and 0 = 71'/2 (0 = 371'/2). First we examine the exceptional direction 0 = 0, in which case we have
H(O) G(O)
au [ J a12)0 2 1 + ....
= (~2 -
From (10.8) we know that if au(b22- a12) > 0, then r--+ 0 as 0 --+ o+ and r --+ oo as 0 --+ o-; then 0 = 0 is the so-called altered ray of the second kind as in Figure 10.3. If au(~2- a12) < 0, then r--+ oo as 0--+ o+ and r--+ 0 as 0 --+ o-; then 0 = 0 is the so-called altered ray of the first kind as in Figure 10.4. Under the above assumptions, system (10.9) becomes
dzjdr
= -z(au + a12Y),
From the theory of higher order singular points we know that (1, 0, 0) is a semisaddle nodal point; when au(~2 - a12) > 0 the half of the positive yaxis of (1, 0, 0) is a hyperbolic region, and when au (~2- a12) < 0 the half of the positive y-axis of (1, 0, 0) is a parabolic region. For 0 = 1r and the singular point ( -1, 0, 0) we have a similar conclusion. For the exceptional direction 0 = 1rj2 (0 = 371'/2), our discussion is similar to 1(i) provided we interchange the coefficients of a and band interchange the subscripts 1 and 2, but let a12 = a21 and b12 = ~1· Hence we conclude that if ~2(a12- ~2) > 0, then 0 = 1rj2 (0 = 371'/2) is a ray of nodal type and the infinite singular point (0, 1, 0) is a saddle point; if ~2 (a12 - ~2) < 0, then 8 = 1rj2 (8 = 371'/2) is a ray of isolated type and the infinite singular point (0, 1, 0) is a nodal point. Combining the two cases au (~2 - a12) ~ 0 and the two cases ~2(a12 - b22) ~ 0, we can get four cases. Through analysis, we find if we disregard different directions of the x-axis, then the two global phase-portraits
THEORY OF LIMIT CYCLES
224
of b22 (a 12 - b22) > 0 are in fact the same, and the two global phase-portraits for b22(a12- b22) < 0 are also the same. Thus we have THEOREM 10.3. For system (10.3) whose right sides do not have a common factor, suppose we can make a22 = 0 and b12 = au. Then for b22 (a12 - b22) > 0 the global phase-portrait is as shown in Figure 10.5(c), below, and for b22(a12 - b22) < 0 the global phase-portrait is as shown in Figure 10.5(d). 3. Now suppose (10.3) has three pairs of single exceptional directions. We may as well assume these three pairs of exceptional directions are 0 = 0 (0 = 1r), 0 = 1rj2 (0 = 311"/2), and 0 = Oo (0 = Oo + 1r), 0 < Oo < 1rj2. That is to say, we may assume tanOo =
b12- au b a12- 22
> 0.
(10.15)
Now we have three integral lines x = 0, y = 0, and y - x tan Oo = 0. From 1(i) we know that when au(b12- au) > 0, 0 = 0 (0 = 1r) is a ray of nodal type and the infinite singular point (1, 0, 0) is a saddle point; when au(b12- au)< 0, 0 = 0 (0 = 1r) is a ray of isolated type, and (1,0,0) is a nodal point. Also from 2 we know that when ~2(a12 - b22) > 0, 0 = 11"/2 (0 = 311" /2) is a ray of nodal type and the infinite singular point (0, 1, 0) is a saddle point; when b22(au- b22) < 0, 0 = 7r/2 (0 = 37r/2) is a ray of isolated type and (0, 1, 0) is a nodal point. For 0 = Oo, we expand G(O) and H(O) into power series of 0- Oo, and get
H(O) H(Oo) G(O) = 01(0- 00 ) [ 1 + .. ·], where 2 2 2 01 = G'(Oo) = - sin0ocos Oo[(a12- b22) + (b12 - a 11 ) ] a12- b22 3 H(Oo) = cos Oo[au + a12 tan Oo + b12 tan 2 Oo + b22 tan3 Oo] =
3 Oo ) (aub22- a12b12) [(a12- b22 )2 ( cos b 3
a12-
22
# O,
+ (b12- au )2] ..J.r 0.
Hence OtH(Oo) and aub22 -a 12b12 have the same sign. From (10.8) we know that if aub22 - b12a12 > 0, then 0 = 00 is a ray of nodal type; then we compute the product of two characteristic roots at the infinite singular point (1,tan0o,O) and get
§10. QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
(a)
(b)
225
tc)
(d)
(e)
(f)
(g) FIGURE 10.5
Hence (1, tan Oo, 0) is a saddle point. If aub22 - b12a12 < 0, then(}= Oo is a ray of isolated type, and AlA2 > 0; then (1, tan Oo, 0) is a nodal point. Thus we have THEOREM 10.4. For system (10.3) whose right sides do not have a com-
mon factor, suppose a22 = 0 and (b12 - au)(a12 - b22) > 0. Then when au(b12- au), B b22(a12- b22), and C aub22- a12b12 are all negative, its global phase-portrait is as shown in Figure 10.5(e); when two of .4, B and C are negative and the other is positive, its global phase-portrait is as shown in Figure 10.5(f); and when two of them are positive and the other is negative, its global phase-portrait is as shown in Figure 10.5(g).
.4
=
=
=
226
THEORY OF LIMIT CYCLES
REMARK. A, B, and C cannot be all positive, because, when A, B > 0, since tan Oo > 0 we know that au and b22 have the same sign. At the same time, if au > 0, then b 12 > au > 0 and a12 > b22 > 01 and so C < 0; if au < 0, then we also have C < 0. Summarizing the above four theorems, we get THEOREM 10.5. The global phase-portraits for homogeneous quadratic systems whose right sides do not have a common factor have altogether seven different topological structures, given in Figure 10.5. The arrows in each figure show only one of the two possible cases; for the other case, the arrows should be reversed. II. Global structure of quadratic systems possessing a star nodal point. A. N. Berlinskil [167] studied a quadratic system having a star nodal point, and through his analysis proved that this system does not have a limit cycle, and there are altogether seventeen global phase-portraits of its topological structures. He constructed the phase-portraits, and gave some methods to distinguish them. We now present his work, but simplify his proof. We first point out (from the theory of Jordan systems) that the following two lemmas clearly hold. LEMMA 10. 1. . The form of the system
+ box 2 + b1xy + b2y 2 , dyfdt = Y + aox 2 + a1XY + a2y 2
dxfdt = x
(10.16)
remains unchanged after any nonsingular real linear transformation.
LEMMA 1 0. 2. For a quadratic system, a necessary and sufficient condition for the origin to be a star nodal point is that the system possess the form (10.16). In the sequel we carry out our discussion separately according to the number of finite singular points. 1. The case of four finite singular points. Suppose a general quadratic system has four finite singular points. The following lemma shows that the question of whether the quadrilateral with the four singular points as vertices is convex or concave has a close relationship with the properties of the singular points [169]. The following simple proof is taken from [168]. LEMMA 10.3. Suppose a quadratic system has four singular points. If the quadrilateral with these points as vertices is convex, then two opposite singular points are saddle points, and the other two opposite singular points are nonsaddle points (that is, nodal points, foci or centers); if the quadrilateral is
§10. QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
227
concave, then the three outside singular points are saddle points (non-saddle points), and the inside singular point is a non-saddle point (saddle point). PROOF. After a linear transformation, we can let these singular points lie at the origin 0(0,0) and the points A1(1,0), A2(a:,,B), and A3(0,1), where a > 0, ,8 > 0, and a + ,8 =1 1. Thus the coefficients of the quadratic system dxjdt = a1x + a2y + aux 2 + a12XY + a22y 2, (10.17) dyjdt = b1x + b2y + bux 2 + b12XY + b22Y 2
should satisfy the following relations:
a1 + au
= 0,
= 0,
a2 + a22 = 0, ~ + ~2 2 a1a: + a2.B + aua: + a12a:.B + a22.8 2 = 0, b1a + ~,8 + bua:2 + b12a:.B + ~2.82 = 0.
b1 + bu
= 0,
Thus (10.17) can be written as
= PtX(x- 1) + P2Y(Y- 1) + P3XY, dyjdt = QtX(x- 1) + Q2Y(Y- 1) + Q3Xy, dxjdt
(10.18)
where
a-1
= - - -,8p 1 -
,8-1
(10.19) a ' Since this system has exactly four singular points, each one is elementary. Computing the constant term of the characteristic equation of the linear part of the point 0, we iihould get Do= PtQ2- P2Ql =I 0. It is easy to prove that for the other three singular points At. A2, and A3, similarly,
P3
Dt
=
--P2
a+!- 1 Do,
D2
=(a+ ,8 -1)Do, D3 = a+;- 1 Do.
Whether the singular point Ai is a saddle point or not can be decided by whether Di < 0 or > 0. If a + ,8 > 1, then the quadrilateral is convex, D 0 and D 2 have the same sign, and D1 and D3 have the same sign, but have sign opposite to Do. If a:+,B < 1, then the quadrilateral is concave, O,At, and A3 are three outside singular points, their Do, Dt. and D3 have the same sign, but have sign opposite to D2. The proof is complete. REMARK. Although this proof is succinct, it does not show why the property of the dynamical system of these four singular points has such a delicate relationship with the geometrical property of the quadrilateral formed by them. To understand this point, please refer to Lemma 11.3 and the remark after it in §11. In the following we return to the quadratic system possessing a star nodal point. As before, we may assume its four singular points are 0(0, 0), At(1, 0),
THEORY OF LIMIT CYCLES
228
A2(o:, /3), and Aa(O, 1), where 0 is a star nodal point, o: '# 0, {3 '# 0, and o: + {3 '# 1. From Lemma 10.2, we know that now P2 = Qt = 0 and Pt = Q2 = -1 in system (10.18), thus the system can be written as dx dt
= X (1 -
Again let (o: - 1) I {3 written as
dxldt
X
+
0: -
{3
1 y) I
dy dt
= y (1 +
Q:
= b and (/3 - 1)I o: = a;
= x(l- x +by),
dyldt
{3 - 1 X
-
y) '
thus the above system can be
= y(l +ax- y),
(10.20)
where a'# -1, b '# -1, and ab '# 1. The four singular points of (10.20) are 0 (star nodal point), At(l, 0), A2((1+b)l(l-ab), (l+a)l(l-ab)), and Aa(O, 1); and the three integral lines are
OAt: y = 0;
OA2: (1 + b)y- (1 + a)x = 0;
OAa: x = 0.
From this we can see that for every singular point there is an integral line passing through it, and so (10.20) does not have a limit cycle;(2) thus whether At, A2, and A3 are saddle points or nodal points depends on whether the quantities
Dt =-(a+ 1),
D _ (1 + a)(1 +b) 2 l-ab'
Da
= -(b+ 1)
are less than zero or greater than zero. Next, if we change (10.20) into homogeneous coordinates, we can easily prove that it has three infinite singular points:
Bt(l,O,O),
B2 (1,
~::.o),
Ba(O,l,O),
and the quantities corresponding to the above-mentioned D, are now
(1 + a)(l- ab) Ja = 1 +b. l+b Since D, = -h we know that if A, is a nodal point (saddle point), then B, is a saddle point (nodal point). The three fixed lines x = 0, y = 0, and x + y = 1 passing through the three singular points O,At, and Aa respectively divide the (x,y)-plane into seven regions. From Lemma 10.3 we know that, given which region A2 is in, we can tell whether A, (hence Bi), i = 1, 2, 3, is a nodal point or a saddle point. Moreover, since other integral curves cannot cross the three lines OA 11 OA2, and OAa outside 0, A1. A2, and Aa, the global phase-portraits can be ( 2 )This
point can also be known from Exercise 9 in §1.
§10.
QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
229
b
Cl
b--1
FIGURE 10.6 completely determined, and easily drawn. The position of A2 in the (x, y)plane can then be determined from the values of the parameters a and b. On the (a,b)-parametric plane, three bifurcation curves ab = 1, a= -1, and b = -1 divide the parametric plane into seven regions (Figure 10.6). The global topological structure of system (10.20) corresponding to the point in every region is the same. But it is easy to see that when the point (a, b) passes through the line a= -1 orb= -1, the figure of the global topological stucture of (10.20) does not change, only a certain pair of finite singular points and the corresponding pair of infinite singular points interchange their relative positions, and at the same time the labels of the saddle points and nodal points are also interchanged; hence the global topological structure is not affected. Thus there are in fact only three different global topological phas&portraits as shown in Figure 10.7(a), (b), and (c) respectively. They correspond respectively to: 1) ab > 1, a< 0, 2) ab < 1, and 3) ab > 1, a> 0, respectively. Thus we get THEOREM 10.6. If a quadratic system possessing a star nodal point has four singular points, then: (i) When these four singular points form the vertices of a concave quadrilateral, and the star nodal point is an outside vertex, the other three singular points are nodal point, nodal point, and saddle point (Figure 10.7(a)). (ii) When these four singular points form the vertices of a concave quadrilateral, and the star nodal point is an inside vertex, the other three singular points are saddle points (Figure 10.7(c)).
230
THEORY OF LIMIT CYCLES
(iii) When the four singular points form the vertices of a convex quadrilateral, the other singular points are nodal point, saddle point, and saddle point (Figure 10.7(b)).
If we instead use the coefficients of the system to indicate our results, then this theorem can be rewritten as THEOREM 10. 6'. For a quadratic system ( 10.20) possessing a star nodal point where (a+ 1)(b + 1)(ab -1) -:f. 0, then corresponding to Theorem 10.6(i), (ii), and (iii), we have (i)' ab > 1, a< 0, (ii)' ab < 1, and (iii)' ab > 1, a> 0, respectively. 2. The case of only three finite singular points. We may as well assume that except for its star nodal point (the origin), there are two finite singular points A1(1,0) and A3(0, 1). Following the method of deriving {10.20), we know that the system to be discussed still possesses the form {10.20), but its parameters a and b correspond to the bifurcation values mentioned in 1; that is, a = -1, or b = -1, or ab = 1 but not a = b = -1; or else they correspond to A2 = A1. or A2 = A3, or A2 = B2 in 1, respectively. It is easy to see that in our present case this system still has no limit cycles. In the following we study these cases separately. (i) Suppose a= -1 (b -:f. -1); that is, A2 = A1. and hence B2 = B1. The two higher order singular points obtained from this are semi-saddle nodal points. It is easy to see that the equator is a middle separatrix of two hyperbolic regions of the singular point B1 (1, 0, 0) and the type of the other singular point remains unchanged. When a = -1 and b < -1, the corresponding global phase-portrait is as shown in Figure 10.7(d); when a = -1 and b > -1, the corresponding global phase-portrait is as shown in Figure 10.7(e).
(ii) Suppose b = -1 (a -:f. -1). Similarly to (i), when b = -1 and a< -1, the corresponding global phase-portrait is as shown in Figure 10.7(d); when b = -1 and a > -1, the corresponding global phase-portrait is as shown in Figure 10.7(e). (iii) Suppose ab = 1, but a = b = -1 does not hold. This corresponds to A2 = B2; B2 becomes a semisaddle nodal point, but the equator separates the hyperbolic region of B2 from its parabolic region. The other singular point keeps the original type. When ab = 1 and a < 0, a :f. -1, the corresponding global phase-portrait is as shown in Figure 10.7(f); when ab = 1 and a> 0, the corresponding global phase-portrait is as shown in Figure 10.7(g).
§10.
QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
231
The above results are written as follows: THEOREM 10.7. For a quadratic system (10.20) possessing a star nodal point, if there are only three finite singular points, then t.he other two finite singular points can only be: (i) a nodal point and a semisaddl~ nodal point; here a= -1 and b < -1, orb= -1 and a< -1 (Figure 10.7(d)); or (ii) a saddle point ant/. a semisaddle nodal point; here a= ~1 and b > -1, or a> -1 and b = -1 (Figure 10.7(e)); or (iii) a saddle point and a nodal point; here ab = 1 and a< 0, a-=/= -1 (Figure 10.7(f)); or (iv) a saddle point and a saddle point; here ab = 1 and a > 0 (Figure 10.7(g)). 3. The case of having only two finite singular points. We may as well assume the origin is a star nodal point and the other singular point is At ( 1, 0). Then the system (10.16) can be written as dxfdt = x- x 2 + btXY + b2y 2,
dyjdt = y(1 + atx + a2y).
(10.21)
In the following we divide the discussion into three cases. (i) Suppose At is a triple singular point. Then the following two subcases can arise: (a) The line Lt : y = 0 and the curve £2 : x - x 2 + bt xy + ~y 2 = 0 intersect at At. and the line La: 1 + atX + a2y = 0 and L2 touch tangentially at At. Then we can deduce that at = -1 and a2 = bt; thus (10.21) can be written as
.
dxfdt
= x(1- x + bty) + ~y 2 ,
dyjdt = y(1- X+ bty).
(10.22)
It is clear that b2 -=/= 0 in (10.22). In order to investigate the characteristics of At, we apply the transformation X= 1 - X+ bty,
y = y,
and still denote X andY as x andy; thus (10.22) is changed to dxfdt
=
-x + x 2 - b2y 2,
dyjdt
= xy.
(10.23)
From the theory of higher order singular points(3) we know that At is a nodal point for b2 > 0, At is a saddle point for ~ < 0, and the system has no limit cycles. Transform the system again in homogeneous coordinates. It is easy to know there is a unique singular point Bt (1, 0, 0) at infinity, and Bt is a saddle point when ~ > 0; Bt is a nodal point when ~ < 0. Moreover, since y = 0 is an integral line passing through 0 and At, it is easy to construct the global (3)For example, see Theorem 65 in [170], §21.
232
THEORY OF LIMIT CYCLES
phase-portrait of system {10.23). When~> 0 it is as shown in Figure 10.7(h); < 0, it is as shown in Figure 10.7(i). (b) L 3 and L2 intersect at At, and Lt and L2 touch tangentially at At. This is impossible. (ii) Suppose At is a double singular point. Following the same discussion as in (i), we know that {10.21) should have at = -1, b2 = a2(a2- bt), and a2- bt 'I 0. Thus {10.21) can be changed to when~
dxjdt = x(1- x + ay),
dyjdt = y(1- x),
(10.24)
where a = bt - a2 'I 0. It is easy to see that At (1, 0) and the infinite singular points (1, 0, 0) and (0, 1, 0) are all semisaddle nodal points, and do not have limit cycles. Its global phase-portrait is as shown in Figure 10.7U). (iii) Suppose At is a simple singular point. Then (10.21) when a2 'I 0 can be changed to
dyfdt
= y(1 +ax- y),
(10.25)
where at= a, b = -bt/a2, and c = b2/a~. Since At is a simple singular point, we can deduce that these coefficients satisfy either o) (b+ 1) 2 + 4(a + 1) < 0, or {3) a2c + ab - 1 = 2ac + b + 1 = 0, c 'I 0. If a2 = 0, then (10.21) becomes
dxjdt = x- x2 + bxy + cy2 ,
dyjdt
= y(1 +ax),
(10.26)
where a = at, b = bt. and c = b2. Since At is a simple singular point, we can deduce that these coefficients either satisfy o) b2 + 4(a + 1)c < 0, or {J) b = c = 0, a -:J -1, or 7) a= 0, b2 + 4c ~ 0. Study case o) of (10.25). It is easy to see at this time that we must have a< -1, the point At(1, 0) is a nodal point, the unique infinite singular point (1, 0, 0) is a saddle point, and the system does not have a limit cycle. Its global phase-portrait is shown in Figure 10.7(h). For the case {3) of (10.25), we must have a 'I 0; the system can be written as dx a+2 a+1 2 dy 2 - = x- x + - - x y - - - y dt = y(1 +ax- y). (10.27) dt a a2 ' It is easy to see that this system has three integral lines, y = 0, y = ax, and y = a(x- 1). Since for each singular point there is an integral line passing through it, the system does not have a limit cycle. When a < -1 (> -1), the singular point At is a nodal point (saddle point), the infinite singular point (1, 0, 0) is a saddle point, and the other infinite singular point (1, a, 0) is always a semisaddle nodal point; its global phase-portrait is as shown in Figure 10.7(k) (Figure 10.7(1)).
§10. QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
233
For system (10.26) we can carry out a similar study, and the results are as follows: In case a), when a< -1 (> -1), its global phase-portrait is as shown in Figure 10.7(h) (10.7(i)). In case {3), there are three integral lines; two of them pass through the origin, and the third does not. When a< -1 (> -1), we get Figure 10.7(k) (10.7(1)). In case-y), there are also three integral lines, the singular point At(1,0) is a saddle point and the system does not have a limit cycle. When b2 + 4c = 0, two of the three integral lines pass through the origin. Two infinite singular points are (star) nodal point and semisaddle point respectively. Its global phase-portrait is shown in Figure 10.7(1). When b2 + 4c > 0, three integral lines always pass through the origin; among the three infinite singular points, one is a nodal point and the other two are semisaddle nodal points. Its global phase-portrait is shown in Figure 10.7(m). Summarizing the above discussion, we get THEOREM 10. 8. If a quadratic system possessing a star nodal point has only two finite singular points and if there is only one integral line passing through the origin, then another singular point is either a nodal point or a saddle point; the global structures of this system are as shown in Figures 10. 7(h), (i). If this system has only two integral lines passing through the origin, then the singular point is a semisaddle nodal point, nodal point, or saddle point; the global structures of the system are as shown in Figures10.7(j), (k), (1). If the system has three integral lines passing through the origin, then the other singular point must be a saddle point, and its global structure is as shown in Figure 10.7(m). In all these cases, there is no limit cycle.
4. The case of having only one finite singular point. In this case the system clearly does not have a limit cycle. Through a suitable rotation of axes, we can make ~ = 0 in (10.16). From the uniqueness of the singular point, we know that a2 = 0. Thus we have
dxfdt = x(1 +box+ bty),
dyfdt = y + aox 2
+ a1 xy.
(10.28)
For the same reason, the coefficients of (10.28) should satisfy one of the following conditions: a) (at - bo) 2 + 4aobt < 0; {3) bt = 0, at = bo =I 0, ao =f:. 0; 1) bt = bo = o, ao =I O; 6) bt = bo = ao = 0.
THEORY OF LIMIT CYCLES
234
(a)
,
(b)
(c)
"
, (d)
(e)
(f)
(g)
(h)
(i)
FIGURE 10.7 When a) holds, there is a unique singular point (0, 1, 0) at infinity, which is a semisaddle point. The equator separates the parabolic region from the hyperbolic region. There is only one integral line x = 0, and its global phaseportrait is as shown in Figure 10.7(n). When (3) holds, (10.28) has an integral line x = -1/b besides they-axis. It is easy to see that the infinite singular point (0, 1, 0) is a semisaddle nodal point. Two hyperbolic regions are all on the same side of the equator, but on this side there is a part belonging to the parabolic region; its global phaseportrait is as shown in Figure 10.7(o).
§10. QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
235
Bli
(k)
(I)
(o)
(n)
(m)
(q)
(p) FIGURE 10.7
When 1) holds, if a 1 =/; 0, then there are two infinite singular points, one of which is (1, -a0 fat.O), a semisaddle nodal point; the equator separates the parabolic region from the hyperbolic region. The other is (0, 1, 0), which is also a semisaddle nodal point. There are two integral lines which pass through the origin. The global phase-portrait is in Figure 10.7(p). If a1 = 0, then there is a unique infinite singular point (0, 1, 0) which is a semisaddle nodal point; the equator separates the parabolic region from the hyperbolic
236
THEORY OF LIMIT CYCLES
region. There is only one integral line passing through the origin, and its global ph~portrait is as shown in Figure 10.7{n). When c5) holds, {10.28) becomes dxjdt
= x,
dyjdt
= y(1 + a1x).
When at i= 0, there are two infinite singular points {1, 0, 0) and {0, 1, 0) which are semisaddle nodal points. Its global phase-portrait is shown in Figure 10.7{p). When at = 0, we have the phase-portrait of Figure 10.7{q), and {10.28) degenerates to a linear system. Summarizing the above disCussion, we get THEOREM 10. 9. If a quadratic system possessing a star nodal point has a unique finite singular point, then there are four and only four cases: {i) it has only one integral line, which passes through the origin; (ii) it has only two integral lines, one of which passes through the origin; {iii) it has only two integral lines, both of which pass through the origin; or (iv) all the integral lines are lines passing through the origin. This system does not have a limit cycle. Phase-portraits are shown in Figures 10.7(n)-(q). Combining the above Theorems 10.6--10.9, we have THEOREM 10. 10. The finite singular points of a quadratic system possessing a star nodal point whose right sides do not have a common factor are only saddle points, nodal points, and semisaddle nodal points. The system does not have a limit cycle. The global phase-portraits have seventeen and only seventeen different topological structures, as shown in Figures 10.7{a)-(q). REMARK. In all the parts of Figure 10.7, s =saddle point, n =nodal point, en =star nodal point, sn =semisaddle nodal point, and db =double singular point. III. Topological classification of structurally stable quadratic systems without limit cycles. Let X denote the Banach space of polynomial differentiable systems of degrees not greater than n, let En be the set of structurally stable systems in X, and let Xs be the set of all systems satisfying the conditions {VI) of §8. We have already seen that {1) Xs is open and dense in X; and (2) a necessary and sufficient condition for a system a to be in Xs is that a is in En.
§10.
QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
237
In this subsection we shall study the problem of topological classification of E2 without limit cycles. The problem was first mentioned and discussed by G. Tavares dos Santos in [171]; and, in the end, he obtained 25 different topological structures. More recently, Cai Sui-lin [172] pointed out that the classification of [171] was incomplete, and added eight more examples; hence, altogether, there should be at least 33 different topological structures. He also studied the direction of a separatrix passing a saddle point and from this determined that under certain conditions there could not be other different topological structures. Independently of [172], Shi Song-ling [173] concluded by studying the logical possibilities that the structurally stable systems E2 can only have at most 65 different topological structures, but he did not prove whether these 65 kinds could be realized in quadratic systems. In fact, we shall see later that among the 65 kinds in [173], there are several which definitely cannot be realized in quadratic systems. Hence the problem of determining how many topological structures of structurally stable quadratic systems without limit cycles there are is up to now unsolved. Now we shall briefly present the above work. In order to-study the topological classification of this system, we first have to study all the possible combinations of its singular points in the interior of the Poincare closed hemispherical surface 0 (equivalent to the Euclidean plane) and on the equator E. In the following we shall adopt the following notation: s: saddle point in the interior of 0; p: c non-saddle point in the interior of (elementary singular point of index +1); 8: saddle point onE; F: source onE; P: (deep) sink on E; o:,/3,"'(,0, and c: elements of E2.
n
THEOREM 10 .11. If o: is in E2, then the combinations of singular points of o: on 0 must be one of the following twelve cases: type a: (1) F; (2) psF; (3) P1P2s1s2F; type b: (1) P1P28; (2) P1P2P3s8; type c: (1) Sls2FlPF2; (2) PSlS2S3FlPF2; typed: (1) SPF; (2) psSPF; (3) P1P2s1s2SPF; type e: (1) P1P28182F; (2) P1P2P3S8182F. PROOF. The proof is divided into four steps: (i) Since Xs = E 2 when we know 0: only has elementary singular points in the interior of n and on E. (ii) The coordinates of a singular point onE are roots of a cubic
n.
= 2,
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THEORY OF LIMIT CYCLES
p
p
FIGURE 10.8
FIGURE 10.9
equation, and so there exist only one or three singular points on E. (iii) Since Xs is open and dense in X, the zero isocline and infinite isocline can be considered as nondegenerate elliptic or hyperbolic curves; thus it is not possible between them to have one or three points of intersection, i.e. the number of finite singular points is 0, 2, or 4. (iv) Since the sum of indices of the singular points on 0 is equal to 1, we can again use Lemma 10.3 to prove the conclusion of this theorem. The proof is omitted and left to the reader as an exercise. In the following, for convenience of presentation, we adopt the notation 8(Pt.P2,P3,P) to represent four separatrices (Lt,Lt,L1,L2) of a saddle point 8 connecting the singular points Pt. P2, p3, and P respectively, in which the two separatrices starting from the first two points Pt and P2 enters s, and the other two separatrices starting from 8 enter P3 and P respectively (Figure 10.8); and we use the notation S(F,p) for a saddle point onE using separatrices [+ and [- not along the equator to connect the singular points F and p respectively (Figure 10.9). The other notation has similar meaning. Since the system under discussion does not have a limit cycle, and it is not possible to use a separatrix to connect two saddle points, hence, from connecting the separatrices of saddle points and the singular points whose indices are +1, we can determine the topological structure of the corresponding phase-portrait. We now carry out our discussion according to the five types and twelve forms in Theorem 10.11. 1. Type a(1). Form F. It is easy to see that there is only one topological structure as shown in Figure 10.10. Its realizable example is 01:
dx/dt = -2xy, dyjdt = x 2
+ y2 + c
(c > 0).
2. Type a(2). Form p8F. It is easy to see that this case also has only one topological structure, whose characteristic is 8(F, F, p, F') or 8(F', F', p, F);
§10. QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
239
F'
F'
F
F
FIGURE 10.10
FIGURE 10.11
the former is shown in Figure 10.11. Its realizable example is o:2: dx/dt = P- OQ, dyfdt = Q
which is obtained by rotating an angle(} (0
< (} « 1)
=
dxfdt = P -y(v + 2x), 0:' { 3 . dy I dt = Q = (X + ~) + y 2
+ fJP,
2
-
T
from (~-t
< 0 < v).
Now we set o: 2 = o: 9. In fact, the singular points of the system o: belong to form psF, but the characteristic root of p has zero real part. Since there exists a first integral 'P(x, y) = x(x 3 /3 + y 2) + ~-tx 2 /2 + vy 2 /2, pis a center. After rotating through an angle (}, the center becomes a focus, and a family of closed trajectories of a becomes an arc without contact of o: 2; hence 0:2 does not have a limit cycle. 02 still belongs to the form psF, and its phase-portrait is shown in Figure 10.11. 3. Type a(3). Form PtP2S1s2F. For this kind we have the following theorem: THEOREM 10. 12. Structurally stable quadratic systems without limit cycles of form P1P2s1s2F have five and only five different kinds of topological structures, whose characteristics are
s1(F,F,Pt.P2),s2(F,F,p2,F'); s1(F,F,p1,F'),s2(F,P2,F',F'); o:s: s1 (F, F, Pt. F'), s2(F, F, p2, F'); o:6: s1 (F,p2, Pt. F'), s2(F, P2• Pt. F'); 0:7: sl(F,p2,Pt.Pi), s2(F, F,p2, F'). o:3: o:4:
PROOF. We first prove there are at most five different kinds of topological structures as mentioned above.
240
THEORY OF LIMIT CYCLES 8'
8'
8
FIGURE 10.12
FIGURE 10.13
FIGURE 10.14
(1) Suppose Pt and P2 are sinks (sources). Then the directions of separatrices of a saddle point 8 1 can only have the following three possibilities (if it is a source, then we just interchange the first two elements and the last two elements in the following):
8t(F, F,Pt.P2),
8t(F, F,pt, F'),
8t(F, F,p2, F').
It is the same for 82. After combinations of 8t and 82, it is clear that we can only have two different kinds of topological structures, a 3 and a 5 • (2) Suppose one of Pt and P2 is a sink and the other is a source. Then the directions of separatrices of 8t can only be of the following five kinds:
8t(F,F,pt,F'),
8t(F,p2,Pt.Pt),
St (F, P2. F'' F'),
8t(F,P2,Pt.F'),
St (P2. P2. Pt. F').
There are the same five kinds for 82. After the combinations of 8t and 82, it is clear that only three different kinds of topological structures a4, a5, and a7 are possible. For details see [172]. On the other hand, the above five different topological structures can all be realized. Examples can be found in [171] and [172]. 4. Type b(1). Form P1P2S. It is easy to see that this case has only one topological structure, whose characteristic is S(Pt. P2), as shown in Figure 10.12. A realizable example is
f3t
= {3:
(0
dxjdt = -2xy, < (} « 1), where {3E:: { d y jdt =y 2 - X 2 + c
(c > 0).
In fact, the singular points of {3E: belong to the form P1P2S, where Pt and P2 are centers and x = 0 is an integral line. After a rotation through an angle 0, its center becomes a focus, and it does not have a limit cycle nor any curve connecting two saddle points. Its phase-portrait is shown in Figure 10.12.
§10.
QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
241
p
p
P'
P'
FIGURE 10.15
FIGURE 10.16
5. Type b(2). Form PlP2P3SS. It is easy to see that this case has only one topological structure, whose characteristics are s(PI.P2.P3,Pa) and S(pt,Pa), as shown in Figure 10.13. A realizable example is /h = tYJ.. (0 < () ~ 1), where
/3>.. { dxfdt = -.Xx + 2xy, ·
dy / dt
= .Xy + y 2 -
x2
(.X> 0).
In fact, the singular points of the system /3>.' belong to the form P1P2P3sS, the value of whose divergence is 4y; but x = 0 is a line without contact (except the origin), and so /3>. does not have a limit cycle. After a rotation through an angle() (0 < (} ~ 1), the separatrix connecting Sand S' (which originally had two singular points s and pa) splits. It still has no limit cycles, as shown in Figure 10.13. 6. Type c(1). Form s 1s2F1PF2. It is easy to see that this case has only one topological structure, whose characteristics are 81 (F1, F2, P, F{) and s2(F1. P', F~, Fi), and its phase-portrait is as shown in Figure 10.14. A realizable example is 11: dxfdt
= x 2 + 2xy,
dyfdt
= -2xy- y2
-
c
(c > 0).
The concrete discussion is omitted and left as an exercise. 7. Type c(2). Form ps1s2saFtPF2. For this kind, we have the following: THEOREM 10.13. Structurally stable quadratic systems without limit cycles of the form ps1s2saF1PF2 have four and only four different topological structures, whose characteristics are 12: Bt(Ft,p,P,F~), s2(P',p,Ff,F~), sa(F2,p,P,Fi); "/3: St(Ft,p, P, P), s2(Ft. P', P, F~), s 3 (F2 , P', P, Fi); 14: s1(F1, F2, P, F~), s2(P',p, Ff, F~), s 3 (F2, p, F{, F~); 1s: s1(Ft.F2,P,F~), s2(F2,P',F{,F~), s 3 (F2 ,p,P,P).
THEORY OF LIMIT CYCLES
242
We briefly indicate the method of proof. First we note that since the system under discussion is quadratic, any line cannot have more than two points of contact with the trajectory of this system (including the singular pointW) unless this line is also a trajectory. Hence the following two cases cannot be realized in the quadratic system: (1) Bt(Ft, p, F~', F~), s2(Ft, P', P, F~), s3(F2, P', P, F{), as shown in Figure 10.15, and (2) St(P',p,P,P), s2(Ft,P',P,F~), s3(F2,P',P,Ff), as shown in Figure 10.16. Next, from a discussion similar to Theorem 10.12, we know that in the present case only four different topological structures 12, "Y3, "Y4, and "'(5 are possible. We can prove that they can be realized by quadratic systems; the examples can be found in [171], and so are omitted here. 8. Type d(1). Form SPF. It is easy to see that in this case there is only one topological structure S(F, F'), and it can be realized. The example is in [171]. 9. Type d(2). Form psSPF. This case can only have three different topological structures, whose characteristics are 62: s(F,P',p,P), S(P',p); 63: s(F,F,p,P), S(P',P); 64: s(F, F,p, P), S(F, F').
They can all be realized; examples are in [171]. 10. Type d(3). Form PtP2B1s2SPF. For this there can only be at most 30 different topological structures,( 5 ) in which there are 8 kinds which have been realized (see [171] and [172]). The remaining 22 kinds are still to be realized or eliminated. Since there are too many kinds, we do not list them here. 11. Type e(1). Form PtP2S1S2F. It is easy to see that this case has only three different topological structures, whose characteristics are ct: St(Pt,P2), S2(F,F'); C2: St(Pt,F'), S2(P2,F'); C3: St (Pt. P2), S2(Pb P2)· They all have realizable examples. See [171]. 12. Type e(2). Form PtP2P3BStS2F. For this there are at most 9 different kinds of topological structures.(6 ) There are four kinds which have been realized, and the remaining five have still to be realized or eliminated. ( 4 )The proof is given ( 5 )See (173], but No.
in §11, Lemma 11.1. 7 and No.8 of that paper and C10 and C13 have the same topt; logical structures. Hence there are at most 30 different kinds of topological structures. ( 6 )See (173]; but we can prove that Nos. 8, 9, and 10 in Table 2 of that paper cannot be realized (see Exetcise 9); hence this form can have at most 9 kinds.
§10. QUADRATIC SYSTEMS WITHOUT LIMIT CYCLES
243
Summarizing the above analysis, we get THEOREM 10.14. Structurally stable quadratic systems without limit cy-
cles can have at most 60 different kinds of topological structures, of which at least 33 can be realized. (7 )
Exercises 1. We have seen that the system dxfdt
= x(A1x + B1y + Cl),
dyjdt = y(Ax +By+ C)
does not have a limit cycle. For the case when the system possesses four finite singular points, use the inequalities among the coefficients to discuss its global topological classification [175]. 2. Suppose a cubic system has integral curves xy = 0 and x 2 + y 2 = 1. Discuss its global topological phase-portrait. 3. We have seen that the quadratic system possessing a fine focus of third order tlzfdt = -y + lx 2 + xy, dyjdt = x + ~x 2 + 3lxy does not have a limit cycle. Discuss its global topological phase-portrait [174]. 4. Use the inequalities among the coefficients of the system (10.28) to present different topological structures of this system. 5. Give a detailed proof of Theorem 10.11. 6. Give a detailed proof of Theorem 10.13. 8'
FIGURE 10.17
7. Suppose the singular points on the equator E of a quadratic system belong to form S P F and S (F, F'). If the system in the region SF' P' S (the region on the side of the figure) and S' F P S' do not have any singular points, then this system does not have a singular point in the interior of 11. Prove it. ( 7 )Recently, D. M. Zhu has shown that the number of possible different topological structures is less than 60, but at least 36 could be realized.
244
THEORY OF LIMIT CYCLES
8. For singular points belonging to type e(2), form PtP2PasS1S2F, prove that the following three topological structures cannot be realized: (1) s(p~, F, F', F'), St (pa, P2), S2(F, F'); (2) s(p~,F,F',F'), St(Pa,F'), S2(p2,F'); (3) s(F,F,pbF'), St(P3,F'), S2(P2,F'). 9. Study the global structure of
dxjdt
= y -lx2 + y 2 ,
=
dyjdt
= -x(1 +ax+ by),
where a < 0, l > 0, D -4(b- l) 3 - 27a3 > 0, -a < 2v'l, a+ bv'l 2 1 < b < 1 + a /4l. For example, take a= -2, b = 1.1, and l = 5. 10. Let
~~ = x( -6x + 4y + 2), ~~ = y(7x + y- 2) + sx, Study the global structures of o8 and o; 6 (0 < (} « 1). Oe:
> 0,
O<s«l.
11. Study all possible cases of global phase-portraits of system (10.1) whose right sides have common linear factors, and draw the figures.
§11. General Properties and Relative Positions of Limit Cycles in Quadratic Differential Systems
For a quadratic differential system
dyjdt = Q2(x, y),
dxjdt = P2(x, y),
(11.1)
except for the questions of the existence of a center and the nonexistence of closed trajectories discussed in the previous two sections, the most important question is the existence of a limit cycle. Its importance is beyond doubt no matter whetlier it is considered from a theoretical or a practical viewpoint. Hence, starting from this section, we formally turn to the study of limit cycles. First we study the general properties and relative positions of limit cycles of (11.1). Some of the so-called general properties discussed here may hold for periodic cycles, and some hold for nth-order differential systems; but, generally speaking, most of the properties only hold for limit cycles of quadratic differential systems. The research covered here first appeared in [11], [12], [14], and [17]; then [19] and [20] obtained a new breakthrough, and, recently, [176] and [177] did some summarization and extension. When we present these results, we shall give the methods of proof that are the simplest and easiest to understand.
LEMMA 11.1. Any line can have at most two points of contact with the trajectory of(11.1) (which can include the singular points of(11.1)) unless the line itself is a trajectory of (11.1). PROOF. Suppose the given line is ax+ by+ c = 0. Then the coordinates of the point of contact should satisfy
-a b
ax+by+c = 0,
Q2(x,y) P2(X, y).
The system in general has two solutions (xi, Yi) (i = 1, 2) unless the first equation is part of the second equation, in which case ax + by + c = 0 is a 245
246
THEORY OF LIMIT CYCLES
trajectory. In particular, if Q2(xi, Yi)
= P2(xi, Yi) = 0,
axi
+ byi + c = 0,
then (xi, Yi) is a singular point of the line ax+ by+ c = 0.' REMARK. This lemma can be generalized to nth order differential equations, to show the number of points of contact is at most n; and if we change the line to an algebraic curve of degree m, then the number of points of contact is at most m(m- 1 + n). LEMMA 11. 2. Suppose 0 and 0' are two adiacent elementary singular
points along one branch of P2(x, y) = 0 (on both its sides, P2(x, y) has different signs) or a branch of Q(x, y) = 0. Then the index of one is +1, and the index of the other is -1, provided that between 0 and 0' there does not exist a point of intersection of this branch of P2(x, y) = 0 (or Q2(x, y) = 0) with its other branch; conversely, if between 0 and 0' there still exists a point of intersection of this branch of P 2(x, y) = 0 (or Q2(x, y) = 0) and its other branch, then the indices of 0 and 0' are either both +1 or both -1. PROOF. Suppose 0 is (xo, Yo). Then the characteristic equation of the linear approximate system of ( 11.1) at 0 is
8P2 _A ax
8P2 8y
8Q2 ax
8Q2 _A By
=0. (xo,yo)
Since we assume 0 is an elementary singular point, ( aP2 BQ2 _ aP2 BQ2) ax By By ax (xo,yo)
#0 •
This shows that P2(x, y) = 0 and Q2(x, y) = 0 have different slopes at 0; hence they intersect, but do not touch each other tangentially at this point. Similarly, it is the same at 0'. Now we study two adjacent elementary singular points 0 and 0' along one branch of P2 (x, y) = 0. Suppose between 0 and 0' there does not exist a point of intersection of this branch of P2(x,y) = 0 with the other branch (Figure 11.1). Construct a smooth simple closed curve C containing 0 and 0' such that the other branches of P2(x, y) = 0 and Q2(x, y) = 0, which do not pass 0 and 0', do not intersect with C, and the interior of C does not contain any singular points other than 0 and 0'. Suppose above this branch of P2 = 0 (or on its left if this branch is a vertical trajectory) P2 > 0, and below it {or on its right) P2 < 0. Moreover, between two branches of Q2 = 0 (they may join in the exterior of C) we have Q2 > 0, and outside these two branches we have
§11.
LIMIT CYCLES IN QUADRATIC DIFFERENTIAL SYSTEMS
FIGURE 11.1
247
FIGURE 11.2
Q2 < 0. Thus we can draw several points Ai, Bi, Di, Ei, Fi, and Gi (i = 1, 2) on C from the directions of the vector fields of system (11.1). From this we can see at once that when a point runs from A1 along A~ towards D1, the direction of the vector field rotates through an angle 1r /2; when the point runs from D 1 along ~ towards F 1 , its direction rotates through an angle 0; when the point runs from F1 along F~ towards A2, its direction rotates through an angle -11"/2. Hence from A1 to A2 along A1D1F1A2 the net rotation of the vector field is 0. Similarly, along A2D2F2A1 from A2 to A 1 , the net rotation of the vector field is 0. In other words, the index of the vector field determined from the closed curve C with respect to (11.1) is zero; hence the indices of 0 and 0' are +1 and -1 respectively because they are elementary singular points. If between 0 and 0' there is a point of intersection of this branch of P 2 = 0 with the other branch (in this case P2 = 0 must be two intersecting lines and OCJi is one of the lines) (Figure 11.2), then, as before, we can prove that the index of the vector field determined from C with respect to (11.1) is +2 or -2; hence the indices of 0 and 0' are both +1 or both -1. REMARK 1. This property can be generalized not only to an nth order differential system, but also to plane autonomous systems. REMARK 2. If the polygonal line OMo" in Figure 11.2 is taken as one branch of P 2 = 0, and the other polygonal line NMO' containing 0' is considered as another branch of P2 = 0, then we can prove as before that the
248
THEORY OF LIMIT CYCLES
indices of the elementary singular points 0 and 0" have opposite signs. Because of this, later on, when the hyperbola degenerates into a pair of lines, we can always take two adjacent half-lines starting from the point of intersection as one branch of the hyperbola, and the remaining two adjacent half-lines are considered as the other branch. LEMMA 11.3. System (11.1) has at most three elementary singular points whose indices are +1 (or -1). PROOF. We may as well assume that P2(x,y) = 0 and Q2(x,y) = 0 do not have a common factor, for otherwise there exists a line or a curve which is filled with higher order singular points, and the other elementary singular point is at most one. It is easy to see that system (11.1) can have at most four elementary singular points. If they are all on the same branch of P2 = 0 or Q2 = 0, then by Lemma 11.2 the indices of two of them are +1, and the others are -1. Hence if P 2 = 0 or Q 2 = 0 is an ellipse or a parabola, the lemma is clearly established. Now suppose both P2 = 0 and Q2 = 0 are hyperbolas, and the four singular points lie neither on the same branch of P 2 = 0, nor on the same branch of Q2 = 0. Then either some two singular points lie on the same branch of P2 = 0 (or Q2 = 0), and the other two on the other branch, in which case there can only exist two singular points of index +1 ( -1); or some three singular points lie on one branch, and the other lies on the other branch, in which case it is possible that there are three singular points with index equal to +1 (or -1), but the fourth singular point must be -1 (or +1). REMARK. Following the method of proof of Lemma 11.3, it is not difficult to give a new geometrical proof to Lemma 10.3 in §10, and from this it is easy to see why there exists such a delicate relationship between the properties of the singular points of a dynamical system and the geometrical properties of the quadrilateral formed by them [176]. LEMMA 11.4. If the line passing through two singular points 81 and 82 of (11.1) is not an integral line, then it must be formed by three line segments without contact ooS1, 8182, S2oo; here the trajectories cross ooS1 and S2oo in one direction, and cross 8182 in the opposite direction. PROOF. Since any singular point can be considered as a point of contact, the first half of this lemma can be deduced at once from Lemma 11.1. In order to prove the second half, we first rotate the axes to make the line 8182 become the x-axis, and suppose that on ooS1 the trajectories of (11.1) all cross it from above to below (Figure 11.3). This shows that ooS1 lies in the region Q 2(x,y) < 0. Now Q2(x,y) = 0 must pass the two points 8 1 and 82, but it cannot have the x-axis as part of it; from the properties of quadratic
§11. LIMIT CYCLES IN QUADRATIC DIFFERENTIAL SYSTEMS
I
I \
249
I I
FIGURE 11.3
curves we know that Q2(x, y) = 0, and the x-axis must cross 81 and 82. Thus ~ must lie in the region Q 2 (x, y) > 0, and S2oo must lie in the region
Q2 (x, y) < 0. The lemma is completely proved. We can further prove [177]
The line connecting a finite singular point and its infinite singular point of system (11.1) is either a trajectory or a line without contact (except that finite singular point). LEMMA 11.5.
PROOF. Suppose this finite singular point is the origin (0, 0). Thus (11.1)
can be
wri~ten
as dxjdt
= awx + ao1Y + a2ox 2 + auxy + ao2y 2,
dyjdt = bwx + bo1Y + b2ox 2 + buxy + bo2y 2 • Its characteristic equation at (0, 0) is
(11.2)
(aw + bol)..\ + (awbo1 - a01bw) = 0. If (0, 0) is not a focus or a center, we have ..\ 2 -
(11.3) (aw + bo1) 2 - 4(awbol- a01bw) = (aw- bo1) 2 + 4aolbw ~ 0. Suppose the infinite singular point of (11.2) is (1, TJ, 0). Then it is easy to see that 11 should satisfy
L
(aij'f/- bij)TJi
= 0,
(11.4)
i+j=2
which is a cubic equation in TJ, and has at least one real root. Thus the line mentioned in the lemma can be written as L = y - TJX = 0. Finding the rate of change of L along the trajectory of (11.2), it is easy to see (since 1J satisfies (11.4)) that
I
ddL = x(bw t L=O
+ (bo1 -
aw)TJ - ao111 2 ).
(11.5)
From this we can see that if the value inside the parentheses on the right side of (11.5) is zero, then y -TJX = 0 is a trajectory; otherwise, it is a line without contact, but on both sides of the singular point the directions of trajectories crossing them are different.
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THEORY OF LIMIT CYCLES
LEMMA 11.6. The interior of the closed trajectory r of system (11.1) cannot contain a nodal point, a saddle point, or a higher order singular point [178], [179]. PROOF. Suppose the interior of r contains a singular point 0. We may as well assume 0 is the origin of the coordinate system, and it is a nodal point, a saddle point, or a higher order singular point. Thus (11.1) can be written in the form (11.2). Since now (11.3) holds, we can pick T/ to make the value of the parentheses on the right of (11.5) equal to zero. Then take the line L = y - qx = 0 passing through 0, and find the rate of change of L along the trajectory of (11.2). It is easy to see that now we have
I
dL dt L=O
= x2
[
I: (bi;- ai;T/ )qi] .
(11.6)
i+j=2
From this we can see that if the value in the parentheses on the right of (11.6) is zero, then y- TJX = 0 is a trajectory, for otherwise the trajectories intersecting the line would always cross it from the same direction. In either case, the vicinity of 0 cannot contain a closed trajectory. This contradicts the assumption. COROLLARY. The interior of a limit cycle r of system (11.1) can only have a unique singular point, which must be a focus. REMARK. Lemma 11.6 and its corollary cannot hold even for a cubic differential system. Li Ji-bin [180] gave an example of a cubic system which has a limit cycle whose interior contains three singular points. LEMMA 11.7. A closed trajectory of system (11.1) or a singular closed trajectory which contains only one singular point cannot contain a line segment. PROOF. Suppose the closed trajectory r of system (11.1) contains a line segment. Then the line l defined by this line segment has an infinite number of points of contact with r. _By Lemma 11.1 we know that l itself is also a trajectory of (11.1). But the points on r leaving l (at least two points) will violate the uniqueness of solution of the differential equations; hence it is not possible. When r contains only one singular point, it is also impossible. (I) REMARK. From the proof of Lemma 11.7, we know that if r contains two singular points, then the conclusion does not necessarily hold. In fact this is easy to prove [14]. (1 )If a singular closed trajectory r can have the equator as its part, then one finite singular point and also a half-line. An example is given in [179].
r
can contain
§11. LIMIT CYCLES IN QUADRATIC DIFFERENTIAL SYSTEMS
D
251
I
FIGURE 11.4
FIGURE 11.5
LEMMA 11. 8. The line segment connecting two saddle points on a singular closed trajectory r must belong to r' but r does not have any other singular point. PROOF. If this line segment does not belong to r, then by Lemma 11.4 we know it is a line segment without contact, and it will form a region with some section of the arc of r whose interior does not contain a singular point and which intersects the line segment. As t increases (or decreases), all the trajectories entering this region have nowhere to go. A concrete example is dx 2 dt = -y- y '
dy = x- x2 dt .
The figure of its global trajectory can be drawn by the reader. THEOREM 11.1. A closed trajectory of system (11.1) or its singular closed trajectory containing only one saddle point and lying in a bounded region must be a strictly convex closed curve which intersects one branch each of P2(x, y) = 0 and Q2(x, y) = 0 at only two points [11]. PROOF. If r is not a strictly convex closed curve, then, since r does not contain a line segment, we can always find a line l which has at least four points of intersection with r, as in Figure 11.4. According to the directions of crossing of r and l, on the line segments AB, BC, and CD, each has at least a point of contact with the trajectory of (11.1) (which may be a singular point). But by Lemma 11.1, we know this is impossible.
THEORY OF LIMIT CYCLES
252
Next, since r does not contain a line segment, r can have one highest point H, one lowest point G, one extreme right point Rand one extreme left point L (Figure 11.5). According to whether the section of arc of is rising or falling, it is clear H and G are the only two points of intersection of the same branch of Q2(x, y) = 0 with r, orR and L are the only two points of intersection of the same branch of P2(x,y) = 0 with r. If two adjacent points out of these four, such as H and R, coincide then H = R is a singular point; thus r becomes singular closed trajectory. At the same time, by the hypothesis of this theorem, L and G cannot coincide with each other, and they cannot coincide with H. REMARK. Similarly we can prove that any singular closed trajectory containing two or three saddle points can also be a convex closed curve, but it must contain a line segment. Moreover, Theorem 11.1 does not necessarily hold for cubic differential systems; for example, the limit cycle of the van der Pol equation is not convex when 11. is rather large.
:r
THEOREM 11.2. For system (11.1), the following two kinds of relative positions cannot exist [11]:
(a)
(b)
FIGURE 11.6 PROOF. Since in Figure 11.6(a) the interior of r1 should contain more than one singular point, this is impossible. In Figure 11.6(b), the line segments 0102,0203, and 0301 connecting the singular points should be line segments without contact. Now suppose the positive direction on r 1 (i.e. the direction of increasing t) is counterclockwise; thus the positive direction on r2 should be clockwise, and so the positive direction on r3 should be counterclockwise. Then D;Ol cannot be a line segment without contact. Similarly, we can prove COROLLARY. The total number of centers and foci of system (11.1) is at most 2.
§11.
LIMIT CYCLES IN QUADRATIC DIFFERENTIAL SYSTEMS
253
THEOREM 11.3. The relative positions of limit cycles of system (11.1) can only have the following two cases [11]: 1) There exist one or more limit cycles in the vicinity of only one focus. 2) Limit cycles appear in the vicinity of each of the two different foci. Based on Theorem 11.3, the further problems are specific realizations of all possible distributions. In this area, [11], [14], [17], [19], and [20] have done some work. Since in [19] we have given examples to illustrate that system (11.1) can possibly have four limit cycles, we should at least discuss whether the distributions of (1, 0), (2, 0), (3, 0), (4, 0), (1, 1), (1, 2), (1, 3), and (2, 2) can indeed be realized. Here (m, n) represents the distribution of m limit cycles in the vicinity of the first focus and of n limit cycles in the vicinity of the second focus (if it exists).( 2 ) For this problem, past research can be divided into two groups: 1) Give an example or prove from theory that the quadratic differential systems can have distributions (1, 0), (2, 0), and (3, 0) [21], (1,1) [11], (1, 2) [14], and (1, 3) [19], [20]; but there is no way to prove whether the limit cycles of the system under discussion have the exact number or have more than that number. 2) Prove rigorously that the limit cycles of some quadratic system have (1,0) distribution (see Example 3) and the limit cycles of another quadratic differential system must have (1, 1) distribution [180). For distributions (4,0) and (2, 2), it is not clear right now whether they can be realized. We conjecture that (4,x), x ~ 0 and (y,z), y ~ 2, z ~ 2 cannot be realized. EXAMPLE 1 [181]. The system
~; = 6x- y- ~x 2 + xy, when, 0
~~ = x ( 1- ~)
(11.7)
< 6 < ! has a unique limit cycle in the vicinity of (0, 0) and of
(2, 2- 26). When 6 ~ 0, there is no limit cycle; when 6 =!,these two cycles expand and become two separatrix cycles formed by the equator and the line X= 1. For the proof we use the Annular Region Theorem, the theory of rotated vector fields, and the uniqueness theorem of [111). The rest of the proof is left to the reader. EXAMPLE 2 [19). For the system dx dt = -y- 62x- 3x 3 + (1 - 61 ) xy + y 2 , when 0 < ~
«
61
«
1, the distribution (1,3) appears.
(2)For quadratic differential systems we pointed out in §9 that the center and the limit cycle cannot coexist, no matter whether they link together or not.
THEORY OF LIMIT CYCLES
254
FIGURE 11.7 PROOF. First consider the system of equations 2 dx dt = -y - 3x
dy ( 2 ) dt = x 1 + gx - 3y .
+ xy + y 2 ,
{11.9)
It is easy to see that {11.9) has only two finite singular points, 0(0, 0) and N(O, 1); the latter is an unstable coarse focus. After a coordinate transformation, we change {11.9) into a standard form of Bautin (9.30). We get
At = 0,
\
1\4
A 2 -
82 = - v. rnn
o~,
77
A -
1152
3 - 41v'82' 41v'82' \ \ _ 3644 0
1\5
= ,
1\6 -
369 J82.
From this, according to the formulas of §9, we can compute that 1h = ii3 = 0 and iis < 0; hence 0(0, 0) is a stable fine focus of second order of {11.9). The infinite singular point {1, .,, 0) of {11.9) can be determined from the equation 'T/ 3 + ., 2 - 2/9 = 0, which has only one real (positive) root. It is easy to prove that the corresponding singular point is a saddle point, and the direction of the separatrix passing through the saddle point is shown in Figure
11.7. Also on the line 1- 3y = 0, we have dy / dt = 2x 2 /9 ~ 0. This indicates that the trajectory of {11.9) always crosses the line from below to above; hence by the Annular Region Theorem we know that system {11.9) has limit cycles in the vicinity of both 0 and N; that is, there exists a (1, 1) distribution {but it is not known whether system (11.9) has exactly two limit cycles). It is easy to see that the limit cycle r 1 closest to N must be an internally stable cycle, and the limit cycle r 2 closest to 0 must be an internally unstable cycle. Then we apply the method of Bautin in §9. We change the right sides of {11.9), to get
dx dt = -y - 3x 2
+ (1 -
£) xy + y 2 ,
v1
dy 2 x - 3y) , dt = x ( 1 + 9
(11.10)
§11. LIMIT CYCLES IN QUADRATIC DIFFERENTIAL SYSTEMS
255
where 0 < 81 « 1. For (11.10), since ii3 = 2St > 0, 0(0, 0) has become an unstable first-order fine focus, and from Theorem 3. 7 in §3 we know that in its vicinity there will appear again a stable limit cycle r3 (c r2). Since when 81 is sufficiently small both r 1 and r2 do not disappear, the (1, 2) distribution exists for system (11.10). Finally, we add one term -82x to the right side of the first equation of (11.10) to change it into (11.8), where 0 < ~ « 81 < 1. Since 0 changes again from an unstable fine focus to become a stable coarse focus, in a smaller neighborhood, there appears again one unstable limit cycle r 4, and at the same time r1.r2, and r3 still exist. Hence system (11.8) has at least three limit cycles in the vicinity of 0, and there exists at least one limit cycle in the vicinity of N. The proof is completed. A similar example can be obtained in [20]. But the starting system of equations dx dt
= -y -
10x2
dy 2 = x+x -25xy dt
+ 5xy + y 2 ,
-
(11.11)
has no cycles in the vicinity of the point 0 and takes 0 as a third order fine focus. Although this example is not as good as Example 2, it yet corrects a mistake of a symbol of ii7 in Bautin's formula (9.40) for the convenience of our readers. More recently, [182] and [183] extended the range of quadratic systems with distribution (1, 3) so that the coefficients of the starting equations are not all fixed numbers. For example, [183] obtained the following result: Suppose the system of equations dx (2l + b)a 2 2 = -y+lx + xy+ny dt l+n '
~~ = x + ax 2 + bxy
-
(11.12)
satisfies the following conditions: 1) a '=I 0; 2) 3n(l + 2n) :$ n(n +b) < 0; and 3) the infinite singular point is unique. Then we can prove that in the vicinity of (0, 1/n) there exists at least one limit cycle and 0(0,0) is a third-order unstable fine focus, or a second-order stable fine focus. Thus if 0 < -..X « -11 « -€ « 1, then the system (2l + b)a
dx dt
= ..Xx - y + lx 2 + (
dy dt
= x + ax2 + (b + e(l + n) + "') xy
l +n
+ e)
2
xy + ny , (11.13)
a
has at least three limit cycles in the vicinity of 0 and at least one in the vicinity of M.
THEORY OF LIMIT CYCLES
256
Similarly, the system of equations in Example 1 has been extended to become the problem of studying under what conditions the system
dxfdt
= -y +ox+ lx 2 + mxy,
dyfdt
= x(1 +ax)
has two (and only two) limit cycles which do not contain each other.( 3 ) Finally, we give a most obvious example of the distribution (1,0) [17]. EXAMPLE 3. If system (11.1) takes a circle or an ellipse r as a limit cycle, then r is the unique limit cycle of system ( 11.1), and is a single cycle. PROOF. First, after an affine transformation, we can changer to the unit circle x2 + y 2 = 1. It is easy to see that (11.1) should be changed to
= -y(ax +by+ c)- k2(x 2 + y 2 - 1), dyfdt = x(ax +by+ c)+ kt(x 2 + y 2 - 1),
dxfdt
(11.14)
where k~ + k~ i= 0. This system has two singular points, which both lie on the line k2x = k 1 y. After a rotation of axes, we make this line the vertical axis; then (11.14) becomes (we still use x and y to denote the rectangular coordinates)
~~ = -y(a'x + b'y + c')- k(x 2 + y 2 -
~~ = x(a'x + b'y + c'),
1),
where k i= 0. The above system can be changed to
:~ = -y(ax +,By+')')- (x 2 + y2 -
1),
dy dr
= x(ax +,By+')').
(11.15)
According to the hypothesis, the unit circle x 2 + y 2 = 1 should be a limit cycle, and it is not possible to have a singular point on it; that is, we should have (11.16) Moreover, the singular point inside the circle cannot be a center; hence (11.17)
Now we analyze the system with center
~~ = -y(,By +I')- (x2 + y2 -
1),
(11.18)
This system and (11.15) have the same closed trajectory x2 + y 2 = 1 and the same singular points. It is easy to draw the complete graph of its trajectory as in Figures 11.8 (.B :::= -1), 11.9 (.8 < -1), 11.10 ( -1 < .8 < 0), 11.11 (.B > 0), and 11.12 (.8 = 0). (3)Simila.r results were also obtained in [185].
§11. LIMIT CYCLES IN QUADRATIC DIFFERENTIAL SYSTEMS
257
.8<-1 FIGURE 11.8
FIGURE 11.9
a:
,8-0 ,B>D
-1<,8<0 FIGURE 11.10
FIGURE 11.11
FIGURE 11.12
Now we consider o in the system (11.15) as a parameter. Then, as ovaries, (11.15) forms a family of equations, and for o = 0 we get (11.18). Computing (11.19) we know that (11.15) forms families of generalized rotated vector fields with opposite directions of rotation outside and inside the circle x 2 + y 2 = 1 respectively, and that x 2 + y 2 = 1 is a closed trajectory of every equation in the family. Since (11.18) has a family of closed trajectories surrounding one or two centers of index +1, from Theorem 3.2 in §3 we know that foro =F 0 the system (11.15) does not have a closed trajectory outside and inside the unit
258
THEORY OF LIMIT CYCLES
circle. Moreover, from (11.19) we can also see that if the trajectory of (11.15) near the outside of the unit circle gets out of (into) the closed trajectory of (11.18) as 1' increases, then the trajectory of (11.15) near the inside of the unit circle should get into (out of) the closed trajectory of (11.1'8) as 1' increases; that is, x 2 + y 2 = 1 must be a stable or unstable cycle of system (11.15), and cannot be a semistable cycle. Finally, in order to prove that x 2 + y 2 = 1 is a single cycle of (11.15), we can assume it has the parametric equations
x=cost(T),
y=sint(T),
where t(T) is easily seen to satisfy the equation
dtjdT =a. cost+ {3sin t + "Y (> 0 when "Y > 0). Thus, along the unit circle, x and y are periodic functions of t with period 211'. We then calculate the value of the integral of the divergence with respect to 1' around the unit circle once, and it is easy to prove it is not equal to zero; hence the unit circle is a single limit cycle. A necessary and sufficient condition for a cubic system to have a quadratic algebraic limit cycle can be seen in [18] and [187]. Xu Shi-long [188] showed that a necessary and sufficient condition for the algebraic curve yn + xm = 1 to be a limit cycle of a quadratic differential system is n = m = 2; a necessary and sufficient condition for the curve to be a limit cycle of a cubic differential system is either n = m = 2 or m = 2n = 4 or n =2m= 4. Moreover, in [189] and [190] the conditions for existence of a quadratic algebraic curve solution for a quadratic differential system were studied. Since in this section we introduce the general properties of limit cycles of quadratic differential systems, we list all the important properties of quadratic systems which have been proved and will be proved later, for the reader's convenience. 1. A quadratic differential system cannot have a center and a limit cycle (§9). 2. A quadratic differential system whose linear part takes the origin (a singular point) as a star nodal point does not have a limit cycle (§10). 3. A quadratic differential system with two integral lines does not have a limit cycle. 4. A quadratic differential system with one integral line and one fine focus does not have a limit cycle (§15). 5. A quadratic differential system with one integral line can have at most one limit cycle (§15). 6. The locus of the points whose divergence apI ax + aQ I ay = 0 is a line, and any closed trajectory, if it exists, must meet the line.
§11. LIMIT CYCLES IN QUADRATIC DIFFERENTIAL SYSTEMS
259
7. If the divergences of two singular points of a quadratic differential system are both equal to zero, then the system does not have a limit cycle (§15). 8. If the number of singular points of a quadratic differential system is more than two, and one of them is a fine focus, then its limit cycle can only appear in the vicinity of one focus [19]. 9. If a quadratic differential system has two fine foci, they can only be fine foci of first order [191]. 10. A quadratic differential system whose vector field has a center of symmetry can have at most two limit cycles, which should have distribution (1, 1)
(§15). 11. A quadratic differential system possessing a third-order fine focus does not have a line solution [176]. 12. If a singular closed trajectory has three saddle points on it, then it must be a triangle with the saddle points at its vertices [14]. 13. If r l and r2 both are singular closed trajectories with one saddle point on each of them, then they cannot have this saddle point as their common singular point [14]. Exercises 1. Prove that when P2(x, y) = 0 or Q2(x, y) = 0 of system (11.1) is an ellipse, its section of arc contained in the interior of a closed trajectory of (11.1) cannot be greater than half the length of the ellipse. 2. Starting from the system of equations
d d; = -338x + 32y + 169x2
-
16y2 ,
d
d~
= -288x + 18y + 144x2 - 9y2
and constructing a family of uniform rotated vector fields, prove that when the angle of rotation passes through 1r/4 the neighborhood of each of the singular points (0,0) and (1, 1) can generate a limit cycle [11]. 3. First analyze the global structure of the trajectory of the system
dx = xy dy = _!(x- 1)(x + 2) + !y2 + !xy- !y dt , dt 3 2 3 3 and then construct a family of uniform rotated vector fields starting from this system. Prove that when 0 < -0 ~ 1, the (1, 2) distribution of limit cycles can appear [14]. 4. Prove the limit cycles of the system of equations dx dy 2 dt =-y+6x+lx 2 +mxy+ny, dt =x(1+ax+by) when 1) b + n =I= 0 and 2) b + n = 6 = m +a= 0, even if they exist, cannot be monotonically close to each other.
THEORY OF LIMIT CYCLES
260
5. Prove the conclusion of Example 1. 6. Prove that a quadratic differential system with two integral lines does not have a limit cycle. 7. Prove that the singular closed trajectory of a quadr~tic differential system with three saddle points on it should be a triangle with these saddle points as its vertices. 8. Prove the second half of Lemma 11.2 and the conclusion of Remark 2 after that lemma. 9. Draw the graph of the trajectory of the system of equations following Lemma 11.8. 10. Prove the conclusion of the system (11.13) in this book. 11. Prove that r in Example 3 is a single cycle. 12. Prove that if in the system of equations dxjdt
= -y +ox+ lx 2 + mxy,
dyjdt
= x(1 +ax)
we have 2a + m = O,a < O,l < O,l 2 - 8a2 < 0, and 0 < 6 < l/2a, then there are two and only two limit cycles which do not contain each other in the whole plane.
§12. Classification of Quadratic Differential Systems. Limit Cycles of Equations of Class I
For a quadratic differential system dxjdt = P2(x, y),
dyjdt = Q2(x, y),
(12.1)
aside from the relative positions of its limit cycles, more important questions are: For a given quadratic differential system, how can we determine whether it has a limit cycle? If it has one, does it have more, and how many? In studying these problems, either in the form (12.1), where P2 and Q2 are general quadratic polynomials or the form of Bautin (9.30), which we have seen in §9, we always run into some inconvenience. This is because we have to find the coordinates of the singular point; sometimes it is very troublesome to solve for x or y from a quadratic equation. Hence we now first introduce a method of classification; that is, we apply some simple transformations to system (12.1), which may have a limit cycle, to bring it into one of three standard forms, and then we proceed to study them one by one [12]. We may assume P2(x, y) and Q2(x, y) do not have a common factor, for otherwise (12.1) can be simplified to a linear system, which obviously does not have a limit cycle. From the theory of quadratic curves, we know there exists at least one real ~ making the equation ~P2(x,y)
(12.2)
+Q2(x,y) = 0
into a degenerate quadratic curve (if Q2(x, y) = 0 is degenerate, then we take ~ = 0; if P2(x, y) = 0 is degenerate, then we can take~= oo; see footnote 1). When this degenerate quadratic curve represents a point or does not have a real locus, the system obtained from (12.1) by the transformation y' = ~x+y, x'=xis dy' dt
= AP2
+ Q2 =
Q2' (X ','Y) ,
dx' dt
nl('') = r2 X ,y ,
( ) 12.3
where Q~(x',y') = 0 represents a point or does not have a real focus. According to the theory of §11, this system cannot have a closed trajectory. Hence, 261
THEORY OF LIMIT CYCLES
262
we may as well assume that >.P2 + Q2 = R1R2, where~ (i ~ 1, 2) is a real polynomial with degrees of x andy not higher than one, and at least one of them is not a constant. It is easy to prove (do so as an exercise) that if for i = 1, 2 the determinant of the transformation y'
= Y + >.x,
x' =~
(12.4)
is always zero, then the system has one or two integral lines, and also does not have a closed trajectory. Hence we only have to discuss the case when the determinant of transformation (12.4) is not zero for i = 1 or 2. In this case the system (12.1) under this transformation(!) becomes dx' fdt
= P~(x', y'),
dy' fdt = x'(ax' +by'+ c).
(12.5)
We still write x', y' as x, y, and, depending on the values of a, b, and c, we can divide the systems (12.5) into three classes: I. a = b = 0, c # 0, dx dt
= k + Dx + ey + lx 2 + mxy + ny
2
,
dy dt =ex.
(12.6)
II. a # 0, b = 0, c # 0,
~~ = k + lJ x + ey + lx2 + mxy + ny2 ,
dy dt = x(ax +c).
(12.7)
dy dt = x(ax +by+ c).
(12.8)
Ill. b # 0,
~~
= k+8x+ey+lx 2 +mxy+ny2,
There is no need to discuss the case when b = c = 0 but a # 0, since in this case Q2(x, y) ~ 0 holds in the whole plane, and it is not possible to have a closed trajectory. According to the theory of §11, if system (12.6), (12.7), or (12.8) has a closed trajectory, then its interior must contain a unique focus or center with index +1. We translate the origin to the singular point and then apply a suitable transformation x = J.LX 1 , y = vy', t = >.t' to change (12.6), (12.7), and (12.8) into I. dy (I) dt = x,
II.
2 { dxfdt = -y +ox+ lx + mxy + ny2, dyfdt = x(1 +ax), a# 0,
=
(II)
( 1 )If.>.. oo, we first interchange x and y in (12.1) and then apply the transformation (12.4) with .>.. = 0.
§12 .. QUADRATIC SYSTEMS OF CLASS I
263
dxjdt = -y + 8x + lx 2 + mxy + ny2 , dyjdt = x(1 +ax+ by), b ~ 0,
(III)
III. {
respectively. Later we shall study systems of classes I, II, and III in detail one by one. Aside from the method of classification, devised by Chinese scholars, there is another method, used in the Soviet Union. Cherkas [192] divided the systems (12.1) into two classes:(2) {
dxjdt dyjdt
=boo+ xy, = aoo + awx + ao1Y + a2ox 2 + auxy + ao2Y2 = Q2(x, y), dxfdt = ~ox 2
+ y,
dyjdt
(A)
= Q2(x,y).
(B)
For systems of classes II and III, under certain conditions, the nonexistence or uniqueness of limit cycles and the interesting property 7 listed at the end of §11 can be proved by first changing the system into (A) or (B), and then changing it again into an equation of Lienard type. Since there are two systems of classification, one naturally asks whether one can once and for all obtain the relationship between the coefficients of these two classes so that we do not have to carry through the transformation from one class to another every time the problem comes up. Part of this work was done in [193], where Liang Zh~jun obtained the formula for the coefficients of (A) or (B) in terms of the coefficients of (III). Specific examples are as follows: (1) n ~ 0. Let k denote a nonzero root( 3 ) of the equation a+ (b- l)k - mk 2 - nk3 = 0,
where o: = k 3 - 8k 2 + k and (3 = kl- a- nk3. When (3 can be changed to (B), where
= -ko: + k2 -
aoo
= 0,
a2o
= - nk3 -
8k 2 - k [ 3 nk3 2a + bk - nk
a01
= 8k 2 ,
a 11 =
aw ao:
n~ 3
[2a
{12.9)
= 0 and o: ~ 0,
(III)
8k 3 = -k4 ,
1 2 +a (a + bk) (8 k -
+ bk +~(a+ bk)(8k 2 -
k
)]
'
k)],
-1
ao2 = k (a+ bk). n 3 o:
- (2)We --shall later give a detailed introduction to the method of changing (12.1) into (A) or (B). ( 3 )The case when (12.9) has only a (real) zero root was not discussed in [193]; this case is left to the reader as an exercise.
THEORY OF LIMIT CYCLES
264
When {3
i
boo=
Q
0, (III) can be changed to (A), where
[~(a- kl) + k- b'k 2] '
1 a00 = ""fi [ko{3 + ao 2 - b10(k{3 + o(2a + bk)) + b~ 0 (a + bk) + boo(a- kl)], 1 a10 = ""fi[-(k{3 + 2ao)- b2o(k{3 + o(2a + bk))
+ b10(2a + bk) + 2b10b2o(a + bk)], a2o
=~[a+ b2o(2a + bk) +(a+ bk)b~ 0 ],
1 aot = ""fi[k{3 + o(2a + bk)- 2b10(a + bk)], -1
an= lf[a + kl + bk + 2(a + bk)b2o],
1 ao2 = ""fi(a. + bk),
in which blO = b'k2 - k- 2o(a- kl)/{3 and b2o =(a...., kl)/{3. Here we do not consider the case o = {3 = 0, since for this case (III) cannot have a limit cycle. (2) n = 0. When m = 0, we leave this case to the reader as an exercise.( 4 ) Now suppose m i 0. Then we can prove that (III) can be changed to (A), where
boo = l + mb', aoo = m 2 + ma- b(2l + mb') + l(l + mb'), a10 = m 2 + 2am- b(3l + mb'), aot = b, a2o = ma- bl,
au = b + l,
ao2 = 0.
Of course, conversely, we can use the coefficients of (A) or (B) to express the coefficients of (III); and work can also be carried out along this line. Next, how to use b', l, m, n, a, and b under our method of classification to express the quantities tit, v3, tis, and 'V7 from §9 is also an important problem.( 5 ) Here we should first find the formulas for At, ... , As from §9. This can be done following the method mentioned in the paragraph before Theorem 9.2 in §9; that is, we first apply the transformation
_.,,
x=~-e,
-2,
y=~,
(12.10)
( 4 )It was incorrectly observed in [193] that the case n = m = 0 did not have a limit cycle, and the formula was not given there. ( 5 }The proof of the last theorem in §16 illustrating Bautin's formulas for 1i3, tis, and 1i7, and the system (12.1} in Bautin's form are sometimes quite convenient for studying certain properties of quadratic differential systems.
§12. QUADRATIC SYSTEMS OF CLASS I
265
and then rotate the axes, taking tan tp =
-(2a+bt5)~ M2 + 26(a- m) - 4{1 + n)
-:-=-':-:-:---'-.,....-~,.---......,..
(12.11)
and e=xcostp-ysintp,
77=Xsintp+ycostp,
r=!v'4-6 2t.
(12.12)
We obtain dxjdt = AtX- y- A3x 2 + (2A2 + As)xy + A6y 2, dyjdt = x + AtY + A2x 2 + (2A3 + A4)xy- A2y 2,
(12.13)
where At= 6/~. Moreover, when 6 ::F 0 the formulas for A2, ... , A6 are too complicated, and there is not much use for them; hence they will not be written out in detail here. Now it is important to know the values of all the coefficients of (12.13) when At = 0; that is, when 6 = 0 and (0, 0) is a fine focus. In this case a tan tp = -1 - , +n
sintp=
cos tp =
Ja2 +~l+n)2'
l+n Ja 2 +(l+n)2'
-;:=;r=::::::;:;=~
At =0,
t
A3 = [a2 + (l + n) 2 312 [l(l + n) 3 + a(m + a)(l + n) 2 + a 2(l + n)(b + n)], A6 = [a2 + (l + n) 2]-312[ma(l + n) 2 + a 2(b -l)(l + n) - a 4 - n(l + n) 3], 2A2 +As= [a2 + (l + n) 2t 312[(2a + m)(l + n)a 2 + ba 3 + a(2l- 2n- b)(l + n) 2 - m(l + n) 3 ], A2 = [a2 + (l + n) 2t 312[na 3 + ma 2(l + n) + a(l- b)(l + n) 2 - a(l + n) 3], 2A3 + A4 = [a2 + (l + n) 2t 312[a2(l + n)(b- 2l + 2n) + a(2a + m)(l + n) 2 - ma3
-
b(l + n) 3 ]. (12.14)
From this we can also obtain A -ma- (l + n)(b + 2l) 4= Ja2+(l+n)2 '
A _ a(b + 2l)- m(l + n) sJa2+(l+n)2 ' ..\a- ..\6 = Ja 2
(12.15)
+ (l + n)2.
Thus, the formulas for At, ... , A6 in terms of 6, l, m, n, a, and b can be completely obtained. From this it is not difficult to further obtain the formulas for ila, Us, and v7.
THEORY OF LIMIT CYCLES
266
Comparing the conditions for the origin being a center in Theorem 9.2 in §9, we get THEOREM 12. 1. The four groups of conditions for system (III) to have the origin as its center are: 1) = 0, m(l + n)- a(b + 2l) = 0, and ma + (l + n)(b + 2l) = 0 (equivalent to >.1 = >.4 = >.s = 0); 2) = 0 and a= l + n = 0 (equivalent to >.1 = >.3- >.s = 0); 3) = 0, m(l + n) = a(b + 2l), and a[a 2(n + b + 21) - (b + n)(l + n) 2] = 0 (equivalent to >.1 = >.2 = >.s = 0); and 4) = 0, m = 5a, b = 3l + 5n, and ln + 2n2 + 2a = 0 (equivalent to >.1 = >.s = >.4 + 5>.3- 5>.s = >.3>.s- 2>.~- >.~ = 0).
o
o o
o
The proof is omitted, and can be used as an exercise. REMARK 1. When we transform the equations, cp can differ by 1r and the corresponding >.2, ... , >.s all change signs, but >.1 and Vi (i = 3, 5, 7) remain unchanged. REMARK 2. Although in theory we can obtain formulas for 'Vs and v7 to distinguish the stability of the origin and the order of its being a fine focus, these formulas are very complicated, and it is not easy to simplify them (however, this is not difficult for v3). Li Cheng-zi [183] first studied necessary and sufficient conditions for the quadratic system of the form dxjdt
= -y + a2ox2 + auxy + ao2Y 2,
dyjdt = x + b2ox 2 + buxy + bo2Y 2
(12.16)
with (0, 0) as its kth-order fine focus (k = 1, 2, 3) and then derived necessary and sufficient conditions for (0, 0) to be a center. These two groups of results are rather simple and easy to apply. In the following we shall introduce several theorems of [183], but their proof is omitted. THEOREM 12.2. For system (12.16) introduce the quantities
W1 = Aa- B/3, W2 = [f3(5A- /3}
+ a(5B- a)b,
(12.17)
W3 = (A/3 + Ba)lo, where
A= a2o + ao2, B = b2o + bo2, a= au+ 2bo2, f3 = bu + 2a2o, 1 = b2oA3 - (a2o - bu)A2B + (bo2 - au)AB 2 - ao2B 3,
o=
a~ 2 + b~o + ao2A + b2oB.
(12.18)
§12. QUADRATIC SYSTEMS OF CLASS I
Then the following assertions are true : 1) (0, 0) is a kth-order fine focus (k kth group of conditions holds: 1°) Wt =1=
o;
2°) Wt =
o, w2 =1= o;
267
= 1, 2, 3)
if and only if the following
3°) Wt =
w2 = o,
w3
=1=
o.
{12.19)
2) The stability of the kth-order fine focus is decided by the sign ofWk: it is stable for wk < 0 and unstable for wk > 0. 3) The origin is a center if and only if Wt = W2 = W3 = 0. In order to study the stability of (0, 0), when only Wt = 0 we have to examine W2, and when only Wt = W2 = 0 we have to examine W3. Hence if we use the quantities Wt =Aa-B/3, when A=/: 0, when B =/: 0, when A= B =0; when A=/: 0, when B =/: 0, when A= B =0
(5A- f3)/3"'f,
W~ = { (5B - a)a"'f,
0, A/316, W~ = { Ba"'fO, 0,
(12.20)
to replace W., W2, and W3 in (12.17), then the conclusion of Theorem 12.2 remains unchanged. COROLLARY. System (12.16) takes the origin as its center if and only if at least one of the following four groups of conditions holds: 1) A= B = 0 (equivalent to At= A3- As= 0). 2) a= /3 = 0 (equivalent to At= A4 =As= 0). 3) Aa- B/3 = "Y = 0 (equivalent to At= A2 = -Xs = 0). 4) 5A- /3 = 5B- a= 6 = 0 (equivalent to At =As= A4 + 5A3- 5-Xs = >.3-Xs - 2-X~- A~ = 0). THEOREM 12.3. For system (III), introduce the quantities
W t = m(l + n)- a(b + 21), W 2 = ma(5a- m)[(l + n) 2 (n +b)- a 2 (b + 2l + n)],
(12.21)
W 3 = ma 2 [2a 2 + n(l + 2n)][(l + n) 2 (n +b)- a 2 (b + 21 + n)]. Then system (12.2) (with Wi changed to Wi) holds for system (III). System
(III) takes the origin as its center if and only if at least one of the following groups of conditions holds: 1) a= l + n = 0. 2) m(l + n) = a(b + 2l) and a[(l + n) 2 (n +b)- a2 (b + 2l + n)] = 0, a =f. 0.
THEORY OF LIMIT CYCLES
268
3) m = b + 21 = 0. = 5a, b = 3l + 5n, and 2a2 + n(l + 2n)
4) m
= 0.
In the following we study nonexistence, existence and uniqueness of limit cycles of equations of class I. First we prove a nonexistence theorem. THEOREM 12. 4. The system of equations
dx/dt = -y + lx 2 + mxy + ny 2 ,
dyjdt = x
(I)o=O
takes the origin as its center when m(l + n) = 0, and does not have a closed trajectory or a singular closed trajectory when m(l + n) -=f. 0. PROOF. We use the method of Dulac functions to prove this theorem. First, for the case of n = 0, we take the Dulac function as
B(x, y) = exp(mx- 2ly- !m2 y 2 ). Then we have
a
ax (BP2)
a
+ ay (BQ2) =
(12.22)
mlx 2 exp(mx- 2ly- !m2y 2).
When ml = 0, the right side of the above formula is equal to zero, and {12.22) becomes an integrating factor of system (I)o=Oi it is clear that the origin is a center. When ml ::I 0, the right side of the above formula keeps a constant sign in the whole plane; hence (I)o=O does not have a closed trajectory or a singular closed trajectory. If n ::I 0 in system (I)o=O• then we take
B(x,y)
= e(o:mn- 2I)Y(x- any+ a)"m,
(12.23)
where a = (m + v4n2 + m 2)/2n2 is a positive root of n 2a 2 - ma- 1 We can compute
a
-(BP2)
= 0.
a + -(BQ2) = am(l + n)x2 (x- nay+ a)mo:-le(o:mn- 2l)y.
ax ay Hence, when m(l +n) = 0, the origin is a center; when m(l +n) -=f. 0, the right side of the above formula keeps a constant sign in the half-plane x-nay+a > 0. Note that the origin lies in this half-plane, and the line x- nay+ a= 0 is crossed by the trajectories of system (I)o=O all in the same direction; hence (I)o=O does not have a closed trajectory. Moreover, since the saddle point (0, 1/n) lies on the line x- nay+ a= 0, if a singular closed trajectory exists, then it must pass this saddle point, and part of it consists of a certain pair of separatrices passing this saddle point. From the previous discussion, we know this is impossible. The theorem is completely proved. The figures showing the global trajectories of (I)o=O which were clearly described in [194] are shown in Figures 12.1-12.7, in which we have assumed
§12. QUADRATIC SYSTEMS OF CLASS I
FIGURE 12.1
FIGURE 12.2
FIGURE 12.3
FIGURE 12.4
m = 1. The case of a center (m = 0 or l and the figures will not be given here.
+ n = 0)
269
can be deduced from §9,
THEOREM 12. 5. System (I) does not have a closed or singular closed trajectory when m(l + n) = 0 but 6 =I 0, nor for 6m(l + n) > 0; however, for Om(l + n) < 0 and 161 sufficiently small, system (I) has a unique limit cycle. PROOF. When 6 varies, system (I) forms a complete family of generalized rotated vector fields with 6 as parameter. Since (I) has a family of closed trajectories when m(l + n) = 6 = 0, it does not have a closed or singular closed trajectory when m(l + n) = 0 but 6 =I 0. Next, if 6m(l + n) < 0, then when 161 increases from 0, the stability of the origin happens to change. lienee from Theorem 3. 7 in §3 we know that limit cycles will appear near the
270
THEORY OF LIMIT CYCLES
FIGURE 12.5
FIGURE 12.6
n>-1>0 FIGURE 12.7
ongm. Moreover from §6, VII we know that when 181 is sufficiently small, there exists a unique limit cycle in the vicinity of the origin. (This conclusion is also true even for system (III) provided that 8[m(l +n)- o:(b+ 2l)] < 0; but for (III), 8 varies and does not form a family of rotated vector fields.) Since this limit cycle monotonically expands and covers some neighborhood as 161 increases from zero (if there is more than one limit cycle, some limit cycle may monotonically contract; but we shall see later that no matter how large 161 is, the limit cycle, if it exists, is unique), from the nonintersecting theorem of §3 (Theorem 3.2) we know that, when 8m(l + n) > 0, (I) does not have a limit cycle. The theorem is completely proved.(6 ) (6)When 6m(l+n) > 0, the fact that (I) does not have a limit cycle can be seen from the comparison theorem for differential equations. For example, suppose m(l + n) > 0; then we have shown that the origin is an unstable focus of (I)o=O• and the system does not have a limit cycle. Hence the origin of (I)o=O is also an unstable focus, and, when t increases, the
§12. QUADRATIC SYSTEMS OF CLASS I
271
The following work is to remove the hypothesis "lhl is sufficiently small" in Theorem 12.5; that is, we have to prove that if system (I) has a limit cycle, it is unique. In China this problem had been completely solved by 1967 (see [16], [195], [196], and [197], although the last two were only published in 1975 and 1978 respectively). In the Soviet Union there were [118] and [198]. Our method of proof shows that in order to solve the problem of uniqueness of a limit cycle for equations of class I, the uniqueness theorem of [111] is all we need. Since m f. 0 is a necessary condition for existence of a limit cycle of equations of class I, without loss of generality, we shall assume that(1) m
= 1,
l + n > 0,
h < 0.
(12.24)
LEMMA 12.1. When 1) n > 0 and h + 1/2n ~ 0; or 2) n ~ 0 and h + l ~ 0; or 3) l ~ 0 and h + n ~ 0, (I) does not have a closed or singular closed trajectory. PROOF. 1) We adopt the method similar to Theorem 1.13 of §1. Take
1- 4nl ( x )] M(x,y) = exp [ 2n y + 2n '
B(x,y) = 2nM(x,y),
P=Q2B-P2M = exp [ 1
~:nl (y + 2:)] {2nx- ( -y + hx + lx 2 + xy + ny 2)},
Q = -P2B
= -2n( -y +ox+ lx 2 + xy + ny 2) exp [ 1 ~:nl
(y + 2:)].
If (I) has a closed or singular closed trajectory r in the neighborhood of the origin, it must be positively oriented; thus from Green's formula we have
But it is easy to compute the value on the left side of the above formula:
1i(BQ2- P2M)P2dt- BP2 · Q2dt =-
i
MPfdt < 0,
trajectories of (1)6>0 starting from some regular point lie on the outside of the trajectories of {1)6=0i hence (1)6>0 also does not have a limit cycle. (1)In order to study any quadratic system, we can in general apply a suitable similarity transformation of :z:, y, and t to three fixed nonzero coefficients of system {III) for some fixed values.
THEORY OF LIMIT CYCLES
272
and the value of the right side:
- j j (1 + 2n6)M(x, y) dx dy > 0. intr
This contradiction shows that r does not exist.( 8 ) 2) When n :::; 0, l > 0 must hold. Taking a Dulac function B(x, y) (1- x)- 1 for system (I) we get !_(BP.)
ax
2
!_(BQ ) = 6 + l + ny2 -l(x- 1)2 2 (1- x) 2 •
+ ay
=
(12.25)
When 6 + l :::; 0, the right side of the above formula keeps a constant sign :::; 0, and the line 1- x = 0 is a line without contact of (I). Hence, the conclusion of the lemma holds. In fact we can relax the condition on 2) to ben:::; 0 and 6(1 + n6) + l :::; 0. But in this case the proof is very complicated, and so we omit it here (it can be found in the first edition of this book). 3) When n > 0 and l :::; 0 but 6 = 0, the trajectories of (I)o=O are shown in Figures 12.5 and 12.7. From this we can see that the separatrix passing through the saddle point (0, 1/n) from the right lies below the line through the saddle point and touches the line tangentially at that point. Now if 6 + n = 0, then the tangent line to the separatrix at (0, 1/n) is ny+x = 1. Let V = ny+x. We compute the rate of change of V along the trajectory of (I), and get
I
dVd = (6 + n)x + lx 2 = lx 2 :::; 0. t V=l From this we see that as 6 decreases from 0 to -n, in Figures 12.5 and 12.7 the separatrix entering (0, 1/n) from its right not only turns to the outside of the!separatrix leaving (0, 1/n) from its left, but also has turned to the upper part of the line ny + x = 1. Thus in this case the limit cycle, of course, does not exist. When 8 < -n, the same is true. The lemma is completely proved.
COROLLARY. When n > 0 and l :::; 0 but n have a limit cycle as long as 6 :::; -1/-/2.
+ l > 0,
system (I) does not
PROOF. This is so because there is· a number not less than 1/-/2 between nand 1/2n. LEMMA 12.2. Suppose 6 system of equations
< 0, l ~ 0, 1- al > 0, and a -l > 0. Then the
dxjdt = -y + 8x + lx 2 does not have
+ xy, dyjdt = x + ay a closed trajectory when 8 + l = 0.
(12.26)
( 8 )This result was first obtained in [194], but the proof was rather complicated, and also there was an additional condition I > 0. Here we adopt the method of proof in [27],
273
§12. QUADRATIC SYSTEMS OF CLASS I PROOF. Apply the transformation x = 1- e-x, y =
y. Then (12.26)
becomes dyjdt = 1- e-x+ ay. Change the system again into a second order differential equation and get
i +(le-x- a)i + (1- al)(1- e-x) = 0. Finally, changing it to the Lienard plane yields
-
~
~
dt = -z + (ax+ le-x - l),
-
dt = (1 - al)(1- e-x).
(12.27)
Now suppose (12.27) has a closed trajectory r. We may as well assume r is the one closest to the origin. Since a -l < 0, it is easy to see that (0, 0) is a stable focus or nodal point of (12.27). However, on the other hand, computing the integral of the divergence along r once, we get
i
(a -le-x) dt
= =
that is,
r
i i
(a -l) dt +
i
l(1- e-x) dt
(a - l) dt < 0;
is also a stable cycle, which is impossible. The lemma is proved.
LEMMA 12.3.
When l > 0, n > 0, and o ~ -1/l, the system of equations dxjdt = -y- F(x),
dyjdt = g(x)
(12.28)
does not have a limit cycle in the half-plane x < 1/n, where -1
F(x) = p[(lx + ol + 1)ePROOF. When
o=
l X-
ol- 1],
-1/l, (12.28) becomes
dxjdt = -y + (xjl)e-lx, Let
dyjdt = (x- nx 2 )e- 21 x.
(12.29)
r
1 H(x, y) = 2y2 + Jo (s- ns2)e-2ls ds.
Then H(x,y) = Cis a family of closed curves containing (0,0) which is a family of closed trajectories of the system of equations dxfdt = -y,
(12.30)
Computing dH/dt along the trajectory of (12.29), we get dH _ aH dx dt - ax dt
+
8H dy _ x 2 (1 ) -3lx O ay dt - l nx e > ,
1 when x < -. n
THEORY OF LIMIT CYCLES
274
Hence (12.29) does not have a limit cycle. Since (12.28) can be obtained from (I) by a change of variables,( 9 ) and (I) forms a family of generalized rotated vector fields with respect to the parameter 6, (12.28) does not have a limit cycle when 6 < -1/l. The lemma is completely proved.(1°} THEOREM 12.6. For arbitrary 6,l, and n, system (I) has at most one limit cycle. PROOF. By the previous discussion, we can assume that (12.24) holds; that is, l + n > 0 and 6 < 0. In the following, we consider several cases. (1) The case when n = 0 or l = 0. When n = 0, (I) becomes
= -y + 6x + lx 2 + xy, dyjdt = x. y = -y', and t = -r. Then (12.31) becomes
dxjdt
Let x
= 1- e-x',
dx' I I dr = -y - [(6 + l)ex - (6 dy' dr
I
+ 2l) +le-x] =
(12.31)
~y'- F(x'),
I
= 1- e-x = g(x'),
where g(x') = 1- e-x' is continuous, x'g(x')
G(±oo)
= fo±oo g(x) dx = +oo.
Moreover, since f(x') = F'(x') continuous, f(O) = c5 < 0, and
=
(c5
_:!:___ [f(x')] = e-x'[-6
dx'
g(x')
> 0 when x' # 0, and
+ l)ex'
-le-x' we see that f(x') is
+ (6 + l)(ex'- 1) 2 ] O (1-e-x')2 > '
since by Lemma 12.1 we know that when a limit cycle exists, c5 + l > 0. Thus, f(x')/g(x') is a nondecreasing function in both (-oo,O) and (O,+oo), and by Theorem 6.4 in §6 we can prove the uniqueness of a limit cycle.( 11 ) When l = 0, (I) becomes dx/dt
= -y + 6x + xy + ny2 ,
Let x' = y, y' = x - 6y dx' /dr
h 2 , and r = = -y'- F(x'),
dyjdt = x.
(12.32)
-t. Then (12.32) becomes dy' /dr
= g(x'),
(12.33)
(9)See system (13.34). (IO)See footnote 6. (11 )In fact what we have used is only a special case of it; that is, the uniqueness theorem in [111], whose conditions can be seen in the remarks after Theorem 6.4. Whenever we mention Theorem 6.4 in this book, we mean the uniqueness theorem in [111] unless stated otherwise.
§12. QUADRATIC SYSTEMS OF CLASS I
275
where F( x') = 8x' + x'2 /2 and g( x') = x' - nx'2. It is easy to see that if a limit cycle exists for (12.33), it must lie in the half-plane x < 1/n. We note that 1 + n8 > 0, and hence, as in the case n = 0, it is easy to verify that f(O) = F'(O) = 8 < 0 and f Jg is a nondecreasing function in ( -oo, 0) and (0, 1/n). This proves that (12.33), and hence (12.32), has at most one limit cycle. (2) The case when l > 0 and n > 0. Apply the transformation X=
Uelx',
dtjdr = e-lx'
y=x',
to (I). We get
au = dr
( -x'
+ nx'2 )e- 21 x' + (8 + x')ue-lx''
dx' dr
=u.
x', we get x' = (-x' + nx'2)e- 21 x' + (8 + x')e-lx' x'.
Changing it again into a second order equation in
Finally transforming it into the Lienard plane, and changing r to-T, we get dx' dr
d
= -y' + lx' + 81 + 1 e-lx' _ 81 + 1 [2
I
= -y' _ F(x')
[2
'
(12.34)
I
_]f__ = (x' - nx'2)e-2lx = g(x'). dr If a limit cycle exists for (12.34), it must lie in the half-plane x Then we have f(O) = F'(O) = 8
< 0,
x'g(x') = x' 2 (1- nx')e- 21 x'
< 1/n.
> 0 when x' =/:: 0.
To prove that (I) has at most one limit cycle, we only have to show that in (-oo, 0) and (0, 1/n), the inequality
_.!!.._ [f(x')] = W(l, x') elx' > 0 dx' g(x') (x' - nx' 2 ) 2
(12.35)
holds, where
+ (l + n- ln8)x' 2 + (2n8 + l8)x'- 8 = ( -8- x')(lx'- 1)(nx'- 1) + (1 + n8)x' = W1(x') + W2(x'), = (-8- x')(lx'- 1)(nx'- 1} and W2(x') = (1 + n8)x'.
W(l, x') = -lnx'3
(12.36)
with W 1(x') In order to prove (12.35), except for the case when x < 1/n, l > 0, and n > 0, owing to Lemma 12.1 and Lemma 12.3, it is sufficient to prove that W(l,x') > 0 under the conditions -1/n < 8 < 0 and -1/l < 8 < 0. i) When x' ~ 0, from the first line of (12.36) we see at once that W(l,x') > 0.
THEORY OF LIMIT CYCLES
276
ii) When 0 < x1 ~ -6, since W1 (x1) > 0 and W2(x 1) > 0, W(l, x1) > 0. iii) When -6 < x1 < 1/n, we note that W(l, -6)
a
= W(O, -6) = -6(1 + n6) > 0, a
ax1W(l, -6) = -l6(1 + n6) > ax1W(O, -6) = 0;
hence there is an
E
> 0 such that when -6 < x1 < -6 + E we have W(l, x1) > W(O, x1).
(12.37)
But W(l, x1) - W(O, x1) = lx1( -6 - x1)(nx1 - 1) has only the three zeros x1 = 0, x1 = -6, and X1 = 1/n. Hence (12.37) holds in the whole interval -6 < x1 < 1/n. It is clear that W(O, x1) = nx12 + 2n6x1- 6 = n(x1+ 6) 2 - 6(1 + n6) > 0;
hence when -6 < x1 < 1/n we have W(l, x1 ) > 0. Combining i), ii), and iii), we get (12.35) at once; (3) The case n < 0. Apply the transformation x = x1+ >..y1,
I
y = y'
(12.38)
= -1+v'1-4nl >0 2l
(12.39)
where the number >.
is a positive real root of
(12.40) Then (I) changes to dx1/dt = (6- >..)x1+ (6>..- >.. 2 - 1)y1 + lx 12 + (2l>.. + 1)x'y1, dy1/dt = x1+ >.y1.
(12.41)
Again let Xt=
2).l+1 I >..2-6>..+1x,
2).[+1
I
Yt= v'>..2-o>..+1y,
7'=VA2-6).+1t; (12.42)
then (12.41) changes to dxl dr = = ~~
lv'>..2- o>.. + 1 2 x1 + 2 ).[ + 1
6- >..
V..\ 2 _ 6..\ + 1 Xt - Yl + -yl + 6'x1 + l'x~ + XtYll
dr = Xt + v..\2
A
_
o).
XtYl
(12.43) I
+ 1 Yl = Xt +a Yt,
§12. QUADRATIC SYSTEMS OF CLASS I
277
where
6'
=
a'=
6 -,\
v'-\ 2 -6,\+1 ,\
y',\2-6,\+1
Note that
<0
'
(12.44)
> 0.
6, + l'
= (,\l +
1)6 + (l + n) (2,\l + 1)y,\2- 6,\ + 1'
(12.45)
and when 6 = 0 we have 6' + l'
a , ,
86 (6
+ l) =
> 0. Moreover, A3 l- A2 l6 + 3,\l + ,\ 2 2(2,\l + 1)(,\2- 6,\
6,\ + 2
+ 1)3/2
> o.
(12.46)
We can see that 6' + l' is a monotonically increasing function of 6; as 6 goes from 0 to negative values, 6' + l' monotonically decreases. At the same time
a' _ l'
=
,\l6 - (l + n) (2,\l + 1)y',\2- 6,\ + 1
< O.
From Lemma 12.2 and (12.45) we know that when 6' = -(l + n)f(,\l + 1) the system (I) does not have a closed trajectory. But (I) forms a family of generalized rotated vector fields with respect to the parameter 6; hence (I) does not have a closed trajectory when 6 ~ -(l + n)f(,\l + 1).(1 2 ) Thus, from (12.46) we know that (12.43) does not have a closed trajectory when {/ + l' ~ 0. Hence, later on, in order to study the uniqueness of a limit cycle of (12.43), we may as well assume that 6' + l' > 0. In order to apply Theorem 6.4 in §6 to prove the uniqueness of a limit cycle of system (I), we can first let Yl = -jj and x1 = x in (12.43), and change the sign of r so that the origin becomes unstable. Thus we get dx/dr = -ii- 6'i -l'x 2
+ xjj,
djj/dr
= i - a'jj.
(12.47)
Then, similarly to Lemma 12.2 for system (12.26), we can apply the same change of variables to (12.47), change it again to a second order differential equation, and finally transform it into the Lienard plane to get
~;
= -y-
((6' + l')eJ: + l'e-x + a'x- 6'- 2l'J = -y- F(x),
~~ = (1- a'l')(l- e-J:) + a'(6' + l')(ex -1) = g(x)
(12.48)
(we still use the independent variable t and the dependent variables x and y). ( 12 )See
footnote 6.
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THEORY OF LIMIT CYCLES
Now we only have to examine whether the conditions of Theorem 6.4 are satisfied. We have
W + l')ex -l'e-x +a',
f(x) =
f(O) = a' + o' < Ol
(12.49)
where f(x) = F'(x). Moreover,
~ [f(x)] dx g(x)
{
(1- 2a'l')(o' + l')(ex/ 2
-
e-x/ 2 ) 2
+ (a'l'- 1)(a' + o')e-x- a'(O' + l')(ai + o')ex [(1- e-x)(l- a'l') + a'(O' + l')(ex- 1)]2
-
}
>0 (12.50)
since
o' +l' > o,
1 1 - 2a'l' = 2Al + 1
-Al-l
> 0' a'l' - 1 = 2Al + 1 < 0' a' + vr:l < 0.
In addition, from the equality g(x) = (1-e-x)(1-a'l' +a'(O' +l')ex) we know that the locus of g(x) = 0 is the vertical line x = 0; hence xg(x) > 0 when x =f. 0. Combining the above results, we know at once that all the conditions of the uniqueness theorem of [111] are satisfied. (4) The case l < 0. Here we still take A ( < 0) as indicated in (12.39). Then (12.41)-(12.49) can be similarly established. For example, the transformation (12.42) still has meaning, because A2 -OA+ 1 is positive when 161 is very small, and when A2 - OA + 1 = 0 system (12.41) has an integral line x' = 0 passing through the origin, and it clearly does not have a closed trajectory. Hence in order to study the uniqueness of a limit cycle, we may as well assume that A2
-
OA + 1 > 0 or
o
> (A 2 + 1)/A.
(12.51)
Using (12.51), it is easy to see A3 l- A2 lo + 3Al + A2
-
OA + 2 =(A- 6)(A 2 l +A)+ 2(1 + 2Al)- Al
= -n(A- 6) + 2(1 + 2Al)- Al = -A(l + n) + 2(1 + 2Al) +no
> -A(l+n)+ = 1 + 2Al
nA 2 + n n -lA 2 A +2(1+2Al) = A +2(1+2Al)
> 0.
Hence (12.46) also holds. Finally, since clearly a' - l' < 0, Lemma 12.2 also holds. By Lemma 12.1, we can assume f/ > -1/2n, so that the factor on the right of (12.50) has a positive minimum; therefore (12.50) also holds. The theorem is completely proved. With this theorem and the theory of rotated vector fields, we know at once that when om(l + n) < 0, as 161 increases from zero the unique limit cycle which is generated because of change of stability of the origin will expand monotonically; finally, when 6 takes the value 6* = f(l, m, n), it will meet
§12.
279
QUADRATIC SYSTEMS OF CLASS I
4 l
FIGURE 12.8 a finite or an infinite saddle point to become a separatrix cycle and then disappear. From the several cases in Figures 12.1-12.7 it is easy to see that for n ~ 0 there are two infinite saddle points and a section of the equator on the separatrix cycle, but for n > 0 there is only one finite saddle point (0, 1/n) on the separatrix cycle. = f(l, m, n) is algebraic or transcendental, and Whether the function whether the 'Separatrix cycle is an algebraic curve or a transcendental curve, are two problems which have not yet been solved. In this area, I. G. Rozet [199] did some calculations. He transformed (I)m=l to
o•
~~
= -y +ox+ px 2
+ qxy- y 2 ,
dy dt
-=X
and then, using the results of many calculations, sketched the branch curve in Figure 12.8 (the case n = 0) and the branch surface in Figure 12.9. However, even in the case n = 0 he did not obtain an approximate representation formula for the curve fJ = f (l). There are a few results on the range of variation of fJ to guarantee existence of a limit cycle of (I) by qualitative methods, some of which can be found in the exercises for this section. REMARK 1. Equations of class I can be directly transformed to a secondorder nonlinear equation; hence their practical value is very high. REMARK 2. By Theorem 12.4 we know that a focus of equations of class I can at most be a first-order fine focus, but for equations of class II it can be a third-order fine focus. (See Exercise 4.) REMARK 3. In [200] and [201], Russian mathematicians gave a new proof of Theorem 12.4. This method of proving nonexistence of a closed trajectory was generalized and applied in [202] and [203]. In [202] a result equivalent to part 1 of Lemma 12.1 was obtained. When l > 0 it is better than 1), but when l < 0 it is not as good as 1).
280
THEORY OF LIMIT CYCLES
FIGURE 12.9
Exercises 1. Prove the inference before formula (12.4). 2. Discuss the relationship between the two methods of classification when (12.9) has zero as its only real root and when m = n = 0. 3. Prove Theorem 12.1. 4. Use Bautin's method of §9 to prove that equations of class II can have three limit cycles near the origin. 5. Prove that (12.17) of Theorem 12.2 can be replaced by (12.20). 6. Prove the validity of Figures 12.1-12.7 and sketch the global figure of (I)h"=O when m(l + n) = 0. 7. Complete the proof of condition 2) of Lemma 12.1. 8. Suppose that in (I), l = 0, m = 1, 8 < 0, and 0 < n ~ 1/3, ..j-8/n ~ n/2; then show that there does not exist an unstable limit cycle near the origin. 9. Suppose 4ln = 1 in (I)m=l· First prove that when 8 = -2l = -1/2n, (I)m=l does not have a limit cycle, and then show that if a limit cycle exists, it must be unique [195]. 10. Prove that when a: = f3 = 0 system (III) does not have a limit cycle, and when m = n = 0 limit cycles may exist. 11. Use a method similar to the proof of Lemma 12.2 to prove that system (I) does not have a limit cycle when n > 0 and 8 ~ -1/n.
§13. Global Structure of Trajectories of Equations of Class II without Limit Cycles
In the previous section we have seen that there are two main problems for equations of class I: one is to determine the range of variation of the coefficient of x (i.e., o) in P2 (x, y) which guarantee the existence of a limit cycle, and the other is the problem of uniqueness of a limit cycle. For the equations of class II, in addition to the above two problems, there are three other important questions: 1) H we already know that the system does not have a closed trajectory, how can we determine the global structure of its trajectory? 2) When the system has two singular points of index +1, how do the generation and disappearance of limit cycles affect each other? 3) When there may be more than one limit cycle in the neighborhood of a singular point, how can we solve the problem of having at most two or at most three limit cycles? In this section we start by studying the first problem. It is easy to establish the following fact. THEOREM.
The system
dx/dt
= -y + mxy + ny2 ,
dyjdt = x(1 + ux)
(13.1)
has one or two centers when mn = 0, and does not have a closed trajectory or a singular closed trajectory when mn # 0.
In the proof we take the Dulac function B(x, y) = 1/(1- mx).
(13.2)
The details are omitted (consider it as an exercise). When mn = 0, the global structure of the trajectory of (13.1) is easily determined (form= 0, see Figure 13.5; for n = 0, consider it as an exercise). Now suppose mn # 0. Then after a suitable affine transformation of x, y, and t, we can assume n = -1 and a < 0. Thus we obtain the system of equations dxjdt = -y(l + y- mx), 281
dyfdt = x(l +ax).
{13.3)
282
THEORY OF LIMIT CYCLES
This system has four singular points: 0(0, 0) is a focus and is stable when m > 0, unstable when m < 0; M(O, -1) is a saddle point, and the others are N( -1/a, 0) and R( -1/a, -(a+ m)/2). It is easy to compute that the characteristic roots of the linear approximate system of (13.3) at N are ±J(a + m)/a; hence when m <-a, N is a saddle point, and so R is a focus or a nodal point, lying below N; when m = -a, R = N becomes a higherorder singular point; and when m > -a, N is a focus and R is a saddle point lying above N. In the following we first study the global structure of the trajectory of (13.3) for the case m < -a. First, we study the direction of crossing of the trajectory on the line MN : x + yja = -1/a. Let U = x + yja. Then on MN we have y -dU = ( x +dt a =
1)
+ -a
(m- ~+a)
(x- ay) +
(m- -1+a) a
xy
xy.
From this we can see that MN is a trajectory for m = 1/a- a, and for # 1/a - a this line is divided into three segments by the two singular points M and N, and on each segment all the trajectories have the same direction of crossing. Next we study the infinite singular points. Transforming (13.3) into homogeneous coordinates and letting x = 1 and dt / dr = z, we get
m
~; = z(yz + y2 -
my),
~~ = z +a+ y2 z + y3 -
my 2.
(13.4)
From this we can see that the y-coordinate of the infinite singular point Ai(1, Yi, 0) satisfies the equation y3 - my 2 + a = -0,
(13.5)
when a(27a- 4m3) < 0 or 27a > 4m3, (13.5) has three different real roots, and the corresponding infinite singular points are Ai(1, Yi, 0) (i = 1, 2, 3). It is easy to see that we should have Yl < Y2 < 0 < Y3. If 27a = 4m3, then A1 = A2 becomes an infinite higher-order singular point; if 27a <4m3, then A1 and A2 disappear, and only A3 remains. It is easy to compute the two characteristic roots of the linear approximate system at the singular point Ai :
Using (13.5), we see that
>.1 = -afyi < 0 if i
= 1, 2;
> 0 if i
= 3.
(13.6)
§13. CLASS II EQUATIONS WITHOUT LIMIT CYCLES
283
Moreover, A2 represents the slope of the curve u = yg- my2 +a (a< 0, m < -a) in the (y,u)-plane at its point of intersection (Yi,O) with the y-axis. Hence, for m < 0,
A2
>0
for i = 1, 3;
< 0 fori= 2.
(13.7)
When m ~ 0, there is only one Ag(1,yg,O), for which we have A2 > 0. From (13.6) and (13.7) we know that At is a saddle point, A2 is a stable nodal point, and Ag is an unstable nodal point. Finally, we study the direction of crossing of the trajectory through the line M'Ai whose equation is YiX- y- 1 = 0. Let V = YiX- y. Then on '1YAi we have dV
dt
= (YiX- y -1) (-~ + YiY) + (myi- ~- Yl) xyYi Yi
(1 +~)X
Yi+a =---x. Yi
Yi
{13.8)
From this we can see that M~ is a trajectory of {13.3) if and only if Yi = -a; that is, yg = -a > 0. At the same time, -a should satisfy {13.5), and so m = 1/a- a; that is, MAg and MN coincide at m = 1/a- a to become a trajectory. In order to determine the sign of the right side of (13.8), we note that equation {13.5), satisfied by }i, can be written as (Yi
+ a)g -
(3a + m)(yi + a) 2 + (3a 2 + 2am)(yi +a) - a 2 ( m-
~ +a) = 0.
Under the conditioll m < -a, we have 3a + m < 0 and 3a 2 + 2am > 0. From this we can see that 1. Yi +a< 0 fori= 1, 2, 3 when m < 1/a- a, and 2. Yt +a< 0, Y2 +a< 0, and yg +a> 0 when m > 1/a- a. Thus the direction of crossing of the trajectory through Mfii can be determined from (13.8). Moreover, when m '# 1/a- a, we can also determine the relative positions of N and R with respect to the line 'MAi as follows: 1. N is between MA2 and MAg when m > 1/a - a; 2. N is above MAg when m < 1/a- a; and 3. R is always below all M'fli. The picture in Figure 13.1 is the distribution of trajectories for the case m < 1/a- a, 27a >4m3 • Using the isoclines P2 (x,y) = 0 and Q2 (x,y) = 0 and the directions of trajectories crossing the lines MN and 'MAi, we can determine completely all the directions of all the separatrices passing the saddle points M, N, A 1 , and A 1 with the exception of two. For example, the separatrix starting from M and entering the right half-plane must lie
284
THEORY OF LIMIT CYCLES
1 m4mP a
FIGURE 13.1
.!.-cs<m4m• G
FIGURE 13.2
between MA3 and MA2, because if it lies above MA 3 (or below MA2), then the adjacent trajectory below it (above it) will also be above MA3 (or below MA2) which is impossible. Thus the separatrix will finally enter A2· Moreover, the separatrix entering N from the upper right side must come from A3, the separatrix entering M from the lower right side and entering A1 must come from R, the separatrix leaving N to the lower right side must enter A2, the separatrix leaving M to the lower left side and the separatrix coming from A1 must all enter A:3 • These are quite obvious. Now we show that the separatrix entering N from the lower left side must come from 0 · If it comes from A3, then the separatrix starting from N to the upper left side
§13. CLASS II EQUATIONS WITHOUT LIMIT CYCLES
285
can only surround 0; but 0 is an unstable singular point when m < 0, and so a stable limit cycle will appear in the vicinity of 0, which is impossible. Next, the two separatrices on the left of N cannot coincide to form a separatrix cycle surrounding the point 0, because as we mentioned in Remark 2 after Theorem 12.1, system (13.3) does not have a singular closed trajectory.(!) Denying the above two possibilities, and noting the direction of the trajectory crossing MN, we see that the separatrix entering N from the lower left side must come from 0. In order to determine the global structure of the trajectory of system {13.3), we still have to know the relative positions of the separatrix from N going to the upper left side and the separatrix entering M from the upper left side. Making use of the method mentioned above, under the conditions m < 1Ja- a and 27a > 4m3 there is no way for us to be sure of their relative positions. Here there exist two possibilities; that is, 1. Under the conditions m < 1/a-a and 27a >4m3 , the relative positions of these two separatrices are in fact completely determined, but we cannot find any method to prove it. 2. Under the conditions m < 1/a- a and 27a > 4m3 , there may appear three different cases for the relative positions of these two separatrices. Which of these is correct? This is the first problem we have to solve in this section. Before we solve this problem, we have to see the global structure of the trajectories of (13.3) when m and a satisfy other similar conditions.(2 ) Using the previous method, we can draw the figure of the global structure (Figure 13.2) under the conditions 1/a-a < m < 0 and 27a >4m 3 , and the problem of Figure 13.1 does not appear. The main reason is that the direction of (1) We explain more clearly the proof of nonexistence of singular closed trajectories in this specific case. Comparing the absolute values of the slopes of the trajectories of system (13.3) at the points Ft(:z:,y) and F2(x,-y), where y > 0 and F1 and F2 all lie above the line 1 - m:z: + y = 0, we get
jdxdyj
F1
l:z:(1 + a:z:)l = y(l + y- m:z:)
l:z:(l + a:z:)l
< y(l- y- m:z:)
Idyl = dx F 2
•
Hence, by the comparison theorem, we see that for the closed trajectory or the singular closed trajectory above the line 1 - m:z: + y = 0 the curve symmetric to the part of r above the :z:-axis with respect to the :z:-axis will completely contain the part of r between points and the :z:-axis and below the :z:-axis. Suppose G is an inner region of r. Then it is easy to see that
II(~~+ aa~2) G
dxdy=
II
mydxdy>O,
G
which contradicts Bendixson's theorem. Hence r does not exist. (2)For the coexistence of these conditions, refer to Figure 13.11.
286
THEORY OF LIMIT CYCLES
m .... .l-a
"
m
"
(a)
27a<4m3
(b)
FIGURE 13.3 the trajectory crossing the line segment M N is different from Figure 13.1. Similarly, we can draw the global figure of the trajectory when m = 1/ a-a < 0 and 27a >4m3 ; this can be done by changing MN in Figure 13.2 to become a trajectory, and the directions of other separatrices are the same as in Figure 13.2. Hence we do not draw this figure. If in the above three cases we keep the remaining inequalities, but change 27a > 4m3 to 27a = 4m3 , then the figure of the corresponding global structure can be obtained from the original figure by letting At= A2 ; here At= A2 is a semisaddle nodal point. If we again change 27a = 4m3 to 27a < 4m3 (but still assume m < 0), then At = A2 disappears, and the problem of how to determine the relative positions of the separatrix coming from N and the separatrix going to M in the fourth quadrant is created. However, since now R is a stable focus or nodal point, and we already know that system (13.3) does not have a closed trajectory, the relative positions of these two separatrices can still be determined when m < 1/a- a, as indicated in Figures 13.3(a) and (b). Only in the case when 1/a- a < m < 0 and 27a < 4m3 does the problem ofrelative positions of the above two separatrices arise. For convenience, let lt, lt, l1, and 12 denote the four separatrices passing through M, and let Lt, Lt, L1, and L2 denote the four separatrices passing through N (when m <-a) orR (when m >-a), as indicated in Figure 13.4. Now we change the condition m < 0 to m = 0. Then system {13.3) has two centers and one infinite singular point A3. The conditions m:;; 1/a- a become a :;; -1. Figures 13.5(a), (b), and (c) show the global structure of system (13.3) in three cases: m = 0 and a < -1, a = -1, and a > -1 respectively.
§13. CLASS II EQUATIONS WITHOUT LIMIT CYCLES
11
Lt
Li
0
:li
L:
Li
li
287
B
z:
FIGURE 13.4
Next, suppose 0 < m <-a. Then 27a <4m3 , but there are three possible ways of ordering between a- 1/a and m. Since the stability of 0 and R has been changed, now the figures corresponding to 13.1 and 13.2 are 13.6 and 13. 7. In Figure 13.6, only the relative positions of Lt and cannot be determined; in Figure 13. 7, only the relative positions of L! and l! cannot be determined. But if m = 1/a- a, then the global structure of the trajectories can be completely determined as indicated in Figure 13.8. H we suppose m = -a, then R = N is a higher-order singular point; then m > 1/a- a and 27a <4m3 must hold. Hence it is only on Figure 13.9 that the relative positions of L! and l! cannot be determined. Finally, suppose m > -a. Then m > 1/a- a and 27a <4m3 still hold. Here we still denote the singular point above the line 1 + ax = 0 by N, which is a saddle point; the singular point below the line is R, which is an unstable focus, the global structure is as shown in Figure 13.10. In Figure 13.10, except for the relative positions of Lt and L! and l! cannot be determined, and we may still have the problem of determining the relative positions of Lt and l! because Lt may possibly cross through the left side of MN. In order to show clearly and completely whether the indeterminate nature of the relative positions of some separatrices in the above figures is due to the nonexistence of this case or the deficiency of the method used, we now introduce an (a, m)-parametric plane, and a bifurcation curve in this plane. The so-called bifurcation curve is a curve in the (a, m )-plane such that for any point (a*, m*) in this curve, the graph of the trajectory of its corresponding system (13.3) is structurally unstable. We should note that by a structurally unstable system in this section we mean that kind of system (13.3) for which the topological structure of the trajectories on the projective plane can change
zt
zt,
THEORY OF LIMIT CYCLES
288
m-o, a<-1
m-0, a--1
(a)
(b)
m ... o, a>-1
(c) FIGURE 13.5
when the coefficients a and m on its right sides vary slightly. Hence the definition is slightly different from the one given in §8. Since we already know that (13.3) cannot have a limit cycle, the unstable structure can only appear in the following cases: 1. System (13.3) has a center. 2. System (13.3) has a higher-order singular point (finite or infinite). 3. System (13.3) has a separatrix connecting two saddle points.( 3 ) We know that case 1 can only appear when m = 0; hence the horizontal axis is a bifurcation curve in the (a, m)-plane. Since we have assumed a< 0, (3)Strictly speaking, under the definition of structural stability in this section, whether case 3 is a sufficient condition for structural stability has not yet been definitely proved.
§13. CLASS II EQUATIONS WITHOUT LIMIT CYCLES
1
O<m< -a, m
O<m-..!.-a<-a CJ
FIGURE 13.8
289
1 O<m<-a, m>--a a
FIGURE 13.7
1 m- -a>-a, 27a<4m' a
FIGURE 13.9
we may as well just confine ourselves to the left half-plane (Figure 13.11). The global figure of system (13.3) corresponding tom= 0 has three different topological structures as seen in Figures 13.5(a), (b), and (c). Case 2 can only appear when m =-a (with R = N) or 27a =4m3 (with At = A2 ). Hence the line m = -a and the curve 27a = 4m3 are bifurcation curves. If we also include system (13.3) for a= 0 in our considerations, then
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THEORY OF LIMIT CYCLES
m> -a>.!.-a, 27a<4m*
a
FIGURE 13.10
FIGURE 13.11
the line a= 0 is also a bifurcation curve, and N = A1 = A2, R = Ag (when m > 0), or N = A2 = Ag, R = A1 (when m < 0). Case 3 is the most complicated one. According to Figures 13.1-13.10, we can see that Lt always comes from Ag, 12 always runs to Ag, and these two separatrices cannot coincide with other separatrices. Also from Figures 13.1 and 13.2 we can see that when A1 and A1 exist, the separatrix entering A1 must come from R. R is not a saddle point, and the separatrix leaving A1 must enter A3 ; hence these two separatrices -eannot coincide with other separatrices. There are six separatrices Lt, L1, Li, zt, zt, and 11 remaining; among them, any two separatrices with different stability (this means leaving or entering the saddle point as t increases) may coincide. Hence we can divide case 3 into two subcases: 3a. A separatrix starting from one saddle point and returning to the same saddle point. 3b. A separatrix starting from one saddle point and returning to another saddle point. The cases Li = Li I Lt = Li I zt = l1, and zt = zt belong to case 3a (for simplicity, from now, = means "coincides with"), but from Remark 2 after Theorem 12.1 we know that all these cases can only occur when m = 0, and so there is no new bifurcation curve to be added.
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291
There are five different possible cases belonging to 3b:
lf = Lz, L! = l!, Lt = lt, Lt = l!, L! = lt. From Figure 13.5, we know that when m = 0 and a = -1 the equalities L! = l!, Lt = zt, Lz = lf appear simultaneously; but when m = 0 and a < -1 these three equalities cannot hold, nor can they form= 0 and a> -1. Moreover, when a< -1, the relative positions of these three pairs of separatrices are just opposite to the case when a> -1. From now on let Ct.C2, and Ca denote the bifurcation curves in the (a, m )-plane such that L! = l!, Lt = zt, and Lz = lf respectively. Then at least one branch of C1, C2, and C3 must pass through the point P( -1, 0) so as to separate the line segment (-oo, -1) on the aaxis from the segment (-1,0) in which two ends of C1 and Ca should go to infinity; and the upper half-branch of c2 must go to infinity;(4 ) for otherwise we can use a continuous curve 8 in the second or the third quadrant to connect a point Q* in the segment (-co, -1) on the negative a-axis to a point Q' on the segment ( -1,0) without intersecting C1 , C2 , or Ca. Thus when a point moves from Q* to Q' along 8, the relative positions of three pairs of separatrices of (13.3) remain unchanged in the corresponding (x, y)-plane, which is impossible. But it is sufficient for the lower branch of C2 to lie above the curve 27a = 4m3 , since for any point (a, m) below the curve 27a = 4m3 its corresponding system (13.3) has three infinite singular points. From Figures 13.1 and 13.2, we see that then Lt enters A2, but zt comes from R. It is clear we can take Lt to surround zt from its outside. From the discussion at the beginning of this section, we see at once that the bifurcation curve Ca which makes Lz = lf has only a unique branch m = 1/a - a (the part of a > 0 is not considered); that is to say, if L2 = lf, then they must coincide on the line segment liN, for otherwise, as in Figure 13.12, a contradiction will arise. Next it is easy to prove that in the unbounded angular region surrounded by the upper half-branch of Ca and the line segment (-co, -1) on the negative a-axis there is no locus of C1 , for otherwise, as indicated in Figure 13.13, L! = l! will appear, 0 will be stable, and all the trajectories meeting the line segment liN will always cross it from left to right, which is impossible. Similarly, we can prove that the unbounded angular region in the third quadrant with the same vertical angle as the above angular region does not contain any locus of C 1 • Also, below the line m = -a, except for the above two angular ( 4 )They
cannot reach the positive m-axis, for the reasons given in Theorem 13.1.
292
THEORY OF LIMIT CYCLES y
N
M
M
FIGURE 13.13
FIGURE 13.12
regions, the remaining two angular regions with the same vertical angle do not contain the locus of c2. Not only this, we can prove still more: the angular region bounded by the lines m =-a and m = -2a does not have C2 in it, and the angular region bounded by the lines m = -2a and the positive m-axis does not have cl in it. To see this, we only have to note that when -a< m:::; -2a, the ordinate of R, -(a+m)ja, is in the interval (0, 1]. Using the method of comparing the slopes of the trajectories (Figure 13.14), it is easy to prove that the curve of symmetry (shown by the dashed curve) of the section of the arc of Lt inside the region {y :::; 0, x ~ -1/ a} with respect to the x-axis lies entirely below the upper half-branch of Lt. From this we can see that the absolute value of the ordinate of the other point of intersection Q of Lt and 1 + ax = 0 is less than 1. However, on the other hand, zt has a negative slope on the left side of 1 + ax = 0, and the ordinate of M is -1; hence the point of intersection P of lt and 1 + ax = 0 must lie below Q; that is, lt and Lt cannot coincide. Similarly, we can prove that the angular region bounded by the positive m-axis and m -2a does not have C1 in it (Figure 13.15).(5 ) Summarizing the above discussion, we get
=
THEOREM 13 .1. The bifurcation curve Ca for L2 = lt has a unique branch m = 1/a- a. The bifurcation curve C1 for L! = l! can only lie in the region {m > 1/a- a,O < m < -2a} and the region {m < 1/a- a< 0}. The bifurcation curve C2 for Lt = zt can only lie in the regions {0 < m < 1/a- a}, {1/a- a< m < 0, 27a <4m3 }, and {0 < -2a < m}. ( 6 )From this we can see that in Figure 13.10, Li = 11 and Lt = It cannot appear simultaneously; that is, the singular closed trajectories formed by these four separatrices cannot exist.
§13. CLASS II EQUATIONS WITHOUT LIMIT CYCLES
FIGURE 13.14
293
FIGURE 13.15
For the branches of C1 and C2 paBSing through P(-1,0), we believe they both exist and are unique, as shown in the graph of Figure 13.11, and both cl and c2 should be smooth curves and not contain interior points when they are considered as point sets in two-dimensional Euclidean space; yet we have no way to prove this. From now on, we assume we only have unique branches ofC1 and C2 passing through P(-1,0). In Figure 13.9 we see that when m = -a and 27a < 4m3 , the relative positions of L1 and 11 cannot be determined. The following shows that this is indeed the situation. THEOREM 13. 2. The upper half-branch of C not only lies above m = 1fa- a, but also crosses through the line m =-a to its upper half. PROOF. In (13.3), let x
d
I
= -x fa, y = -y' fa,
...J!.... = x1(1- x 1 ) dt '
1
1
and m
dX = -yI (1 Tt
= -a.
1 1- x1) . ay
Then (13.9)
Now we prove that when lal is sufficiently large, for system (13.9), L1 should run to the right of 11; this kind of relative positions is the same as the relative positions of L1 and 11 corresponding to the point on (-co, -1) on the negative a-axis. Hence this shows that the point on m = -a must lie below
Cl-(6) ( 6 )Does the following possibility exist: C1 has an even number of branches passing through P(-1,0), but all lie below m = -a and moreover there are an odd number of branches lying in the angular region -a< m < -2a? According to the previously mentioned theory that cl has to separate the two line segments (-oo,-1) and (-1,0) on the
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294
y'
FIGURE
13.16
First, in the region {y' > 0, 0 < x' < 1} we compare the slopes of the trajectory of (13.9) and the trajectory of the system
dx' jdt = -y',
dy' fdt
= x'
(13.10)
(Figure 13.16), and we have
-x'(1- x') y'(1- y' fa-x')
x' -x'y' = > 0. y' ay'(1- y' fa-x')
---:--"""-.,..--'---:- + -
Hence the trajectory of system (13.10) passing through N'(1, 0) (a circle with center at the origin) is above the trajectory of (13.9) passing through the same point. Suppose they meet the positive y 1-axis at A and B respectively. Next, in the region {x' < 0, a < y' < 0}, we compare the slopes of the trajectory of (13.9) and the trajectory of the system
dx' fdt = -y1 (1- y' fa),
dy' fdt
= x'(1- x')
(13.11)
and get
-x1(1- x') y'(1- y' fa-x')
x'(1- x')
-x'2 (1- x')
+ y'(1- y' fa) = y'(1- y' fa- x')(1- y' fa) > O.
negative a-axis from each other, we know that the above possibility does not exist. In other words, there must exist an odd number of branches of C passing through P( -1, 0) to cross through the line m = -a and then running to the upper half of this line.
§13. CLASS II EQUATIONS WITHOUT LIMIT CYCLES
295
Hence the trajectory of (13.11) passing through M'(O, a) lies on the right of the separatrix of (13.9) passing through the same point. Suppose they intersect the negative x'-axis at C and D respectively. It is easy to compute the equation of the trajectory M0 of (13.11):
x'2
x'3
y'2
2-3+2-
y'3 3a
a2
=6.
When iai > 1, the equation x'2 f2- x'3 f3 = a2f6 has a unique (negative) real root; that is, the abscissa of C, X6. From this we can see that if ial is very large, then (13.12) X~ :::: - {la272. But xi:, < x0; hence as long as ial is sufficiently large, lxiJI can be greater than any given positive number. Finally, in the second quadrant, we compare the slopes of the trajectory of (13.9) and that of the system
dx'fdt=a-y',
dy' fdt
= x'
(13.13)
and get
-x'(1- x') x' y'(1- y' fa-x') --a--y'
=
-=-x'[a2 (1- x')- y'2 ] ay1 (1- y' fa- x')(a- y') > O.
The term inside the square brackets on the right of the above formula can be assumed to be positive, since a 2 (1- x')- y'2 = 0 is a parabola with its vertex at N'(1, 0) and passing through the points (0, ±a). We have mentioned previously (see Figure 13.15) that the separatrix M' DH passing through M' of system (13.9) must lie in the region a 2 (1- x')- y'2 > 0, and now we have to compare the trajectories CF and CE of (13.9) and (13.13) passing through C respectively; they must also lie in the region a 2 (1- x')- y'2 > 0, and CF should be above CE. System (13.13) has a first integral
x'2 +(a- y')2 = k2; hence we know the equation of 6E is
x'2 +(a- y')2 =
Xc + a2,
and so the ordinate of E is
Yk = Ja'J. + x'6 +a. From (13.12) we know that, when Ia! is very large,
Ys::::
Ja2 + V4 a{0,4 +a:::: _1_1all/3 2~ ·
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296
When a varies, the ordinate of A remains unchanged, but y~ can become greater than 1 provided that lal is sufficiently large. Hence E is above A, and therefore His above B, which is what we had to prove. We can prove that as lal- oo, Ct must enter the infinite singular point G on m = - 2a. For the lower branch of C 1 we can also prove that along any line a= ao < -1, when m < 0 and lml is sufficiently large, the point (ao, m) must lie below C1 [204]. Moreover, we can also prove, going to infinity along the lower half-branch of Ct. that mfa- oo [205]; the proof is omitted. From this we can see that Ct cannot be symmetric with respect to the a-axis, because the upper half-branch of Ct must remain below the line m = -2a, and along this branch, as a- -oo, lm/al cannot approach oo. AB for the bifurcation curve C2 , not only must its upper half-branch remain below the hyperbola m = 1/a- a going to infinity, but also we can show that mfa- 0 along C2 as a- -oo (the proof is omitted). On the other hand, the lower half-branch of C2 must lie between the curve 27a = 4m3 and the negative a-axis and must approach the origin, because for the system (13.3) corresponding to the point (a,m) which makes 27a ~4m 3 , Lt all enter A2, but zt all come from R. From this we see that when (a, m) and 27a = 4m3 are very close but satisfy 27a <4m3 , Lt must run below zt; conversely, for the system (13.3) corresponding to the point on the segment (-1,0) of the negative a-axis, its Lt should run above lt. Hence there must exist C2 which separates the negative a-axis from the curve 27a = 4m3 • In Theorem 13.1 we proved that the angular region bounded by the line m = -2a and the positive m-axis cannot have C1 in it, but it can have C2. Now we shall prove that in this angular region there exists a branch of C2 which connects with the C2 passing through P( -1, 0) at the origin a = m = 0. For convenience, from now on we denote this branch by 02. THEOREM 13.3. There exists a C2 in the angular region -m/2
= -y,
dyjdt
=X
in the region x ~ -1/a, 0 :5 y :5 -(a+ m)/a, and get
-x(l+ax) +~-x[y-(a+m)x] + y- mx) y - y(l + y _ mx) > 0.
Y(l
(13.14)
§13. CLASS II EQUATIONS WITHOUT LIMIT CYCLES
297
FIGURE 13.17
From this we can see that the circular arc iB with its center at the origin and passing through R is below the separatrix Lt of (13.3). It is easy to compute XB
= _!J1 +(a+ m)2. a
Let the point of intersection of Lt and the x-axis be D. Then x D > x B. Next, we compare the slopes of the trajectory of system (13.3) and that of the system
dxjdt =my, in the region x
~
-1/a, -1
~
y
~
dyjdt = 1 +ax
(13.15)
0, and get
-x(1+ax) _1+ax y(1+y-mx) my
= -(1+y)(1+ax)
>O·
my(1+y-mx)
'
hence the trajectory of (13.15) passing through B lies on the left side of the trajectory of (13.3) through B. It is easy to see that the former trajectory has the equation
and the abscissa x E of the point of intersection E of the trajectory and y = -1 satisfies the equation (13.16)
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298
Again in the region y :5 -1, 0 :5 x :5 -1/a we compare the slopes of the trajectory of (13.3) and that of the system
dxjdt
= -y(1 + y),
-x(1 +ax)
x(1 +ax)
dyjdt = x(1 +ax) ,
(13.17)
and get -mx 2(1 +ax)
0
= y(1+y-mx)(1+y) > ·
y(1+y-mx) + y(1+y)
From this we can see that the separatrix of (13.17) entering M(O, -1) from It is easy to see that the former has the lower right part must be below the equation !2 x2 + ~3 x3 + !2 y2 + !3 y3 1 and the ordinate YA of its point of intersection A with 1 +ax = 0 satisfies the equation(T)
zt.
=s
12 -YA 2
13 = + -yA 3
-1(1 - -1) 6 a2
•
(13.18)
Finally, we compare the slopes of the trajectory of (13.3) and that of the system (13.19) dyjdt = -ay~(1 +ax) in the region x
~
-1/a, YA :5 y :5 -1, and get
-x(1+ax) +ay~(1+ax) y(1+y-mx) m2y = (1 + ax)[-m(m + ay~)x + ay~(1 + y)] > O, m2y(1 + y- mx) provided m is sufficiently large. Thus, the arc AS of the trajectory of (13.3) must be on the left side of the arc AQ of the trajectory of (13.19). It is easy to see that the equation of AQ is y~(1
thus the abscissa
XQ
+ ax)2 + m2y2 = m2y~;
of Q should satisfy
(1 + axQ) 2 = m 2 (1- 1/y~).
(13.20)
Now if we can prove that XE > XQ for sufficiently large m, then the separatrix Lt of (13.3) will run below lt; this kind of relative positions of Lt and is just opposite to their relative positions when -a < m < - 2a. From this
zt
(7)From this we can see that when a is fixed, the value of YA is also fixed, independently ofm.
§13. CLASS II EQUATIONS WITHOUT LIMIT CYCLES
299
a-o FIGURE 13.18
we can deduce that C2 exists in the angular region m > -2a > 0. In order to get XE > XQ, we know by (13.16) and (13.20) that we only need show that [1-
Ji +(a+ m)2] 2 +am> m2 ( 1- ~);
that is, 2 + (a+m)2 +am- 2J1 + (a+m)2
m
2
1
> 1 -2· YA
This formula clearly holds when a is fixed and m is sufficiently large, since its right side is a fixed number less than 1, whereas its left side approaches 1 as m--+ oo. Finally, we have to prove that C2 passes through the origin. For this, we note that when a= 0, system (13.3) becomes dxfdt
= -y(1 + y- mx),
dyfdt
= x.
The global structure of its trajectory is as shown in Figure 13.18; here N = A1 = A2 is an infinite higher-order singular point (1, 0, 0) and R = A3 is a semisaddle nodal point (1, m, 0). From Figure 13.18, we see that Lt will run below zt; their relative positions are just opposite to the relative positions of Lt and zt which correspond to the points to the right of the point (-1, 0) on the negative a-axis, and so C2 must separate the positive m-axis from the negative a-axis; that is, it should pass through the origin. The theorem is completely proved. We can further prove m/a--+ -2 along C2 as a--+ -oo; that is, C2 must pass the infinite point of the line m = -2a. The proof is omitted (see [204]). In the following we discuss again the possibility of establishing the equalities Lt = l'l and Li = lt.
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300
'Y
FIGURE 13.19
zt
FIGURE 13.20
THEOREM 13. 4. There does not exist a point (a, m) which makes L1 = hold.
PROOF. If there exists a point (a*, m*) such that its corresponding system (13.3) has L! = then, as indicated in Figure 13.19, 0 should be an unstable singular point and R should be a stable singular pojnt; this is not possible under any circumstances when a < 0.
zt,
THEOREM 13. 5. The bifurcation curve C4 for Lt = l1 is a part of the curve which lies above the line m = -a, and connects the origin and the point of intersection S of Ct and m = -a. PROOF. As in Figure 13.20, when Lt = l1, first we know from the direction of the trajectory crossing MN that R must be unstable and 0 must be stable. Hence C4 must be above the line m = -a.( 8 ) We note again in Figure 13.20 the relative positions of Lt and and the relative positions of L1 and 11; we know that C4 must lie below 02 and on the right side of Ct. Now from Figure 13.21 we see that corresponding to the points of 02, we have Lt = lt. Since 0 is stable, if we consider to be a continuation of lt, then Lt must run to the right side of l!. In fact, according to the continuity of a solution with respect to its initial value, for the points below and close to 02 this is actually the case. Again from Figure 13.22, we see that, corresponding to the points on the line segment m = -a on the right side of C1, R = N is a higher-order singular point, Lt = L2 = 0, and L1 will
zt
zt
( 8 )When 0 < m <-a and m > 1/a- a, Figure 13.7 shows that R is a stable singular point with no closed trajectory near it; hence Lt must enter R and cannot cross through the left side of MN.
§13. CLASS II EQUATIONS WITHOUT LIMIT CYCLES
301
N
------~~----~~;r--~
a;
M
FIGURE 13.21
FIGURE 13.22
FIGURE 13.23
run to the left side of 11. Hence from the continuity of a solution with respect to its initial value, for the points close to and above this finite line segment, Lt of the corresponding system (13.3) will run to the left of 11. Thus, there must be a c4 separating from m = -a; that is, c4 must pass through the origin. Also, for the points of C 1 above m =-a we have L1 = l1 in the figure of the trajectory of the corresponding system (13.3), and Lt crosses through M N and enters 0 (Figure 13.23). The relative positions of Lt and 11 are just opposite to the relative positions corresponding to the points close to and above the line m = -a and on the right side of C1 ; hence C4 must terminate at a point of intersectionS of C 1 and m =-a. In fact, the system
ct
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302
(13.3) corresponding to S has R=N, Considering Lt as a point and Li as a continuation of Lt: we then have The theorem is completely proved. We believe the bifurcation curves on the parametric (a, m)-plane have all been found (Figure 13.11), and they are all differentiable curves. For C~, C2 , and C4 , we still do not know whether they are algebraic or transcendental curves, nor by what formulas they are represented. Going back from Figure 13.11 to Figure 13.1, we find that under the conditions 27a >4m3 m < 1/a-a,
Lt = l1.
the relative positions of L1 and li have in fact three possibilities; which one appears depends on whether the point (a, m) is on the left side of C~, on the right side of C1 or above C1. For Figures 13.6, 13.7, 13.9, and 13.10, the situations are the same. Up to here, the initial steps in the problem of the global structure of the trajectories of system (13.3) have been taken. Now we have some results on whether there exists a unique branch for each C1 , C2 or C4 as shown in Figure 13.11. In [206], after transformation, system (13.3) becomes dx/dt
= -y(1 +y- mx) + mx,
dyjdt
= x(1 +ax).
Then, using the theory of rotated vector fields, we prove that there is only one C2 which does not have an interior point. Moreover, if in (13.3) we let m = ka (k > 0 is fixed), and let the parameter vary, then we can use the theory of rotated vector fields again to prove that in Figure 13.11 there is only one C1 below the a-axis, and that it does not have an interior point. The problem of uniqueness of C~, C2, and C4 can be solved by using the formula for differentiating the solution with respect to its parameter and the implicit function theorem. Cao Yu-lin has given a careful proof of this assertion. Also, for the problem of bifurcation curves of equations of class II with two fine foci (hence without a limit cycle) dx m(m+ 2a) 2 dt = -y + 4 x
2
+ mxy + Y
'
~~ =
x(1 +ax)
(13.21)
for most cases, quite satisfactory results have been obtained. Note that Figure 13.10 also belongs to the case of two fine foci, but does not belong to (13.21). Also the bifurcation curve corresponding to this figure is just the curvilinear triangle formed by c~. C2, and 04 above the line m = -a in Figure 13.11.
§13. CLASS II EQUATIONS WITHOUT LIMIT CYCLES
303
There is also the following work on the study of global structures of trajectories and bifurcation curves of quadratic systems (not necessarily equations of class II): For the system
dxfdt
= -y + lx 2 + ny2 ,
dyjdt
= x(1 +ax+ by)
Luo Ding-jun [208] and Sun Kai-jun [209] studied the global structure and bifurcation surface when -y+lx2 +ny2 = 0 is an ellipse, parabola or hyperbola with a(b + 2l) ::/: 0 (hence the system does not have a closed trajectory, and the proof can be seen at the beginning of §15), but their results are not as complete as in this section. Cao Zhen-zhong [206] studied the global structure and bifurcation surface of the system of equations
dx/dt = -y +ox+ mxy- y 2 ,
dyjdt
= x(1 +ax).
Cao Xian-tong [210] studied the same problem for the system of equations
dxjdt = -y + lx 2 + mxy- y 2 ,
dyjdt = x(1 +ax).
Ren Yong-iai and Suo Guang-jian [211] studied the global structure of a quadratic system with three straight line solutions. Liang Zh~jun [174] studied the global structure and phase-portrait of the system of equations
dxjdt
= -y + lx 2 + 5axy,
dyfdt
= x + ax 2 + 3lxy
without cycles, but with a third-order fine focus. Exercises 1. Prove Theorem 13.1. 2. Construct the figures of all the trajectories of system (13.1) when n = 0. 3. Prove the uniqueness of the lower half-branch of C1 and C2 at the end
of this section. 4. Prove if there is no higher-order singular point except a fine focus (0, 0) for system (13.3), then it must have a fine saddle point (that is, the value of its divergence at this saddle point is zero) or a fine focus. 5. Use the transformation
= -a'y', y = a'x' to change (13.3) into equations of class m, and use the method of analysis on a'= 1/a,
b'
= m/b,
x
global structure to prove that along the lower half-branch of the bifurcation curve cl going to oo, m/a-+ +oo. 6. Prove that along the upper half-branch of the bifurcation curve C2, mja--+ 0 as a--+ -oo.
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304
7. Prove that in the angular region 0 < -a < m in Figure 13.11 we can construct a topological transformation T, which does not keep a constant orientation, to change every half-ray starting from the origin to another halfray, to change an open arc §G of C 1 into C2, and to change C4 into itself; and prove T is an identity transformation on the line m = - 2a. 8. For the system of equations
dxjdt
= -y + lx 2 + mx + ny2 ,
dyjdt
= x(1 +ax)
find necessary and sufficient conditions for a line connecting two saddle points to be an integral line. 9. Prove that when 8 satisfies
a8 3
-
2am8 2
+ (am2 + 3a- m)8 + 1- am- a2 = 0
the system of equations
dx/dt
= -y + 8x + mxy- y 2 ,
dyjdt = x(1 +ax)
takes a certain M~ as an integral line; hereM is a saddle point (0, -1) and ~ is an infinite singular point. Moreover, the condition for this system to have three real roots is 27a- 4m3 > 0 (suppose a< 0). 10. Find a parabola solution of system (13.3) passing through M and N with its principal axis through A2, and prove this solution exists only when a= and m = -5/v'B (hence the lower half-branch of Ct passes the point ( -5/.;6)).
-J312 -J312,
§14. Relative Positions of Limit Cycles and Conditions for Having at Most One and Two Limit Cycles in Equations of Class II
In this section we shall study the second and third problems mentioned at the beginning of §13. We first look at the simpler case. Suppose that in equations of class II,
~; =
-y + 8x + lx 2 + mxy + ny 2 ,
dy dt = x(1 +ax),
(14.1)
two of the coefficients of the terms of second degree on the right side of the first equation are zero. Then we have
dxjdt = -y + 8x + ny 2 , dxjdt = -y + 8x + mxy, dxjdt = -y + 8x + lx 2 ,
dyjdt = x(1 +ax); dyjdt = x(1 +ax); dyjdt
= x(1 +ax).
(14.2) (14.3) (14.4)
It is easy to see that (14.2) and (14.3) can be integrated when 8 = 0, and they take (0, 0) as their center; and when 8 :f. 0 they have no limit cycles because the divergence of (14.2) is a constant 8, and (14.3) can be proved to have no limit cycles by the Dulac function (1 - mx)- 1 • For (14.4), the situation is not the sante. We may as well assume l > 0 and a > 0; and it is not difficult to use the well-known method to prove that when 8 ~ 0 or 6 2:: lja, (14.4) does not have a limit cycle, but when 8al > 0 and 8 lies in some interval (0, 8*), (14.4) has a unique limit cycle. In the following we study mainly the case where the coefficients of the quadratic terms on the right side of the first equation of (14.1) have only one zero. (I) l = 0. In this case we have the system (we may as well assume n = -1)
dxjdt = -y + 8x + mxy- y 2 ,
dyjdt = x(1 +ax).
First we prove a useful theorem for nonexistence singular closed trajectory. 305
(14.5)
of a closed trajectory and a
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306
THEOREM 14 .1. System (14.5) (we may as well assume a < 0) cannot have a closed trajectory or a singular closed trajectory passing a saddle point in either of the following cases: 1) m8 $ 0, lml + 181 -::/= 0; 2) 8(m- 8) $ 0, lml + 181 i= 0.( 1 ) PROOF. When the first group of conditions holds, it is easy to see that any closed trajectory or any singular closed trajectory passing a saddle point of (14.5) cannot intersect the line 1- mx = 0. Now we take the Dulac function to be B = 1/(1- mx); then we have
a a 8 -my 2 ax (BP) + ay (BQ) = (1- mx) 2 • The right side of this formula always keeps a constant sign on any side of the line 1 - mx = 0; hence the theorem is proved. Now we suppose the second group of conditions holds. We translate the x-axis to the line y = -8/m (the case of m = 0 has been seen in (14.2); hence we may as well assume m i= 0) and keep they-axis unchanged. Then (14.5) becomes dx = !._ ( 1 + ( 1 + 28) y+mxy-y2 , dt m m m (14.6) dy dt = x(1 +ax).
!._)
The system of equations whose vector field is symmetric to that of (14.6) with respect to the new x-axis is dx dt
= !._ ( 1 m
!._) _(1 + 28) y _ mxy _ y2, m m
(14. 7)
dy = -x(1 +ax). dt The locus of points of contact of the trajectories of these two systems is easily seen to be x = 0,
1 + ax = 0,
and
8 (1 - m8) -
m
y 2 = 0.
(14.8)
Since the divergence of (14.6) is only zero on the x-axis, any closed or singular closed trajectory r must intersect the x-axis. Also, from the theorems of §11, we know that if r appears in the vicinity of (0, 0), then it cannot meet 1 +ax= 0. Moreover, for 8(m- 8) < 0 the last equation of (14.8) does not have a real locus, and for 8(m - 8) = 0 its locus is the x-axis. From this (1 )This theorem was first obtained in [212]. But here we use the method of proof in [124], which was first seen in [16].
307
§14. CLASS II EQUATIONS WITH 1 OR 2 CYCLES
we can see that the curve symmetric to r+ (the part of r above the x-ax:is) with respect to the x-ax:is, and r- (the part of r below the x-ax:is) do not have a common point except on the x-ax:is; that is, the curve symmetric to r+ lies entirely above (or below) r-. Thus, for any closed or singular closed trajectory r of (14.6), we must have
11 (~= + ~~)
11
dxdy =
intr
mydxdy #
o.
int r
This contradiction shows that r does not exist. Similarly, we know that the vicinity of another singular point with index +1 on 1 + ax = 0 cannot contain a closed or singular closed trajectory. The theorem is completely proved. The following transformation of coordinates is useful for our later discussion. Note the two singular points on the line 1 + ax = 0 are
R( N
-~,4 [-1-:- J(u ~)'- ¥]),
of
+ J(u :)'- ~]),
of
H· ~ [-1-:
index index -1. +1;
Moving the origin toR, we get
~~ = dy = dt
(8
+my2)x- ( 1+: +2y2) y +mxy- y3,
~x+ax 2
(14.9)
,
where Y2 represents the ordinate of R. Now we apply the transformation x=- [
m 2 48] 3 / 4 (1+a) -~ x,
m 2 48] l/ 2 (1+a) -~ y, 48] - t m t=- [ (1+a) -~ 1
y=-
[
1
1/ 4
2
(14.10)
1
to (14.9) and get dx1/ dt 1 = -y' + 81x' + m 1x' y' - y12 , dy 1jdt' = x'(1 + a 1x 1),
(14.11)
where
,
8 = [(1
-(8 + my2) + mja)2- 48/ajl/4' ml = m [ ( 1 +
1
[(
a +a
:f- ~]
m)2 1 +-; 1/4
48] 3/ 4
- ~
'
(14.12)
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308
According to Theorem 14.1, it is only possible for system (14.1) to have limit cycles near the two singular points of index +1 when
m6 > 0 but 161 < lml.
(14.13)
In the following we want to explain whether limit cycles can coexist near both singular points of index +1. This problem is closely related to the order between m and a. 1. m > -a > 0. As shown in Figure 13.10, for 6 = 0, 0 is stable, R is unstable, and M and N are saddle points. Let 6 increase from zero. Then 0 becomes unstable. By the proof of Theorem 3. 7 of §3 and Theorem 12.5 of §12, there exists a unique stable cycle near 0. On the other hand, the two singular points on the line 1 + ax = 0 move far apart (we denote them by R' and N'). Now we prove that the stability of R' is different from R, since near R' there also appears a unique unstable cycle.(2) Hence we note that for m > -a> 0 and 161 sufficiently small,
H m112
~ 6-
; [ 1+ :
= 6- m [1
2
+
J(
1+ : )' -
+ m- (1 + m) (1a
a
~
l
26 )] a(1+m/a)2
+ 0(62)
a8 + 0(6 2 ) < 0, a+m where Y2 is the ordinate of R'. Hence, from (14.11) and (14.12) we know that R' is a stable focus of (14.9).(3) Hence when m > -a > 0 and 0 < 6 « 1, limit cycles coexist near the singular points 0 and R', but their stability is different. When 6 continuously increases, it is not certain whether these limit cycles disappear simultaneously. Now suppose near 0 the limit cycle, it exists, is always unique, and at 6 = 6 it expands and becomes a separatrix cycle passing through M (note that when 6 varies, (14.5) forms a family of generalized rotated vector fields on-each side of the line 1 + ax = 0; hence, when 6 increases, the two separatrices lt an~ li in Figure 13.10 can certainly come close together and coincide). Thus 6 should clearly be a function 6 = f(m,a) of m and a. HE>nce from (14.11) we know that the value 6' of 6' which makes the limit cycle near R' expand and becorne a separatrix cycle through N' must be the same function of rn' = --
--;--
lc51. ( )The Uniqueness of the limit cycle indicated here is only limited to sufficiently small
l'<Jt)~~ fact can also be seen from the figure of isoclines, since now the upper branch of th x, yh roug
A{ 0 Passes through 0
and N', and is above c5 +my = 0; its lower branch passes and R', and is below c5 + my = 0.
§14. CLASS II EQUATIONS WITH 1 OR 2 CYCLES
309
and a'; that is, 'E' = f(m', a'). Whether this function f can be determined is a problem worthy of our consideration. Of course, the uniqueness of a limit cycle in the vicinity of every singular point has to be proved. In [215] the uniqueness of a limit cycle is proved only when m > - 2a and 1 + 4am ~ 0. Aside from this, there is no other result. 2. 0 < m ~ -a. First we examine the case when 0 < m < -a.(4 ) When 6 = 0, N( -1/a, 0) is a saddle point and R( -1/a, -(a+ m)fa) below N is a focus. If 6 increases from 0, then N' and R' move far apart. Since for 6 = 0
6 + my2
= -m ( 1 + :) < 0,
initially R' remains unstable. Now we ask: Does the stability of R' change as 6 continuously increases? When does it happen? Clearly we can see that a necessary condition for R' to change its stability is 6' = 0; that is, li + ffl!h
= li -
;
[1+ :
+
J(
1+ : )' -
~] = 0.
Solving this equation, we get 6 = m, and 86' 88
I
= _ [a- 38 + 3m/2 + m2 /2a + (m/2)J(1 + m/a) 2 a[{1 + mfa)2- 48jaj31 2
.5=m
48/al 6=m
-1
= {1- m/a)3/2 < 0. {14.14) Hence when 6 increases from m, 8' decreases from 0; that is, R' as a singular point of (14.11) changes from stable to unstable, and so, as the singular point of the original system {14.5), it must change from stable to unstable. However, according to Theorem 14.1, we know that for 8 ~ m there is no closed trajectory in the vicinity of R'. Hence from Theorem 3. 7 in §3, we know that as 8 increases from less than m to m, there is an unstable limit cycle which shrinks (not necessarily monotonically) and approaches R'. How is this unstable cycle generated? There are several possibilities:( 5 ) (i) It is generated from a separatrix cycle through N' and surrounding R'. (ii) It is generated from a separatrix cycle passing through M and N' and surrounding R'. (iii) It is generated by splitting a semistable cycle which suddenly appears in the vicinity of R'. ( 4 )The global figure of the trajectory at 6 = 0 has three possible cases as in Figures 13.6, 13.7, and 13.8. ( 5 )This unstable cycle cannot be generated from a separatrix cycle passing through M and surrounding R'. The reason is seen in formula (14.16), below.
THEORY OF LIMIT CYCLES
310
FIGURE 14.1
FIGURE 14.2
FIGURE 14.3
Which case it belongs to depends on the order relation between m and
1/a- a and the order relation between m and 1 +a. In fact, when 6 = m = 1 +a, the coordinates of R' are ( -1/a, -1), and at this time the coefficients in (14.11) are
m'
= mv'-1/a = (1 + a)v-1/a,
a'= a(-1/a) 3 12 = -~.
From this we can see that m' = 1/a'- a'; that is, in this case MN' has become an integral line, and when o= m, from m' < 1/a' -a' we can deduce that m > 1 +a, and from m' > 1/a'- a' we can deduce that m < 1 +a. Thus similarly, as before, we can draw three figures (Figures 14.1-14.3) for 0 < m < -a, which correspond to m = o < 1 + a, m = o = 1 + a, and m = 8 > 1 + a, respectively.
§14. CLASS II EQUATIONS WITH 1 OR 2 CYCLES
311
Moreover, it is easy to see that when 0 < m $ 1/a- a we must have a< -1, whence m > 1 +a; and when m > 1/a- a, and m > 0, (but 1/a- a is not necessarily positive), then the three relations m > 1 +a, m = 1 +a, and m < 1 + a can all possibly exist. Now when Figures 13.6 and 13.8 for the case 0 < m $ 1/a-a, 6 = 0 change to Figure 14.3 for the case m = 6 > 1+a, we observe the change of limit cycles and separatrices in the neighborhoods of the two singular points. Since when 6 varies, (14.5) on any side of 1 +ax = 0 forms a family of generalized rotated vector fields, hence when 6 increases from 0, on the one hand, the point 0 changes from stable to unstable, and a stable limit cycle is generated near it; on the other hand, the two separatrices passing through N' and surrounding . 0 then approach each other and coincide. Notice that for
aQ2)
( aP2 ax + ay
N' =
6 + my1
> 0,
(14.15)
where Y1 > 0 is the ordinate of N', we can see that when two separatrices coincide and form a singular closed trajectory, this trajectory should be internally unstable. Hence it cannot be formed by the expansion of a stable cycle in the vicinity of 0 after arriving at N'. On the contrary, the separatrix passing through N' is first formed; then the two separatrices interchange their positions and an unstable cycle is generated from the separatrix cycle, which shrinks inwards as 6 increases, and finally coincides with the limit cycle, which lies inside and expands outwards to become a semistable cycle, and then disappears. Of course, here we assume that during the process of increase of 6 there do not suddenly appear one or more semistable cycles in the neighborhood of 0 which then split, some expanding and some shrinking (we conjecture this case does not happen). Hence in the neighborhood of 0, there must exist two values of 6, 0 < 61 < ~ < m, in the interval (0, m) such that when 6 = 61 a separatrix cycle passing through N' is formed, but when 6 = ~ a semistable cycle is formed. Hence for 6 in (0, 61), the vicinity of 0 has a unique stable cycle; for 6 in (61, 62) this vicinity has two and only two limit cycles, the outside one unstable, and the inside one stable; for 6 > 62 , there is no limit cycle in the neighborhood of 0. Next we observe how the separatrices in the vicinity of R' vary. Since the direction of the trajectory crossing M N' moves from left to right, it is easy to see that during the process of change of 6 from 0 to m there exists at least one value 6i < m such that the two separatrices passing through M and surrounding R' coincide and become a separatrix cycle. On the other hand, we can compute = 6 -m
ax
ay
(0,-1)
312
THEORY OF LIMIT CYCLES
Hence this separatrix cycle must be internally stable. From this we can see that as 6 j m, the unstable cycle shrinking towards R' cannot be generated by the above separatrix cycle. On the contrary, there must exist a 62 (< 6i) for which the vicinity of R' suddenly generates a semistable cycle, which splits into at least two limit cycles when 6 > 62, the outer one being a stable cycle which expands gradually and becomes a separatrix cycle passing through M at 6 = 6i, and then disappears; the inner one is an unstable cycle which shrinks into one point R' (-1 I a, -1) and changes the stability of R'. · However, the above analysis cannot determine whether limit cycles can coexist in the vicinity of 0 or R'; it is useless to compare the values of ~ and 62 since even though system (14.11) has the same form as (14.5), the values of m', a' and m, a, and the values of m' I a' and ml a are not the same. In order to solve this problem, note that when 2 6 = m +2am 4a
(14.17)
we have J(1 + mla)2- 46la = 1; hence a'= a, m' = m, and 6' = 6; that is, at this time (14.11) and (14.5) are identical. Hence the structure of the trajectory in the vicinity of 0 is the same as in the vicinity of R. But this fact can only occur in the following four cases: a) The vicinity of 0 does not have a limit cycle, and the periphery of R has not yet generated a limit cycle. b) The vicinity of 0 and the vicinity of R have a unique single cycle. c) The vicinity of 0 and the vicinity of R have a semistable cycle. d) There are two cycles each in the vicinity of 0 and R, and their forms are identical. If case a) occurs, then it is clear that the limit cycles cannot coexist in the vicinities of 0 and R for any 6; if case c) or d) occurs, then we have an example of (2.2) distribution since even in case c) a semistable cycle can be considered as a combination of two single cycles just as the multiple root of an algebraic equation. This problem has recently been solved in [216], and the answer is that a) holds. The author first transformed (14.5) into equations of the class (A) of Cherkas [192] introduced in §12
dxldt = 1 + xy,
(14.18)
and then according to the method of [192] applied the transformations Y = y 3 jx + 1/x2 , = 1/x and y = lell-ao 2 z, = to change (14.18) into an
e
e e
§14. CLASS II EQUATIONS WITH 1 OR 2 CYCLES
313
equation of Lienard type:
dz/dt
= P4(~)1~12ao2-3sgn ~ + zP2(~)1~1ao2-2sgn ~.
dUdt = z,
(14.19)
where P2(~) and P4(~) are quadratic and quadric polynomials respectively. Finally, [216] made use of the condition (14.17) to show that in this ca.se the divergence of (14.19) ha.s constant sign and hence (14.19) does not have a limit cycle. This explains that for any 8, limit cycles cannot coexist in the vicinities of both 0 and R. Next we see, for 0 < 1/a- a< m < -a (here we must have a< -1 and hence must have m > 1 +a), a.s 8 increa.ses from 0 tom, how the figure of the trajectory changes from Figure 13.7 to Figure 14.3. At this time a stable cycle is generated from 0, and continuously expands. On the other hand, the two separatrices through M and surrounding 0 come close together and coincide. Since the divergence of (14.5) always takes negative value at the point M (when 0 :5: 8 < m), there must be an odd number of limit cycles in the vicinity of 0. We conjecture that when a limit cycle exists, it should be unique; that is, the stable cycle generated from 0 continuously expands and becomes a separatrix cycle through M, and then disappears. Afterwards, the two separatrices lf and l1 through M interchange their positions, and in the vicinity of 0 there is no closed or singular closed trajectory. But the uniqueness ha.s not yet been proved. The situation is more complicated in the neighborhood of R'. When 8 increa.ses from 0, the two separatrices through N' and surrounding R' rotate in different directions; they may coincide to form a separatrix cycle through N', or it may be possible that the separatrix going directly to the left side coincides with another separatrix starting from M going to the right side to become an integral line M N', the separatrix from the right not yet having arrived to coincide with it. In the former ca.se, from (14.5) we know that the separatrix cycle will generate an unstable cycle, and when 8 j m, it shrinks towards R'; this is case (i) mentioned previously. In the latter ca.se, there can be two different situations: One is: when M N' becomes an integral line, the separatrix from N' going to the lower right side ha.s not yet interchanged its position with the separatrix entering M from the lower right side; then the two separatrices Lt and lf all enter R', but tt is still on the outside of Lt, a.s shown in Figure 14.4. As 8continues to increase, Lt and tt coincide, and then interchange their positions; hence lt and lf together surround R'. According to (14.16), as discussed previously, we know that now in the vicinity of R' a semistable cycle, externally stable but internally unstable, will suddenly appear, and this is case (iii) as previously mentioned. The other situation is
THEORY OF LIMIT CYCLES
314
Cl'
0
FIGURE
FIGURE
14.4
14.5
zt
that when M N' becomes an integral line, Lt and also just coincide. Then as 8 continues to increase, the separatrix cycle through the two saddle points generates an unstable cycle, and this is case (ii) mentioned previously. We can give examples showing that the above three different cases can appear [21], and it is clear that case (ii) is the transitional case from case (i) to case (iii). It is obvious that Figure 13.7 with m > 1/a- a, m > 0, and 8 = 0 can change into Figures 14.1 and 14.2 form= 8 ~ 1 +a in case (i) as previously mentioned. In particular, when 1/a- a > 0 (i.e., a < -1}, Liang Zhao-jun [217] used the previous method to prove the limit cycles cannot coexist in the vicinity of both 0 and R'. But whether this conclusion holds when a ~ -1 is still unknown.( 6 ) The above demonstration is also suitable for m = -a, because when 6 = 0, even though R = N is a higher order singular point, as soon as 8 becomes positive, N' immediately separates from R', and R' of index + 1 lies below and it is a stable singular point; moreover, when 8 = m, it becomes unstable again. The above method of discussion by dividing into the three cases m > 1 +a, m = 1 + a, and m < 1 + a can also be applied to the case m > -a, because for 8 = m the figures of its global structures are exactly the same as Figures 14.1-14.3, save that the starting figure of the global structure is not Figure 13.6, 13.7 or 13.8, but Figure 13.10. Readers can analyze for themselves when ( 6 }If we limit ourselves to the case 0 < 6 < m < -a, then, using the method of [27], we can prove that when m/2 ~ 6 < m or 0 < 6 < !m + !m2 /a, the limit cycles of (14.5) are centrally distributed; here there is a gap between these two intervals.
315
§14. CLASS II EQUATIONS WITH 1 OR 2 CYCLES
this change takes place. We should note that when 6 = 0 and 6 = m, we have = 0, and also
6'
how 6' changes with 6 is shown in Figure 14.5. 3. a< m < 0. When 6 = 0, 0 and R(-1la, -(a+m)la) are unstable foci. If we let 6 decrease from 0, then the point 0 becomes stable and a unique unstable cycle appears near it. As before, the stability of R' begins to change at 6 = m, and R' and N' following the decrease of 6 come close together; at 6 = m, R' arrives at ( -1la, -1), and there is a stable cycle which shrinks and approaches R'. In the following we prove that the limit cycles cannot coexist in the vicinities of 0 and R'. LEMMA 14 .1. When 6I m ~ ~, the vicinity of the point 0 does not con-
tain a closed or singular closed trajectory. The method of proof is almost the same as the second part in the proof of Theorem 14.1, and is omitted. Now we note that by (14.12) we have ~-!+ (1+mla)l2-6lm m'- 2 J(1+mla)2 -46la'
(14.20)
From this we can see that 6' I m' > ~ for 6I m < ~. Hence applying Lemma 14.1 to system (14.11), we see that here the vicinity of R' does not contain a closed or singular closed trajectory. In other words, if there may exist a limit cycle near the point 0, the vicinity of R' must have a limit cycle. The case of an even or odd number of limit cycles in the vicinities of these two singular points can be discussed as before. In short, for system (14.5), we conjecture that if there exist an odd (even) number of limit cycles in the vicinity of a singular point, there must be at most one (at most two). 4. m ~ a. Similarly to 3, if we let 6 decrease from 0, then an unstable cycle appears near the point 0. Moreover, R' and N' come close together as 6 decreases. If m = a, then at 6 = m, R' and N' arrive simultaneously at ( -1la, -1) and become a higher-order singular point; when 6 decreases again, R' and N' disappear. If m < a, then at 6 = m the saddle point N' arrives at (-11a, -1), which corresponds to 6' = 0, but R' corresponds to o' = m(l- mla) > 0, and so R' does not change its stability. As ocontinues to decrease, R' and N' come into coincidence, and then disappear. From (14.20), since now mla ~ 1, it is easy to see that we have o' lm' ~ for all 6 in [m, 0]; that is, a limit cycle never appears near R'.
!
THEORY OF LIMIT CYCLES
316
Synthesizing the above analysis, we get THEOREM 14. 2. For system (II)t=O (a < 0, m f. 0), we have: (i) If m > -a > 0, limit cycles can coexist in the vicinities of two singular
points of index +1. (ii) IfO < m < 1/a-a, then, for 8 in some interval (81,82), in the vicinity of the point 0 there are at least two limit cycles; for 8 in another interval (62,61), the vicinity of R' has at least two limit cycles. However, limit cycles cannot coexist in the vicinity of 0 or R' if 0 < m ~ -a and -a > 1. (iii) If a < m < 0, then limit cycles cannot coexist in the vicinity of 0 orR'. (iv) If m ~ a, then there does not exist a limit cycle in the vicinity of R'.
(II) m = 0. Here we have the system dx dt
= -y +ox+ lx 2 + ny2 ,
dy dt
= x(1 +ax),
a f. 0.
(14.21)
We may as well assume a< 0 and taken= 1; and from now on we only have to study( 7 ) dy dt = x(1 +ax),
a< 0.
(14.22)
LEMMA 14.2. When lo ~ 0 but Ill+ 181 f. 0, system (14.22) does not have a closed trajectory or a singular closed trajectory. PROOF. We can take the Dulac function as e- 21 Y, and the proof is omitted.
It is easy to compute that when 8 = 0, we have v3 = -!7l'al for system > 0 (< 0) the origin is an unstable (stable) focus, and when 8 < 0 (> 0) and 181 is sufficiently small there exists a unique unstable (stable) limit cycle near the origin. (14.22); that is, when l
LEMMA 14.3. When
8 ~ l/a (l > 0)
or
8 ~ 1/a (l < 0)
the limit cycle in the vicinity of the origin disappears, and the case for the neighborhood of another singular point of index +1 is similar but the inequality is reversed. PROOF. When 8 = lja, from (14.22) we get
dx -y+ y 2 dy = x(1 +ax)
l
+ ~·
(14.23)
(1)At this point we mainly present several theorems from (179]. For system (14.22), L. I. Zhilevich (218] did similar research, but the results were not the same.
§14. CLASS II EQUATIONS WITH 1 OR 2 CYCLES
317
but the family of integral curves of the equation dx -y+y 2 dy = x{1 +ax)
in the vicinity of each of the two singular points of index +1 is a family of closed curves {see Figure 13.5 of §13). From {14.23) we can see that they are arcs without contact for the trajectory of system {14.22). Hence in this case {14.22) cannot have a limit cycle. Then from the theory of rotated vector fields we know at once that when 8 ~ lfa (l > 0) or 8 ~ lfa (l < 0) the vicinity of the origin does not have a limit cycle. In order to consider the other singular point of index +1, we first apply a transformation similar to {14.10) and then proceed with our discussion, and, similarly to the case of 0 < m < 1/a- a in {I), the following theorem will hold: THEOREM 14.3. Limit cycles cannot coexist in the vicinities of two singular points of index +1 of system {14.22).
The detailed proof is left to the reader. System {14.22) has at most four singular points, and they are denoted by 0{0, 0), N(O, 1), R( -1/a, yt), and A( -1/a, Y2), where Y1,2
=
a=FJa2-4(l-ao) · 2a
It is easy to see tkat N and A are saddle points, and 0 and R are nonsaddle points of index +1. In the following we use Theorem 6.4 (still its special form, i.e. the uniqueness theorem of [111]) to discuss the uniqueness of a limit cycle of system {14.22). From Lemma 14.3 we know that we only have to consider the interval (lja, 0) or {0, lfa) of o. If we rewrite {14.22) as dxfdt = -y + y 2 +ox+ lx 2 = -cp(y)- F(x), dyfdt = x + ax 2 = g(x),
where cp(y) = y- y 2 and cp'(y) = 1- 2y, then we can see that cp(y) monotonically increases in the interval -oo < y ~ !· Let G denote the plane region {-oo < x < -1/a, -oo < y ~ 1/2}, and F(x) =-ox -lx 2, f(x) = F'(x) = -8- 2lx. Then we have f(x))' _ 1 2 ( g(x) - (x + ax 2 ) 2 [2lax + 2a8x + 8].
THEORY OF LIMIT CYCLES
318
Hence when l > 0 and 6 < 0 we have (f(x)fg(x))' ~ 0 provided that 6 ~ 2l/a, but when l < 0 and 6 > 0 we have (f(x)fg(x))' ~ 0 provided that 6 ~ 2lfa. Hence, provided there exist parameters (a, l, 6) that make the limit cycles of the corresponding system (14.22) lie entirely in the region G, we are allowed to use Theorem 6.4 to prove the uniqueness of a limit cycle. In the following we shall give some conditions to be satisfied by the parameters (a, l, 6) in order to guarantee that the limit cycles of system (14.22) surrounding the origin must lie entirely in the region G.(8 ) THEOREM 14.4. When l > 0 and a 2 (a 2 - 4l) -16 ~ 0 (i.e., a 2 ~ 2l + 2y'i2TI), system (14.22) has at most one limit cycle (unstable cycle). PROOF. Let the two lines without contact passing through the saddle point A( -1/a, Y2) be Lt ,2
= y- Kt ,2
K1,2
=
(x + !) a -
Y2
= 0,
2~ ( -{3± Jf32 -4a),
where a= 2y2 -1 and f3 = 6- 21/a; K 1 and K 2 are two roots of the equation aK2 + {3K + 1 = 0. Since a< 0 and f3 > 0 (by Lemma 14.3, when l > 0, we may as well assume 6 > l/a; hence f3 > -l/a > 0), we have Kt < 0 and K2 > 0. Also it is easy to see when 6 E (l/a, 0), we have Y2 ~ 0. Suppose the line Lt without contact and they-axis intersect at Ytoi clearly Yto > 0. Since the highest point of the limit cycle in the vicinity of the origin must lie on the y-axis, and it cannot intersect Lt, hence if we require Yto ~ we can assure that the limit cycles surrounding the origin must lie entirely in the region G. For Yto ~ ! , it is easy to see that we only have to require
!,
H( 6) = 63
4l) - 16 < O. (14.24) 16a Let H'(8) = 0. We get its two roots to be 81 = (2l- a 2)/3a and 82 = -a/2; and, under the conditions of the theorem, 81 ,82 > 0, and since H(O) ~ 0, for all8 < 0 inequality (14.24) holds. The theorem is completely proved. For the case of a 2 (a2 -4l) -16 < 0, since at this time H(O) > 0, inequality (14.24) cannot hold for all 8 > 0. Suppose 83 is a unique negative root of H(6) = 0. Then when 8 < 83 we have H(6) < 0. Using a suitable estimate on 83, we can get the following theorem:
+
5a2 - 4l 62 4a
+
a2
-
2
21 6
+
a 2 (a 2
-
THEOREM 14.5. When l > 0 and 6 -l/a ~ -a(1 + (l- 4)/(a2 system (14.22) has at most one limit cycle (unstable). The proof is omitted; see the original paper of the author [179]. (B)The following five theorems are given in [179] and [218].
+ l))/8,
§14.
CLASS II EQUATIONS WITH 1 OR 2 CYCLES
319
FIGURE 14.7
FIGURE 14.6 THEOREM 14.6. When l one limit cycle (stable cycle).
< 0 and a2 +4l ~ 4,
system (14.22) has at most
PROOF. Let the line L1 without contact and they-axis intersect at 'fhi it is easy to see that 'fh > 0. As before, we only have to prove that 'fit ~ ~. By the proof of Theorem 14.4 we only have to establish inequality (14.24), and 6 ~ min(61. ~). Under the conditions of this theorem, it is clear that a 2 (a 2 - 4l)- 16 > 0; hence H(O) < 0. Hence H(6) has at least one positive root; we suppose its smallest positive root is 64. From the conditions of this theorem, we know that H (l /a) < 0, and
l
;i <
21- a 2 3a <
a
-2,
and it is easy to compute H( -a/2) > 0; thus -a/2 is a minimum point of H, and (21- a 2 )/3a is a maximum point of H; hence 64 should be between l/a and (21- a 2 )/3a (see Figure 14.6). Hence when 6 < 1/a, inequality (14.24) must hold, and 6 ~ min(61. ~). Again by Lemma 14.3, the conclusion of this theorem holds. Estimating suitably the approximate value of 64, we can also get THEOREM 14. 7. When l < 0 and 6 + a/8 ~ (1 (14.22) has at most one (stable) limit cycle. The proof is omitted. ·
+ Jl2"+4)j4a,
system
THEORY OF LIMIT CYCLES
320
The uniqueness Theorems 14.4 and 14.6 obtained above can only solve part of the problem on the number of limit cycles for system (14.22). Considered from the parametric (a, l)-plane, the above two theorems only prove that when (a,l) is in region I (Figure 14.7), the limit cycle of (14.22) is unique. Nothing is known outside this region. But we can definitely say that the points in (a, i)plane not in the region I, the uniqueness of a limit cycle does not necessarily hold. THEOREM 14. 8. When l ~ 2a2 , if there exists a limit cycle surrounding the origin, then for some 6 there are at least two [179]. PROOF. Suppose r is a limit cycle of (14.22) surrounding the origin. Calculating the integral of divergence along r' we get
J(f)
= =
i (~: i
[6
+ ~~)
dt
=
i
+ l ~a!6(y2- y)]
=~
1
(6
+ 2lx) dt =
i
dt
[y2-y+6(l-a6)] dt
!r 2al = ~ 1 [( _ !) 2 26(l- a6)- all l - a6 !r Y 2 + 4al l- a6
(6- 2alx 2) dt
d t.
Hence when 6 (we may as well assume 6 > lja) satisfies 26(l- a6)- al ~ 0, we have J(f) < 0. From this we can see that in this case r in fact does not exist, since. if r exists and is not unique, then we can always find one r such that J(f) < 0 does not hold; conversely, if r exists and is unique, the stability of r and the stability of the origin are the same, which is not possible. On the other hand, in order to make 6 satisfy 26(l- a6)- al ~ 0, we only require that
6
~ 6' = 2~ ( 1 -
J
7
1- 2
2
)
0
Also, only when 6 ~ "G = (4l- a 2 )/4a do the singular points Rand A exist. When l ~ 2a2 , it is clear that 6' > 8. Hence in the range of variation of the parameter from generation of a limit cycle to its disappearance, R and A do not exist. Now suppose the limit cycle r exists and is unique. Then it must be an unstable cycle, since the origin is stable. r continues to expand as 6 decreases, until it passes through the saddle point N and becomes an internally unstable separatrix cycle. But we compute that the divergence of N equals 6 < 0, and this requires that the separatrix cycle should be internally stable. The
321
§14. CLASS II EQUATIONS WITH 1 OR 2 CYCLES
above contradiction explains why for some 6, the number of limit cycles in the vicinity of 0 is at least two. From this we can see that if the number of limit cycles in the vicinity of 0 can be at most two, then this case is the same as the case of 0 < m < 1/a- a in (I), when the stable cycle in the vicinity of the origin has not yet expanded to arrive at the point N, but the two separatrices passing through the point N have already come close together and coincide to form a separatrix cycle. Afterwards, the separatrix cycle disappears, and generates a stable cycle which shrinks inwards; here the vicinity of the origin has exactly two limit cycles, and they finally coincide to become a semistable cycle and disappear. Theorems 14.4, 14.6, and 14.8 show that in the (a, i)-parametric half-plane, the problem of the number of limit cycles has been solved only in regions I and n, but the problem in region n has not yet been completely solved since the property of having at most two limit cycles has not yet been proved. Up to now, those theorems in §7 have not been used for system (14.22), since its function ~(y) is a monotonic function only when y < For anywhere outside regions I and IT, how many limit cycles does the cor~ responding system (14.22) have? This is still an unsolved problem. However, we still believe that if system (14.22) has limit cycles, it should not have more than two. (Ill) n = 0. Here we have the system
!·
dx dt
= -y+6x+lx 2 +mxy,
dy
dt = x(1 +ax).
(14.25)
Without loss of g~nerality, we assume that a = 1 and l > 0. Thus we have
dx
dt = -y + 6x + lx
2
+ mxy,
dy
dt = x(1 + x),
l > 0.
(14.26)
It is easy to see that this system has two singular points 0(0, 0) and R( -1, (l- 6)/(m + 1)). 0 is a nonsaddle point; R is a nonsaddle point when m + 1 :::; 0, and is a saddle point when m + 1 > 0. · LEMMA 14.4. When 6 = 0 and 0:::; a limit cycle in the whole plane [219].
m:::; 2, system (14.26) does not have
PROOF. We note that 1- mx = 0 is a line without contact. Construct a Dulac function
B(x,y)
= (1- mx)2/m
2
+2/m-tex 2 -my 2 +2(1/m+t)x-2l 11 •
Then when 6 = 0, we have
!_(BP2 ) + 88 (BQ2)
ax
y
=
-l(2mx2 + 2- m)x 2 (1- mx)- 1B(x,y).
THEORY OF LIMIT CYCLES
322
It is obvious that the right side of the above formula is negative when 0 $ m $ 2. The lemma is proved.(9 )
LEMMA 14. 5. When 6 = 0 and m $ 0, system (14.26) does not have a limit cycle surrounding the origin. PROOF. Construct a Dulac function B(x,y)
= (1/(mx -1))e- 21Y.
Then when 6 = 0, we have :x (BP2)
+ :y (BQ2)
= -l(2mx- 2 + m)x2(mx- 1) 2 e- 21 Y.
Clearly the right side of the above equation has a constant sign when x > 1/m. Since 1 - mx = 0 is a line without contact, any closed trajectory surrounding the origin must lie in the region x > 1/m. The lemma is proved.
LEMMA 14.6. When l(l + m6) $ 0, system (14.26) does not have a limit cycle. PROOF. Construct a Dulac function B(x,y) = (1- mx)- 1 . Then
:. (BP2 ) +
~ (BQ,) = ! [-1m
2 ( •-
! )' + + l
l
rM (1 -
=)-'.
Under the condition of the lemma, the right side of the above formula has a constant sign. The lemma is proved.
LEMMA 14. 7. When 6/l ~ 1, system (14.26) does not have a limit cycle surrounding the origin.(1°) PROOF. Consider the system dxjdt
= -y + mxy,
(14.27)
which takes the origin as its center. Comparing the slopes of the trajectories of (14.27) and (14.26), we get -y + mxy + lx 2 + 6x -y + mxy lx + 6 X + x2 X+ x 2 = 1 +X • From this we can see that the locus of points of contact of the trajectories of these two systems consists of 1 + x = 0 and lx + 6 = 0. Any closed trajectory r of system (14.26) cannot intersect 1 + x = 0; if 6/l ~ 1, then the line lx + 6 = 0 is on the left side of x + 1 = 0, and cannot intersect r. But this means that r cannot exist. ( 9 )[210] gave another method of proving this lemma. (1°)Since :Vs and l(m- 2) have the same sign, we only have to prove Lemma 14.7 for m < 2; when m ;::: 2, the conclusion is obvious. Moreover, this lemma also holds when l < o.
§14.
CLASS II EQUATIONS WITH 1 OR 2 CYCLES
323
LEMMA 14.8. When 6 ~ l, there does not exist a limit cycle of system (14.26) surrounding the singular point R. PROOF. Move the origin of the coordinate system to R. The system becomes dx ( m(l- 6) ) 2 dt = -(1 + m)y + 6 + m + 1 - 2l x + lx + mxy, (14.28) dy dt = x(x -1). We may as well assume 1 + m < 0, for otherwise R is a saddle point or does not exist. Apply the transformation y , t = t/ v' -1 - m; v'-1- m then (14.28) becomes (we still write x, y, and t as x, y, and t) x
= -x, y =
dxfdt = -y + 6'x + l'x 2 + m'xy,
dyfdt = x(1 + x),
(14.29)
where
6- (2 + m)l l' _ -l 6, _ - (m+1)../-1-m' - v-1-m' (14.30) m' = -m(1+m)- 1 • Then we obtain the proof of this lemma from Lemma 14. 7, since from 6' /l' ~ 1 we can deduce 6 ~ l. We have seen from the previous discussion that when 6 = 0 the focal quantity of the origin has the same sign as l(m- 2). Also from {12.21) it is easy to see that when 6 = 0 and m = 2, the focal quantity of the origin has the same sign as -l; hence from the theory of Bautin we can get THEOREM 14.9. (i) When 6 = 0, m > 2, and m- 2 « 1, there exists at least one limit cycle in the vicinity of the origin. (ii) When m > 2, 6 < 0, and 0 < 161 « m - 2 « 1, there exist at least two limit cycles in the vicinity of the origin. (iii) When m < 2, and 0 < 6 « 1, there exists at least one limit cycle in the vicinity of the origin. (iv) When 6 = 0, m < -2, but lm + 21 « 1, there exists at least one limit cycle in the vicinity of R. (v) When m = -2 and 0 < 6 « 1, there exists at least one limit cycle in the vicinity of R. COROLLARY 14.1. By (iii) and (v) of the theorem, when m = -2 and 0 < 6 « 1, limit cycles can coexist in the vicinities of 0 and R. In the following we shall study the uniqueness problem of a limit cycle in several cases. Since we already know that the line 1 - mx = 0 is without
THEORY OF LIMIT CYCLES
324
contact, we can introduce the change of variables dtfdr = 1/(1- mx). Hence system (14.26) becomes (we still denote r as t):
dx lx 2 +ox dt = -y + 1- mx '
dy x'+ x 2 dt = 1- mx ·
( 14·31 )
In order to study the uniqueness of a limit cycle, we apply the transformation _y
=-
y + lx 2 + ox 1-mx'
x=x
to system (14.31). Then the system becomes (we still denote andy) dxfdt = y, dyfdt = -g(x)- f(x)y, where
x + x2 ( ) gx =1-mx'
f(x)
x andy
as x
(14.32)
= lmx 2 -
2lx - o (1~mx) 2
System (14.32) has two singular points: 0(0, 0) is a focus when R(-1,0) is a saddle point when 1 + m > 0.
161 < 2, and
THEOREM 14.10. When m < 0, system (14.26) has at most one limit cycle surrounding the origin [220]. PROOF. We shall use Theorem 6.4 of [112}, §6, to prove this theorem. Clearly we only have to verify that when m < 0, f(x)fg(x) is a nondecreasing function in the intervals (1/m, x 1 ) and (0, +oo ), where Xt
= ( 1 - J1 + 6mjl) jm
is a zero of f(x) (from Lemma 14.6, we know that when 1 + 6m/l ~ 0 the system does not have a limit cycle; hence we may as well assume 1+6mfl > 0). Here we can consider two cases: when m ~ -1, we consider the intervals (1/m,xt) and (O,+oo), and when 0 > m > -1, we consider the intervals (-1,x 1 ) and {O,+oo). It is easy to compute
d (l(x)) dx g(x) where
=
P4(x) (1- mx)2(x + x2) 2 '
+ (21- ml- 3mo)x2 + (26 - 2m6)x + o. Since we have assumed l > 0, we may as well assume o > 0 when m < 0, for P4 (x) = m 2lx 4
-
4mlx 3
otherwise the vicinity of 0 cannot have a limit cycle. Obviously P4(x) > 0 when x > 0. In the following we only have to prove that in the interval (1/m,xt) (when m ~ -1) or in (-1,x 1 ) (when 0 > m > -1), P4(x) > 0.
§14. CLASS II EQUATIONS WITH 1 OR 2 CYCLES
325
First we consider x $ -6 jl, i.e. x E (1/m, -6 fl) or x E ( -1, -6 /l). Rewrite P4(x) as P4(x) = (x + 6/l) lx( -m + 2- 4mx + m 2 x 2 ) - 6m(x- 1/m)(mx2 + 1); then it is easy to see that no matter what the value of x E (1/m, -6/l) (when m $ -1), or x E ( -1, -6/l) (when 0 > m > -1) is, we have P4(x) > 0. In order to determine the sign of P4(x) in (-6/l,xt), we can rewrite it as
2lx- 6)x(mx- 1)- mlx 3 + (-ml- 2m6)x 2 + (6- 2m6)x + 6.
P4(x) = (mlx 2
-
Suppose
cp(x)
= -mlx 3 + (-ml- 2m6)x 2 + (6- 2m6)x + 6.
Then
= (1 -
+ m6 /l) > 0, cp'(x) = -3mlx 2 + 2( -ml- 2m6)x + 6- 2m6.
6 fl)8(1
(14.33) (14.34)
Form< 0, if cp'(x) = 0 has a real root, they must be two negative roots, the larger of which is
xo = 1 + 26/l + J(1- 6~~
+ 36(6/l + 1/m)/l
<
-~.
(14 _35)
From (14.34) and (14.35) we can see that cp(x) is an increasing function in (-6/l,xt), and from (14.33) we know that cp(x) > 0 in this interval. Hence when m < 0 and x.E (-6/l,xt), P4(x) > 0. The theorem is proved. From Corollary 14.1 and Theorem 14.10, we again consider system (14.29), and get COROLLARY 14.2. When m = -2 and 0 < 6 « 1, system (14.26) has one unique limit cycle each in the vicinity of 0 and R. Cherkas and Zhilevich [221] modified the uniqueness theorem of [118] to obtain the following Lemma 14.9, which is used to study the uniqueness of a limit cycle of system (14.26) when m > 0. LEMMA 14.9. Suppose in the strip Xt < 0 < x2 we are given an equation dy g(x) (14.36) dx = F(x)- y"
J; g(x) dx,
Introduce Filippov's transformation z = into dyfdz = 1/(Ft(z)- y), dyfdz = 1/(F2 (z)- y),
to change the equation
when x > 0, when x < 0.
(14.37) (14.38)
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326
Equations (14.37) and (14.38) are defined in the intervals (0, zot) and (0, zo2) respectively, where Zoi = J;• g(x) dx. Suppose the following conditions are satisfied: 1) xg(x) > 0 when x '=I 0, and g'(O) '=I 0. 2) Fi(z) < 0 when 0 < z < zot· 3) There exist two numbers z 0 and z* (0 ~ zo < z* < Zoi) such that (a) FHz)(z- zo) < 0 when z '=I zo, 0 < z < zo2; (b) Ft(z") = F2(z*); (c) F~'(z) < 0 when z" < z < Zo2i (d) either F~'(z) < 0 when z0 < z < z"; or when zo = 0 and limz--+0 o:i(z) = o:0 > 0 (where O:i ( z) = Fi ( z) - 1/ FI( z)), the following inequalities hold: F~'(z)(z-
F~'(z)
<0
z) < 0 when z '=I z < z*, 0 < z < z*, when 0 < z
< zoti
(e) (djdy)[F2(y)- Ft(Y)] > 0 when (3 < y < Ft(z"), where (3
=Max [ lim Fi(z)] . 1=1,2 Z--+Zo\
Here Fi(Y) is the inverse function of Fi(z). Then equation (14.36) has at most one limit cycle; if this cycle exists, it must be a single cycle.
The proof is similar to Theorem 6.11, and is therefore omitted. THEOREM 14.11. Let one of the following three conditions hold:
1) o~ = 0·,
2) 0 < m
-1 + v'4m - 7 <_ 2·, 3) m > 2' 1~ > ---::---2m ·
Then system (14.26) has at most one limit cycle; if it exists, it must be a stable cycle [221].
PROOF. From Lemma 14.7 we know that when~~ l, system (14.26) does not have a limit cycle; also from Lemma 14.4 and the theory of rotated vector fields we know that when ~ ~ 0 and 0 ~ m ~ 2, the system does not have a limit cycle. In the following we may as well just consider the system of equations dx lx2 +~x dy x+ x 2 (14.31) dt = -y + 1 - mx ' dt = 1-mx·
§14. CLASS II EQUATIONS WITH 1 OR 2 CYCLES
327
Comparing with (14.36), we know now
F( ) _ lx 2 + 6x x - 1-mx' z x+x 2 z= 1 -mx dx, 0 = F~ x' = (lx~ + 6xi)' 1-mxi az z 1- mx·I :l:i Xi +x~
( ) _ x + x2 gx -1-mx'
1
F~u
_ -mlx~(z) + 2lxi(z) + 6 - (1- mxi(z))(xi(z) + x~(z)) ·
'
Here Xt(z) (x2(z)) is the inverse function of z(x) for x > 0 (x < 0). From the above formula we get that zo = 0 when 6 = 0, and in the region in which a limit cycle may exist we have
F'(z) r=
l(-mx1 + 2) (1- mxt)(1 + ~1?·
When 5 > 0, the sign of F~ (x2 (z)) at the point X2(ZO) =
Xo
=
1- J1-m6/l
----!....----:.....
m
changes from positive to negative, and Fi(z) < 0 for 0 < z < ZOl· In the following we have to prove that the curves Ft(z) and F2(z) have a common point in the interval 0 < z < min(zot, zo2); for this we only need to show that the system of equations
lx~ + 6x2 _ lx~ + 8x1 1 - mx2 - 1 - mx1 ' :~:2 x + x2 x + x2 dx = dx 10 -:--1-mx 1-mx 0
1zt
(14 .39 ) (14.40)
has a unique nontrivial solution when x2 < 0 and Xt > 0. We note that if Ft(z) and F2(z) do not have a common point, or (14.39) and (14.40) do not have a real solution for x 2 < 0 and Xt > 0, then system (14.31) does not have a limit cycle. From (14.39) we can solve Xt
-6 -lx2 mx2)'
= cp(x2) = 1{1 -
and again in (14.40) we consider Xt as a function of x 2. From this we get
X~
= '1/J(Xt, x2) = X2(X2 + 1)(1- mxt). Xt(Xt + 1)(1- mx2)
THEORY OF LIMIT CYCLES
328
Study the difference of the two derivatives:
where
P4(x2) = m 2 l 2 x~
+ (-2ml 2 + m2 l 2 )x~ + (2l 2 + (2lo + ml8)x 2 + 62 - 8l.
ml 2 )x~
In order to prove that the system of equations (14.39) and (14.40) has a unique nontrivial solution, we only have to prove that P4(x2) has only one zero for -1 < x2 < 0. Now suppose not, i.e. P 4 (x2) has at least two zeros in -1 < x2 < 0. Clearly this is not possible at o = 0. Hence we can assume o > 0. Since P4( -1) > 0 and P4(0) < 0, P4(x2) has at least three zeros in (-1,0). Thus P4(x2) has at least two zeros in (-1,0). But from P4(-1) < 0 and P~(O) > 0, we can deduce that P4(x 2) has at least three zeros in (-1,0). Thus P~'(x 2 ) must have at least two zeros in (-1,0); but this contradicts the direct calculations. Thus the system of equations (14.39) and (14.40) has a unique nontrivial solution when x2 < 0 and Xt > 0. In the following we verify (c) and (d) of condition 3) in Lemma 14.9. We can directly compute
where
Q4(xi) = m 2lx1 - 4mlx~ + (2l- ml- 3mo)x~
+ 2(1- m)8xi + 0. It is easy to see that when 8 = 0 for all z E (0, zot) we have FJ.' < o. and as for F~'(z), there exists a unique z < z• such that when z = z, F~'(z) changes from positive to negative. When 8 > 0, we only have to study F~' (x2 (z)) for x2 E ( -1, xo). First we consider x2 E (-8/l,xo), and rewrite Q4(x2) as
Q4 (x2) = (mlx~ - 2lx2 - h')x2(mx2 -
1)
+ (x 2 + 1)( -mlx~- 2mox2 + h'). It is easy to see that Q4 (x 2 ) > 0 when F~'(x2(z))
< 0.
X2 E
(-ojl, xo), and consequently
§14. CLASS II EQUATIONS WITH 1 OR 2 CYCLES
329
Next consider -1 < x2 ~ -6/l, and rewrite Q4(x2) as Q4(x2) = ( x2
+ ~) [m 2 lx~- (4ml + m 2 6)x~ +
m2 62) mt52 m2t53] ( mt5 + 21 - ml + - 1- X2 - mO - - 1- - l l
mt52 +o + -~-
mt53
m2t54
+ l2 + ---za· It is clear that when mo + 2l - ml + m 2t5 2/l (-1 + v'4m -7)/2m) we have Q4(x2) > 0.
~ 0 (that is, 6 /l
>
In order to verify condition 3)(e) of Lemma 14.9, it suffices to prove that the equation
has only one zero when condition (14.39) holds. It is easy to compute that
Ll _ 1 -
where -y(u)
( -l- m0)(x1 - X2)'y(x1 + X2) ( -mlx~ + 2lx2 + o)( -mlx~ + 2lxt + o)'
= u 2 + (1- 2/m)u- 2ojml.
Since (14.39) holds,
-8 -lmx~
Xt
+ X2 = l(1- mx2) < 0,
and so Llt has only one zero in the region under study. The conditions of Lemma 14.9 have all been verified. THEOREM 14. 12 [222]. When t5 ~ 0 there exists at most one limit cycle of system (14.26) surrounding the origin. PROOF. By Lemma 14.7 we only have to study the case when m l
> 2 and
> o > 0. Changing the signs of y and t in (14.31), we obtain dy dt
=
x + x2 1 - mx'
dx dt
lx 2 + ox = -y - 1 - mx ·
We have to use Theorem 6.11 to prove this theorem. By the proof of Theorem 4.11, we now only have to verify the condition
2F{'(z)z + F{(z) < 0, since
z=
1 x
0
-1
x+x 2 dx 1-mx
1( m1) x -m1- ( m1) lnll-mxl,
2 -=-X
2m
when 0 < z < zot.
m
1--
2
1+-
THEORY OF LIMIT CYCLES
330
F'( ) -mlxHz) + 2lxt(z) + 6 1 z = (1- mxt(z))(xt(z) + xHz))' where Xt(z)
> 0 is the inverse function of z
=
1 z
0
x+x2 1 -mx dx,
and
where
P4(xt) = m 2 lxf - 4mlx~ + (2l- ml- 3m6)x~
+ 26(1- m)xt + 6. Hence we only have to prove that when 1/m > x
Q(x)
>0
= 2P4(x)z(x) + (mlx 2 - 2lx- 6)(x + x2 ) 2 < 0;
here z(x) can be expanded into a power series
z=
1
m+1
m(m+1) 4 x 4 mn- 3(m+ 1) xn+···, n
2 3 2x + 3 -x +
+···+
and so the above formula can be written as
Q(x)
= x3 { Q1(x) + Qo(x)- 2m6- 2l + i(m + 1)6 + x(ao(26- 2m6) + at6) + x2[at{26- 2m6) + a26] + x3[a2{26- 2m6) + aa6] + · · ·},
where
Q1(x) = [m2 lx 4 - 4mlx 3 + (2l- ml- 3ml)x2 ] · [ 32 (m + 1) + -m+1 4 -2mx + ... + 2(mn+ 1) mn-3xn-3 + .. ·] < 0, Q0 (x)
= x(-2l- 6- 3mo) + x 2 ( -2ml- 2l) + ml(m + 1)x2 < 0,
an= !(m + 1)mn-axn. n Since 0 < x < 1/m, an < an-t· From this it is easy to prove that Q(x) < 0, and the theorem is thus proved.
§14. CLASS IT EQUATIONS WITH 1 OR 2 CYCLES
331
Starting from the above three theorems, many cases of the problem of uniqueness of a limit cycle of system (14.26) have been basically solved, but no results on the problem of having at most two solutions have been obtained. The problem oflimit cycles of system (14.26) is also studied in [223] and [224]. The complete study of equations (14.1) of class II has been given in [209], as mentioned in §13; that is, the global structure and bifurcation curves of the system possessing two fine foci of third order (hence without cycles) was studied. Also, the author of [223] used the transformation x = x, y = y - (JX, but {3 there was a coefficient to be determined. After the above transformation, system (II) becomes (we still denote x and y by x and y)
dxjdt
=-
dyjdt =
y + (t5 + f3)x + (l- mf3 + nf3 2 )x2
+ (m- 2nf3)xy + ny2 = {3P(x, y) + x(1 +ax).
P(x, y),
(14.41)
= x(1 +ax),
(14.42)
Comparing this with the system
dxjdt
= P(x,y),
dyjdt
we can derive from the qualitative properties of (14.42) some qualitative properties of (14.41). For example, we can prove THEOREM 14.13. If (14.42) does not have a limit cycle in the vicinity of 0(0, 0), and 0 is a stable (unstable) singular point of (14.42), then when {3 > 0 ( < 0), (14.41.) does not have a limit cycle in the vicinity of 0. Suppose the vicinity of another singular point M of index +1 of (14.42) does not have a limit cycle, and M is an unstable (stable) singular point of (14.42). Then when {3 < 0 (> 0), (14.41) does not have a limit cycle in the vicinity of M. Also if (14.42) does not have a limit cycle in the whole plane, and 0 and M have the same stability, then limit cycles of (14.41) cannot coexist in the vicinity of 0 or M.
[223] also applied the above transformation to equations of class I, and chose a suitable {3 to prove THEOREM 14.14. 1) Suppose l
< 0, l + n > 0, 0 < 3n ~ y1- 4nl, and
~ =!4 ( v'1-n 4nl ) 3 +
1 2n (1- v'1- 4nl) > 0.
Then (I)m=l does not have an unstable limit cycle for 0 < -D ~ ~. 2) Suppose one of the following conditions holds: (i) n > 0, t5 + 1/2n ~ 0.
THEORY OF LIMIT CYCLES
332
(ii) 0 < 4n(l + n) < 1 and 8 + [1- y'1- 4n(l + n)]/2n ~ 0. (iii) n < 0 and 8 + 21/(1 + v'1 - 4nl) ~ 0. Then (I)m=l does not have a limit cycle. Also if in (14.41) we choose (3 = m/2n and choose a Dulac function e-2(1-m 2 / 4 n)y for (14.42), and apply Theorem 14.13, then we can prove that if a
(8 + ;:) (1-7:) ~ o
then limit cycles of (II), if they exist, must be centrally distributed. Moreover, Chen Lan-sun [224] also studied (14.1) and obtained theorems on nonexistence, existence, and number of limit cycles, for examples, THEOREM 14.15. I/8
m
'# 0,
= m(1+n)-2a = 0, n '# ~1, n(4-m2)-2m2 '# 0,
then system (14.1) has at most one limit cycle in the whole plane.
THEOREM 14.16. When 5n + 3 > 0, m(1 + n)- 2a > 0, 8 < 0, and l5n+ 31, lm(1 +n)- 2al, and lSI are suitably small, then system (14.1)1=1,6#0 has at least three limit cycles near the origin. THEOREM 14. 1 7. I/ 8 = l + n = 0 and a '# 0, then the vicinity of 0 does not contain a limit cycle. If 0 < -a- (m 2 + 4)/2m ¢:: 1, then the vicinity of R has at least one limit cycle. If a < 0 or -a- (m 2 + 4)/2m < 0, then the vicinity of R does not have a limit cycle.
Exercises 1. Prove the conclusion on the uniqueness of a limit cycle of system (14.1) at the beginning of this section. 2. Prove that when m < 0, n > 0, and a > 0, system (14.1)6=0 does not have a limit cycle in the vicinity of 0 [224]. (Hint. Introduce the time transformation dt/dr = e- 2 '11 /(mx- 1).) 3. Prove Lemma 14.1. 4. Prove Lemma 14.2. 5. Prove that the conclusion of Lemma 14.3 also holds for the singular point of index + 1 other than 0. 6. Prove Theorem 14.3. 7. Prove that Y10 ~ in Theorem 14.4 is equivalent to the inequality in (14.24). 8. Prove the equality in (14.30). 9. Prove that the system
!
dxjdt = -y + x 2 + mxy- y 2 ,
dyjdt = x(1 +ax)
§14. CLASS II EQUATIONS WITH 1 OR 2 CYCLES
333
does not have a closed or singular closed trajectory near the origin, and that near the other singular point R of index +1 when a ! -(4 + m 2 )/2m the system has an unstable cycle which shrinks and approaches R, and when a~ -(4 + m 2 )/2m there does not exist a closed or singular closed trajectory near R. 10. Prove that if in the system of equations
dxjdt
= -y +ox+ lx 2 + mxy + y 2 ,
we have a < 0, l > 0 and 82 + m8 + l in the vicinity of the origin.
dyjdt = x(l +ax)
< 0, then there exists at most one cycle
§15. Some Local and Global Properties of Equations of Class III
In this section we study some local and global properties of the system
dxjdt
= -y + 6x + lx2 + mxy + ny2 ,
dyjdt = x(1 +ax+ by),
(15.1)
which may have a limit cycle. H we do not require that b '::/: 0, then (15.1) represents the most general quadratic system which may have a limit cycle, and the equations of classes I and IT discussed in §§12-14 are in fact special examples of (15.1). AB we mentioned in §12, Soviet mathematicians have another way of classifying quadratic systems. No matter whether they are equations of Class A or B, after transformation into our classification, generally speaking, they all possess the characteristic of b '::/: 0. Hence in this section we shall also introduce some results on quadratic systems obtained by Soviet mathematicians in the seventies and shall see that their method of classification has its own good points. First we introduce a simple but important theorem on the nonexistence of a limit cycle. THEOREM 15 .1. The system
dxjdt
= -y + lx2 + ny 2 ,
dyjdt = x(1 +ax+ by)
(15.2)
has one or two centers when a(b + 21) = 0, and does not have a closed or singular closed trajectory when a(b + 21) '::/: 0. PROOF. For system (15.2) we can take(l)
B(x, y) = ( 1 + by)-21/b-1., (1 )Note that the right side of (15.3) can take a complex value when 1 +by < 0. 335
(15.3)
THEORY OF LIMIT CYCLES
336
Then we can compute :x (BP2)
+
:y
'
(BQ2) = -a(b + 2l){1 + by)- 21 /b- 2 .
{15.4)
When a(b + 2l) = 0, the right side of (15.4) is always zero, and system (15.2) has a first integral; hence it has one or two centers. When a(b + 2l) ::/: 0, the right side of (1.54) keeps a constant sign in the half-plane 1 +by > 0, and 1 +by = 0 is a line without contact for the system (15.2); hence (15.2) does not have a closed or singular closed trajectory in the vicinity of the origin. In order to prove that (15.2) is the same in the half-plane 1 +by < 0, we only have to move the origin to the singular point of index +1 in the half-plane 1 + by < 0 (it is easy to see that if this singular point exists, it must be (0, 1/n)); then, using the new function B(x, y) of the form (15.3), we can get the proof as before. When a( b + 2l) ::/: 0, results on the global phase-portrait and bifurcation surface of system (15.2) are given in [208] and [209], as mentioned at the end of §13. Later, Ju Nai-dan attempted in 1965 to prove for special equations of class 111(2)
~~
= -y + 8x
+ lx 2 + mxy + ny2 ,
~~ = x(1 +by)
(15.5)
some results completely parallel to Theorems 12.4 and 12.6 for equations of class I, but his conjecture was only partially proved. He obtained the following two theorems. THEOREM 15. 2. The system of equations
dx dt
= -y + lx 2 + mxy + ny2 ,
~~ = x(1 +by)
(15.6)
has a center when m(l + n) = 0, and if the algebraic equation n(n + b)fP - mO- 1 = 0
(15.7)
has a real root for 9, then (15.6) does not have a limit cycle when m(l+n) ::/:0. THEOREM 15.3. When one of the conditions 1) n = 0, 2) l = 0, 3) b = -n, or 4) b = l holds, the limit cycle of system (15.5), if it exists, must be unique. (2)His results given here were not published by him, but were given in §14 of the first edition of this book.
337
§15. EQUATIONS OF CLASS III
The first theorem can still be proved by the method of the Dulac function, and will be discussed in the next section. The second theorem can be proved by the uniqueness theorem of [111]. During the ten-year period after 1966, our work was stopped completely. But during the same period, Soviet mathematicians made remarkable progress in studying quadratic systems. At the end of §12 we mentioned the new method of Kukles and Shakhova [200] for proving Theorem 12.4. In 1970, Cherkas [203) used the method of [200) to prove Theorem 15.2, and did not need to add more conditions to system (15.6). After that Cherkas, Zhilevich, and Rychkov (see [225], [221] and [292]) also used [112], [128] and a uniqueness theorem similar to that in [118] to prove that if system (15.5) has a limit cycle, then it must be unique. Hence Ju Nai-dan's conjectures have been completely proved. Now we introduce new equations with center constructed in [203] to compare a given quadratic system, and use it to prove the nonexistence of a closed trajectory of system (15.6) (Theorem 15.2), in which we do not demand that (15.7) has a real root for 0. For convenience, we adopt the method of the original paper to prove that the equation dy -x+ax 2 +bxy+cy2 (15.8) = dx y(cx + 1) (which is the system (15.6)) does not have a limit cycle. We may as well assume c ¥- 0, and then assume c = 1. Also, we may as well assume a ~ 0 or -1 < a < 0. If a ~ -1, then we can use the transformation
1+xt
1 = -1 - ,
y Yt = 1 +x
+x
to change (15.8) to dy1 dx1
in which we have -(a+ 1) dy dt
+ bx1Y1 + (1- c)y~ Yt(Xt + 1)
-x1 - (a+ 1)x~
=--~~--~~--~~~--~~
~
·
0. Rewrite (15.8) as a system of equations
= -x + ax 2 + bxy + cy2
'
dx dt = (x
+ 1)y.
(15.9)
The stability of (0, 0) can be determined from the sign of b(a +c). When b(a +c)> 0 (< 0), (0, 0) is an unstable (stable) focus, and when b(a +c)= 0, (0, 0) is a center. It is clear that a limit cycle can only appear in the half-plane :t > -1. First we prove that when ac ~ 0, a 2 + c2 > 0, and b ¥:- 0, system
338
THEORY OF LIMIT CYCLES
(15.9) does not have a limit cycle. This because (here T is a period of the cycle)
h=
~loT [bx + {2c + 1)y] dt
=~loT [b (ax 2 +bxy+cy2- ~~) + (2c+ 1)y] dt = !!._
{T (ax 2 + cy 2 ) dt, T lo and the right side of the above equation has the same sign as b( a + c); hence there is no limit cycle. Hence from now on we assume one of the following two conditions holds in order to prove our conclusion: a
> 0, c < 0, b '!- 0;
-1 < a < 0,
c > 0,
(15.10)
b '!- 0.
{15.11)
Consider the equation
(x + 1)yy1 = -x + ax2 + [bx + 1/J(x)]y + cy2 •
(15.12)
We choose a suitable 1/J(x) to make {0, 0) become a center of (15.12). In {15.12) we apply the transformation y = cp(u), where cp(x) is also a function to be fixed. We get
(x + 1)cp2uu' = -x + ax2 + [bx + 1/J(x)]cpu + [ccp- (x + 1)cp']cpu2. Now choose
+ 1)cp']cp = a( -x + ax2), (bx + 1/J)cp = [3( -x + ax 2),
[ccp - (x
{15.13) (15.14)
where a and {3 are constants. From (15.13) we see that cp(x) satisfies an equation of the form
(x + 1)yy' = cy2 + ax(1- ax) whose general integral is 2
2c [
=
(x + 1Y [ M
y = (x + 1)
M
(15.15)
r (x +
+ Jo
2ax(1- ax) ] 1)2c+1 dx .
(15.16)
Now we take
cp(x)
p. ( )
+ 2a (x; ~) 2 c
]
- 2aP2(0)
1/2
,
§15. EQUATIONS OF CLASS III
339
where P2 (x) is the numerator of the rational function obtained from the integration of {15.16), which equals 1 +a
1 + 2a
a
~ + 1- 2c (x + 1) - 2 - 2c (x + 1)
2
1 +a+ {1 + 2a)(x + 1) ln(x + 1)- a(x + 1) 2 1+a - 2 - - (1 + 2a)(x + 1) - a(x + 1) 2 In(x + 1)
(c = 1).
M is a constant to be determined later. Then from {15.14) we get
'1/J(x)
X
= cp(x) [.8(-1 + o:x)- b
Note that o:, ,8, and M have not yet been fixed. H we can make '1/J(x) have a constant sign in some interval [m, n], except for a finite number of points, then comparing the equation (15.12) with center and the given equation {15.9), we see that {15.9) does not have a limit cycle in the strip m ~ x ~ n, -oo < y < +oo. In order to make the function '1/J(x) have the same sign in a neighborhood of x = 0, we require that
.8 = -b 0, the function P2(x) is obviously always negative. In fact, let x + 1 = t; we get P2(t- 1), whose discriminant is
(a+ c)(1 +a- c) (1- 2c)2c(1- c) Now we take an arbitrary o:
0
< ·
< 0 and take M
= 2o:P2(0); then the graph of
cp(x) is a branch of a hyperbola. Thus '1/J(x) has a constant sign in (-1, 1/a). At this time, the qualitative figure of the integral curve of equation (15.15) between the branches of the hyperbola is as shown in Figure 15.1. Note that the hyperbola is a separatrix passing through a saddle point on the line X= -1. (ii) When a+ c < 0 we still take o: < 0, and we choose
P2(1ja)
M = 2o:P2(0)- 2o: ( 1/a + 1)2c,
so that y = cp(x) is a separatrix of equation (15.15) passing through (1/a,O). Thus the line y = VM (1 - ax) intersects the separatrix at two points, whose
THEORY OF LIMIT CYCLES
340
y, cp(z)
FIGURE
15.1
y.
FIGURE
15.2
abscissas are 0 and 1/a, and there are no other points of intersection because the number of contacts of any line with the vector field of (15.15) is not more than two. The qualitative figure of the integral curve of (15.15) is as shown in Figure 15.2. From this we can see that ,P(x) keeps a constant sign in ( -1, 1/a). Now we look at the case (15.11). Take a> 0; then under condition (15.11), equation (15.15) has two saddle points (0, 0) and (1/a, 0) on the x-axis, which are on opposite sides of x = -1. It is easy to prove that the infinite point on the y-axis is a saddle point. At the same time there are still two finite nodal points, lying on the line x = -1. One separatrix starting from (0, 0) enters the nodal point Pt(-1, y'o(a + 1)/c), and another separatrix runs to the infinite nodal point. From this we can see all the trajectories above these two separatrices all connect the nodal point ( -1, y' o( a + 1)/c) and the infinite nodal point; that is, the graph of the single-valued function y = cp(x) obtained previously when M > 0. Now take M = afc(a + 1). Then the line y = v'M(1- ax) will pass through the saddle point (1/a, 0) and the nodal point (-1,y'o(a+l)/c), and this line is not tangent to the vector field of (15.15) when x > -1. If we take the function cp(x) determined by M (see Figure 15.3) and substitute it in the formula for ,P(x), we can see that ,P(x) keeps a constant sign when x > -1. Thus, we have already proved (15.8) does not have a limit cycle. In the following, we again prove the uniqueness of a limit cycle of equation
(15.5). THEOREM
15.4. If equation (15.5) has a limit cycle, it has at most one.
§15. EQUATIONS OF CLASS lli
341
FIGURE 15.3 PROOF. For the convenience of those readers who wish to refer to the original papers, we shall use their notation; that is, we rewrite {15.5) as
du dx
=
-x + ax 2 + (bx + c)u + du 2 u(x + 1)
{15.17)
Using the transformation u = (x + 1)dYl {when d = 0, it is an identity transformation), equation {15.17) can be changed to
dy1 -x + ax 2 + (bx + c)(x + 1)dY1 dx = (x + 1)2d+ly1 ' or to the equivalent system of equations
dy
dt
= (x- ax 2 )(x + 1)-'Jd-l = g(x),
~: = -y +
Lz (bx + c)(x +
1)-d- 1 dx
= -y
-foz f(x) dx.
(15.18)
First we study the uniqueness of a limit cycle in the vicinity of the origin. From Theorem 15.2 and the theory of rotated vector fields, we know that for cb(a +d)> 0 there does not exist a limit cycle surrounding the origin; hence in the following we may as well assume cb( a + d) < 0. Also, without loss of generality, we assume that b < 0, and (as in the proof of Theorem 15.2) that a > -1. Hence, from now on, we always have
b < 0,
c( a + d) > 0,
a > -1.
In the following, we carry out our discussion in six different cases: 1) a ~ 0, c > 0, d ~ 0; 2) a > 0, c > 0, d < 0, a + d > 0; 3) a > 0, c < 0, d < 0, a + d < 0;
THEORY OF LIMIT CYCLES
342
4) -1 0, a+ d < 0; 6) -1 < a ~ 0, c > 0, d > 0, a+ d > 0. I. First we study cases 1) and 4), which are characterized by ad~ 0. From the position of the singular points and integral lines of {15.17), we see that limit cycles can only lie in the strip -1 < x < 1/a (or x > -1, for case 4)) or, when a= 0, in the half-plane -1 < x. Next, on computing the divergence for (15.18), we get 8Pf8x + 8Qj8y = (bx
+ c)(x + 1)-d-l,
and any limit cycle cannot intersect with the integral line x + 1 = 0; hence it must intersect the line bx + c = 0. Thus only for b + ac < 0 does there exist a limit cycle. From this it is not difficult to prove that for system (15.18), in case 1), when x E (-1,0) or (-cjb, 1/a), we have( 3 ) d (') (x dx =
g
+ 1)(-abx2 - 2acx +c)+ dx(bx + c)(ax- 1) (ax2- x)2(x + 1)1-d
(15.19)
>0. Use x2 to denote the abscissa of the extreme right point of a limit cycle L1 near the origin. Then we must have X2 > -c/b. Construct a function !1(x) = f(x)- f((x 2)) g(x). g
X2
In the region surrounded by L1 we can find a point (x 1 < 0, 0) such that ft(xt) = 0, and it is clear that ft(x2) = 0. Since ft(x)fg(x) is monotone in the intervals (-1,0) and (-c/b, 1/a), we know that ft(x) < 0 when x E [x1,x2]. Now for system {15.18) we apply Theorem 6.4 of §6 to obtain the uniqueness of a limit cycle. In case 4) it is not hard to see that a limit cycle surrounding the origin can exist only when c > b,( 4 ) and as before it must contain the point ( -cjb, 0) of the x-axis. Here we again use (15.19) to prove that in the intervals (-1, -c/b) and (0, +oo) we have d(f fg)fdx < 0; the rest of the proof is similar to case 1). II. Next we study cases 2) and 3), the ones in which a> 0 but ad< 0. As before, we still assume b < 0. Hence we have to slightly modify the uniqueness theorem (Theorem 6.11) of Rychkov before we can use it (see [221]). that here -aln: 2 - 2acx +cis positive definite. first use the transformation dt/dT = 1/(1 + x), and prove that the integral of the divergence along any closed trajectory is always negative when c = b. Thus the origin is also stable, and there does not exist a closed tr~ectory. ( 3 )Note
( 4 )We
343
§15. EQUATIONS OF CLASS III
LEMMA 1. Suppose that for d1 < x < d2, dtd2 equation dy g(x) dx = F(x) -y'
<
0, we are given an
which can be transformed to dy dz
1 F1 (z ) - y
when x
> 0,
dy dz
1
= F2(z)- y
when x
< 0,
J;
by means of the Filippov transformation z = g(x) dx. The above two equations are defined for 0 < z < Zot and 0 < z < zo2 respectively, where Zoi = J;• g(x) dx. Suppose the following conditions are satisfied: 1) xg(x) > 0 when x =/:- 0, and g'(O) =/:- 0. 2) FHz) < 0 when 0 < z < zot· 3) There exist unique numbers zo and z• (0 ~ zo < z• < ZOi) such that a) F~(z)(z- zo) < 0 and z =/:- zo, 0 < z < zo2i b) Ft(z•) = F2(z•); c) F~'(z) ~ 0 when z• < z < Z
!~ (F,(z)- Ff~z)) =no> 0 the inequality F~'(z)(z- z) < 0 holds when z =/:also, Ff'(z) < 0 when 0 < z < zot; e) ddY [F2(Y)- Ft(Y)] > 0 for Ft(z*) > y > {3
z < z•, 0 < z < z•; and
=~ax [ lim t=1,2
Fi(z)] ,
Z-+Zoi
where Fi(Y) is the inverse function of Fi(z). Then the equation has at most one limit cycle; if it exists, it must be a single cycle.
The proof of this lemma, except a few individual cases, differs very little from Theorem 6.11; hence it is omitted. In the following we use this lemma to prove for a> 0 (i.e. cases 2) and 3)) the uniqueness of a limit cycle of system (15.18) surrounding the origin. We only prove case 2); the proof of 3) is similar. In the (y, z)-plane, we examine the curves y = Fi(z) (i = 1, 2) and use the following parametric equations to represent them: (i=1,2).
THEORY OF LIMIT CYCLES
344
Here
F(x)
=-
foz f(x) dx
c - b] b c- b = - (x + 1) -d [ d-b 1 (x + 1) + -d+ d- 1 + -d-, G(x)
= foz g(x) dx = (x + 1)
_ 2d [
1
1]
-1
< X2 < 0 < Xl
a 2a + a+ 2 2d - 2 (X + 1) + 1 - 2i X + 1) + 2d
a 2a + 1 a+ 1 - 2d- 2- 1- 2d- 2d'
1
The point of intersection of these two curves can be determined from the equations F(xt) = F(x 2) and G(xt) = G(x2). We can prove existence and uniqueness of the solution of this system as in [118], i.e., the curves y = F1 (z) and y = F2 (z) have only one point of intersection (z*, F 1 ( z*)). We omit the proof. From the method of differentiation of compound functions, we know that F{'(z) < 0 (when 0 < z < zot) in the lemma is equivalent to
.!!._ dx
(L) > o g
when 0 < x < 1/a. Now we rewrite the numerator of the fraction of (15.19) P(x) = (x + 1)( -abx 2 - 2acx +c)+ d(bx +c)( -x + ax 2) as
P(x)
= -a+d - ( x + 1)(-abx2 a
2acx +c)+ d(b + ac)(a + 1)x
3
de
--(ax -1) 2 [(a + 1)x + 1]
a
when a+ d > 0, d < 0, b + ac < 0, a> 0 and c > 0. From 0 < x < 1/a, we can deduce that P(x) > 0. Hence we obtain the proof that Fi'(z) < 0 when 0 < z < ZOl· In order to verify the condition
:y[Ft-l(y)- F2-t(y)] < 0 in the corresponding region, we only have to prove that the two curves z = [Fi- 1 (y)]~ (i = 1, 2) have at most one point of intersection for y > 0. The point of intersection can be determined from the system of equations G'(xt) G'(x2) F'(xt) = F'(x2)' F(xt) = F(x2),
§15. EQUATIONS OF CLASS III
345
which is equivalent to
A ( X1 + ~) ( X2 + ~)
= ~ + 0,
(15.20)
where
A= b(-ba d-1
+
B=ac(d~1 +c~b),
c-b) d
+
b(b+ac) d-1 '
C=c(d~1 +c~b)·
If we consider condition 2) and the necessary conditions for existence of a limit cycle (b + ac < 0, b- c < 0), we see that
B 2 + AC = c(b + ac)[ac(d- 1) + b(a + d)][c(d- 1) + bJ, d- 2 (d- 1)- 2 < 0, i.e., the first equation of (15.20) determines the increasing function x2 = cp(xl). Next, starting from the second equation of (15.20), we compute dx2/dx1 at z = z•, and get
dx2 dx1
F'(xl)
Fi(z*)dzl/dx
= F'(x2) = F~(z•)dz2/dx'
where
.
dzl/dx = g(xl) > 0, and Fi(z•) < 0 by condition 2), while FHz*) < 0 by condition 3a). Hence dx2/dx1 < 0, and from F(xt) = F(x2) we determine the decreasing function x2 = .,P(xt). Thus it is clear that (15.20) has a unique solution. The proof for case 3) is similar. III. Finally, there remain cases 5) and 6), in which ad < 0 but d > 0. In order to prove the uniqueness of a limit cycle of system (15.81) for this case, G. S. Rychkov [292] improved Lemma 1 and obtained the criterion for uniqueness:
zo, z• and y• (0 < zo < < zo1. /3 < y• < 0), such that the following conditions hold: (5 ) 1) F~(z) < 0, 0 < z < zo2, and F1(0) = F2(0) = 0. 2) (zo- z)Fi(z) > 0 when z '# zo,O < z < zo1i and F1(z*) = F2(z*). 3) 2F~'(z)z + FHz) < 0, 0 < z < Zo2· 4) (y•- y)d(F1(y)- F2(y))jdy > 0 when y # y•, /3 < y < 0, LEMMA 2. Suppose that there exist unique points
z•
( 5 )The
meaning of {3, zoi. and Fi(Y) is the same ss before.
THEORY OF LIMIT CYCLES
346
Then the equation dy g(x) dx = F(x)-y
cannot have more than one limit cycle; if it exists, it must be unique. There is a great difference between the proof of this lemma and the proof of Lemma 1 or Theorem 6.11. In order to verify that system (15.18) in cases 5) and 6) satisfies condition 3) of the lemma, the calculations are very complicated; hence the details are not given here. Interested readers can refer to Rychkov's paper [292]. The above analysis only proves the uniqueness of a limit cycle of system (15.5) in the vicinity of the origin. Note that if (0, 0) and another singular point (0, 1/n) are on the same side of 1 +by = 0, then (0, 1/n) is a saddle point; but if 1 +by = 0 separates (0, 1/n) from (0, 0), then (0, 1/n) is also a singular point of index +1. The uniqueness of a limit cycle in its vicinity can be solved by the method of translation of the origin, because at this time the form of system (15.5) does not change. Finally, take B(x, y) = (1 + by)- 21 /b-l; then we can compute
a
ax (BP)
a + ay (BQ) =(my+ o)B(x, y).
But y = -8/m is a line parallel to 1 +by= 0 which does not coincide with 1+by = 0; therefore it must be on one side of 1+by = 0, and any limit cycle of (15.5) cannot intersect the line 1 +by= 0. Hence limit cycles cannot coexist in the vicinities of (0,0) and (0, 1/n). Theorem 15.4 is completely proved. REMARK. The reader has already seen (Theorem 12.6 in §12) that in order to prove the uniqueness of a limit cycle of class (I), we have to consider four different cases, using different transformations of variables to change them into equations of Lienard type, and then apply the uniqueness theorem of [111] to solve the problem. For Theorem 15.4 of this section, the situation is similar. However, the form of equations, no matter whether it belongs to class (I) or (Ill)a=O (i.e. system (15.5)) is much simpler than the form of equations after transformation; moreover, the difference between these two systems is only one factor (1 + by) on the right side of the second equation, and the conclusions for the nonexistence and uniqueness of a limit cycle are exactly the same. Hence we have reasons to mention the following questions, which deserve the reader's consideration for further deep investigation. PROBLEM I. Can we prove the uniqueness of a limit cycle of class (I) directly without transformation of variables? PROBLEM II. Can we deduce from the uniqueness of a limit cycle of class (I) the uniqueness of a limit cycle of class (Ill)a=O?
347
§15. EQUATIONS OF CLASS III
Now we present two important corollaries of Theorem 15.4: COROLLARY 15 .1. If a quadratic system has an integral line and a fine focus, then it does not have a limit cycle.
This holds because we can change the system into (15.6) by transformation of coordinates. COROLLARY 15. 2. If a quadratic differential system has one integral line, then it has at most one limit cycle.
This holds because we can change the system into (15.5) by transformation of coordinates. The above two corollaries were mentioned at the end of §11. For the uniqueness of a limit cycle of equations of class m, we also have the following results. Chen Lan-sun [226] proved that the system
dxjdt = -y + 6x + lx 2 ,
dyjdt = x(1 +ax+ by)
has one and only one limit cycle when 6a(b + 21) > 0 and 6 varies in some interval (0, 6*) or (6*, 0). Zou Ying [227] proved the uniqueness of a limit cycle of the system
dxjdt
= -y + 6x + bx 2 + Mxy- by2 ,
dyjdt
= x(1 +ax+ by).
Liu Jun [228] extended these results, but assumed the first equation does not contain the y 2 term. Ren Yong-tai [229] studied the existence, uniqueness and stability of a limit cycle of the system
dxfdt = ly + mx -ly2
+ nxy- mx2 + a(x + bny- x2 ),
dyjdt = x(l +by- x). For the problem of centralized distribution, except the equations of class
(ill) discussed in §14, we still have the method of proof first used in [25]: when [2am- b(b + 2l)][m2 - 2n(b + 21)] ~ 0
(15.21)
limit cycles of system {15.1) must be centrally distributed (i.e. it is not possible that there exist limit cycles in the vicinities of two singular points of index +1). Yang Zong-pei [27] extended the method of [25] and also improved the result, showing that when (15.21) does not necessarily hold, the limit cycles of system (15.1) must be centrally distributed provided that [2am- b(b + 2l)][m2
-
2n(b + 2l)]
+ m 2 b2
~ 0.
(15.22)
348
THEORY OF LIMIT CYCLES
Li Xiao-gui [230] applied the transformation x = z, fj = ax/b+y to system (15.1) to obtain the system of equations (we still denote x and ii by x andy)
= -y + 6'x + l'x::~ + m'xy + ny2 = P(x, y), dyjdt = (a/b)P(x, y) + x(1 +by) = Q(z, y),
dxjdt
(15.23)
where
d'
= d + ~b'
l'
= l- ~b (m- ~n) b
(15.24)
I
Then comparing (15.23) with the system
~~ = -y + 6' z + l' x::~ + m' xy + ny2 ,
dy dt = x(1 +by)
(i5.25)
he obtained many results about the existence and centralized distribution of limit cycles. For example, he proved
15.5. If system (15.25) has a limit cycle or a separatrix cycle r surrounding a singular point 0, and if, when a6/b > 0, system (15.23) has a limit cycle in and only in the interior of r (this cycle must be stable (unstable) if a/b > 0 (< 0)), and (15.25) does not have a cycle in the vicinity of 0, then when (ajb)(6 +a/b) < 0 (15.23) does not have a cycle in the vicinity of 0. If (15.25) has a limit cycle or a separatrix cycle f' in the vicinity of another singular point M of index +1, then when (a/b)(6 + m/n) < 0 system (15.23) has a limit cycle in and only in the interior of r' (this cycle must be stable (unstable) ifa/b < 0 (> 0)), and if(15.25) does not have a cycle in the vicinity of M, then when THEOREM
i (6+ ~ -i) >0
(15.23) does not have a cycle in the vicinity of M. In addition, in the same paper Li also obtained some results on the centralized distribution of limit cycles. There is a very good result on the centralized distribution of limit cycles of class III obtained by Chen Lan-sun and Wang Ming-shu [19]. They proved THEOREM 15.6. If, in addition to two singular points of focal type of index +1, (15.1) has a third finite singular point, then its limit cycles must be centrally distributed; conversely, if (15.1) has only two finite singular points of focal type, then its limit cycles cannot be centrally distributed.
The proof is by the method of the Dulac function, whose details are in §16. In addition, in [231] Liu Nan-gen discussed the problem of centralized distribution of equations of class II.
§15. EQUATIONS OF CLASS III
349
The range of variation of 6 (the coefficient of x on the right side of the first equation of (15.1)) to guarantee existence of a limit cycle is an important problem of theoretical significance and practical value. We have already given a simple example in §11 (181]. Also in §12 we pointed out that for equations of class I, if the range ofvariationoU is (0,6*) or (6*,0), then 6* = f(l,tn,n), but this function and the equations of its separatrix cycles are unknown to us. For system (15.5), Yu Shu-xiang [179] obtained under given conditions four results on the precise interval of 6 to guarantee existence of a limit cycle, as follows: THEOREM 15.7. 1) /fn+l > 0, m > 0, l < 0, 0 < n S 1 andmll4n(l + 1) > 0 in system (15.5), then there exiats a limit cycle surrounding the origin if and only if
0 > 6 > 2:(::1) (-m + .../m2- 4n(l + 1)). 2) Ifl + n < 0, m > 0, l > -1, 0 < n < 1 and mil + 4l(1 - n) > 0 in (15.5), then there exists a limit cycle surrounding the origin if and only if
0 < 6 < 2:(::1) ( -m + .../m2 - 4n(l + 1)). 3) lfn = 0, m > 0, -1:::; l < 0 and mil+ l > 0 in (15.5) then there exists a limit cycle if and only if 0 < 6 < -1/m. 4) lfn = 0, m > 0, -1:::; l < 0 and mil+ l:::; 0 in (15.5), then there exists a limit cycle if and only if 0 < 6 < -m + 2A. Similar results were also obtained in [232] and (233]. For example, I. G. Rozet (233] proved THEOREM 15.8. When l = 0, b > -n, 6 = mjb, and mil+ 4nb < 0, system (15.5) has two separatrix cycles formed by the line 1 +by = 0 and the semi-equator, which surround two finite singular points of index +1, respectively. Similarly to [119], Kukles and Rozet (232]showed that the separatrix cycle corresponding to the endpoint of the interval of existence of 6 is also formed by a section of the arc on the equator and two half-ray segments. Wang Huifeng (234] studied the range of variation of 6 for the existence of a limit cycle and obtained the following succinct result: THEOREM 15.9. Suppose that b = -1 in (15.5}, and suppose that m > 0, 2l + 1 > 0, and n = 1 + m 2 /(2(21 + 1)). Then the system has a limit cycle if and only if (1 - n)/m < 6 < 0.
THEORY OF LIMIT CYCLES
350
Here when 6 = (1-n)/m, the separatrix cycle obtained is no longer formed by a line segment or a section of arc on the equator, but is an elliptic trajectory through the saddle point (1/n, 0). Moreover, Suo Guang-jian and Du Xing-fu [235] also studied the conditions for system (15.5) to take a curve formed by one branch of a hyperbola and an arc of the equator as its singular closed trajectory (they also obtained an admissible interval of variation for some varying parameter to guarantee existence of a limit cycle) and the conditions which take a curve formed by a parabola and one line segment to be its singular closed trajectory. Suo [236] studied similar problems for a quadratic system with symmetric center dy dt =
b
+
""' W
bijX
i
.
y3.
(15.26)
i+j=2
For example, he obtained THEOREM 15.10. If in the system of equations
dx 2 dt = 1- x +xy,
~~ = l-lx2 + 2xy + ny2 + a(1- x2 + xy)
(15.27)
we have l - 2 > n ~ 0, ln - l + 2 < 0, and ln - l + n + 3 > 0, then (15.27) has two limit cycles which do not contain each other if and only if l-ln- 2 . n+ 1 In addition, there are several papers which study the qualitative properties of a quadratic system with a third-order fine focus (0, 0):
O
~~ = -y + lx 2 + 5axy + ny2 ,
~~ = x + ax 2 + (3l + 5n)xy.
(15.28)
Cai Sui-lin [237] and Wang Ming-shu and Lin Ying-ju [238] proved independently that when n = 0, (15.28) does not have a limit cycle in the whole plane.( 6 ) Chin Yuan-shun et al. [182] further proved THEOREM 15 .11. If (15.28) has only one infinite elementary singular
point, then this system has only two finite singular points, one of which must be a coarse focus. There are an odd number of limit cycles surrounding the coarse focus, and an even number surrounding the origin.
In the following we present some nice results obtained by Soviet mathematicians on the study of quadratic systems according to their method of classification. ( 6 )Recently Li Cheng-zhi proved that when n =f. 0, (15.28) does not have a cycle in the whole plane (the paper will be published in Chinese Ann. Math.)
351
§15. EQUATIONS OF CLASS III
Consider the equation ' - Q2(x, y) - Lo
=
= y -TJX,
Yl
(15.29)
= y; then the coefficient
(~, ~) -17P2 (~. ~)] ele=o
~[(a~ob~oTJ) + (a~l- b~117)17 + (a~2- b~217)TJ 2 ]. 1]
We take 17 to be the zero of the quadratic expression of 17 in [ ] on the right side of the formula, and we still denote x1 and Yl by x andy. Thus we obtain
dy Q2(X, y) dx = boo+ b10x + ~ox 2 + buxy + bo1Y ·
(15.30)
1. If bu =/: 0, then we can further use a linear transformation through x to make bo1 ;:= 0, and assuming y' = b10 +~ox+ buy and x' = x, we get (we still denote the numerator by Q 2, and x' andy' by x andy) dy dx
Q2(x,y)
= boo+xy·
(15.31)
Here if boo = 0, then the equation and (III)a=O are essentially the same; if boo=/: 0, then (15.31) can be changed to
dy dx
a~ + a10x + a01y + a2ox 2 + auxy + ao2Y 2
=------~--~~--------~~--~
1+xy
If bu = 0, then (15.30) can have a limit cycle only when bo1 case (15.30), after a linear transformation of x, becomes
dy Q2(x,y) dx = ~ox 2 +y"
(15.32)
=/: 0. In this (15.33)
Here if ~o = 0, then the equation is of class I, and there is no need for further consideration. If ~o =/: 0, then (15.33) becomes
dy Q2(x,y) dx= y+x2 •
{15.34)
(15.32) and (15.34) are the two main classes in the Soviet method of classification mentioned in §11. Now we derive some useful results for (15.32) In (15.32) we make the transformation y 1 Y=-+-2
x
x
'
(15.35)
THEORY OF LIMIT CYCLES
352
Then we get (15.36) where P4({) = -a2o- a10{ +(au- aoo)e P2({) = (1 + 2ao2){2 - aot{- au.
+ ao1e- ao2e\
(15.37)
The system of equations corresponding to (15.36) is d{ = cy: dt ...
{15.38)
From (15.32) we can see that x = 0 is a line without contact of the system; hence it cannot intersect any limit cycle. Thus, when it is changed to (15.36), the number of limit cycles and their positions do not change. Since { = 0 is an integral line of (15.38), it cannot intersect any limit cycle; thus we can make a transformation {15.39) and (15.38) becomes Y ~~
= (P4(x)l{l 2ao 2- 3 + P2({)1elao 2- 2 y)sgn {,
{15.40)
or the corresponding system of equations d{/dt
where
= y,
dyfdt = -g({) -f({)y,
= -P4({)1{12ao2-3sgn {, /({) = -P2({)1elao 2sgn {.
g({)
(15.41)
{15.42)
2-
The divergence of system {15.41) is - f({). From this we can see that if P 2({) keeps a constant sign in the region { > 0 or { < 0, then system (15.41), and hence (15.32), does not have a limit cycle in that half-plane. In fact, this conclusion is also valid in the strip {15.43) For this we first prove THEOREM 15.12. If P4({) = Q2({)P2({) in (15.37), i.e. P2({) is a quadratic factor of P4({), then equation (15.32) does not have a limit cycle. PROOF. We first assume P2({) has two nonzero roots, i.e. au "I 0 and 1 + 2ao2 -I 0. Then Q 2({) = -a02 (1 + 2ao2)- 1 + ao1(1 + ao2)(1 + 2ao2)- 2{ + a2oail· (15.44)
e
§15. EQUATIONS OF CLASS III
353
Now we transform (15.41) into the Lienard plane, and obtain drJ/dt
= -g(~),
(15.45)
where F(~) =If.: J(€) de, and (eo, 0) is a singular point offocal type of (15.45) or (15.41). In order to prove (15.45) does not have a limit cycle in the vicinity ~f ( ~o, 0), we may as well confine our discussion to the strip < here eo_(€o) is greater than (less than) ~0 and is a root of g(e) = 0 closest to eo; if (eo) does not exist, we just take -oo (+oo) instead. Using Filippov's transformation z =If.: g(€) de, (15.45) can be transformed to two equations
eo e<eo;
eo
1 dTJ dz - F1(z)- TJ
(corresponding to
~0 ~ ~ < eo),
{15.45)1
1 drJ dz - F2(z) - TJ
(corresponding to
eo < ~ ~ ~o).
(15.45)2
= F1(z)
In the following we shall prove that the curves TJ has one common point (0, 0) in the region
O$z:$olln
[t
and TJ
= F2(z)
g(€)d€,t g(€)d€].
From this, by Theorem 5.4 of §5, we know at once that (15.45) does not have a limit cycle in the vicinity of (~o, 0); to see this we only have to prove that the quantity (15.46) (where Fi(TJ) is the inverse function of Fi(z)) does not change sign in the corresponding region, or( 7 ) that the two simultaneous equations
1
(1
Q2(6)~fo2-1 = Q2(~2)~~o2-1, P2(~)eao2-2 de=
f.o
1(2
(15.47)
P2(~)eao2-2 d~
(15.48)
f.o
eo·
to
have a unique system of solutions 6 = 6 = ~0 when > 6 ~ ~0 ~ 6 > It is easy to see that when ao2 ::F -1, 0, 1, condition (15.48) can be changed to 1 + 2a02 (€002+1 _ ~Go2+1) _ ao1 ( ~002 _ ~002) ao2 + 1 1 2 ao2 1 2 (15.49) _ au (CGD2-1 _ ~ao2-1) = 0 ao2
( 7)I.e., g(e<J)/ /({<J)
el = e'l =eo.
-1 "' 1
2
'
= g(6)//(6) and F(el) = F(e2) has only one system of solutions:
THEORY OF LIMIT CYCLES
354
and conditions (15.47) can be written as
- ao2(1 + 2ao2)(e~o 2 +1
e;o2+1) + ao1(1 + ao2)(e~oa - e;02 )
-
+ a2o (1 + 2ao2)2(e~o2-1- e;o2-1) = 0.
(15.50)
au Substituting the formula for eio 2+1- e;o 2+1 of (15.49) into (15.50), we get (eio2-1 - e;o2-1 )6.
= 0,
(15.51)
where
6. = -ao2au (1 + ao2) + a2o (1 + 2a 02 )2. ao2 -1 au Since except when 6 = 6 = eo we always have 6 < et. equation (15.51) has a unique solution 6 = 6 = when 6. ::/:- 0, i.e., in the vicinity of (eo, 0) there is no limit cycle. H 6. = 0, then (eo, 0) is a center and hence does not have a limit cycle. For the case ao2 = -1, 0 or 1, we can make a similar transformation and get an equation equivalent to (15.51) as (a2o/au)[e1 2 - G 2] = O,
eo
(a2o/au)[e1 1 - e2 1] = 0 or - 2auln{6/6) hence we can get the same conclusion as before. For the case au = 0, Q2 (e) has the formula
= 0;
Q2(e) = -ao2(1 + 2ao2)-le + a01(1 + ao2)(1 + 2ao2)- 2e + alO j aOl
the remaining calculations are similar to the case au ::/:- 0. Moreover, if P2(e) has a pair of zero roots, then from the fixed sign of the divergence we know at once that it does not have a limit cycle. The theorem is completely proved. From this theorem we can immediately deduce an important property of a quadratic system mentioned at the end of §11: THEOREM 15.13. If the curve of system (15.38) whose divergence i3 zero passes through two elementary singular points,( 8 ) then it does not have a limit cycle. PROOF. By our previous discussion, the elementary singular points should have the forms of singular points (6,0) and (6,0). Also from (15.38), we can see that the characteristic equation of these points is
>.2 ( 8 )In
-
P2(ei)>.- ei1'4(ei) =
o.
this case these two singular points are either fine foci or fine saddle points, or one is a fine focus and the other is a fine saddle point; and their common characterizing feature is: the linear part of the characteristic root has the same absolute value.
§15. EQUATIONS OF CLASS III
355
By the hypothesis of the theorem we must have P2 (6) = P2 (6) = 0. But on the other hand, 6 and 6 must be the roots of P4(€) = 0; hence P2(€) is a factor of P 4 (€). Application of Theorem 15.12 will complete the proof of this theorem. Now we return to (15.43). Suppose P2(€) has a constant sign in this strip region. Then it is clear that system (15.41) does not have a limit cycle in this strip. Hence if P2(€t) = P2(6) = 0, can the system have a limit cycle surrounding (6,0) or (€2,0)? From Theorem 15.13 we know immediately that the answer is negative. Finally, we note that when Bol = a10 = 0, the phase-portrait of system (15.38) has symmetric properties with respect to = 0; in this case in order to study the existence and number of limit cycles, it is sufficient to confine ourselves to the half-plane € ~ 0. Applying the transformation €2 = X, (15.36) can then be changed to a simpler form: 2XY(dY/dX) = - a2o +(au- aoo)X- ao2X2 (15.52) + [(1 + 2ao2)X- au]Y + (1- ao2)Y 2.
e
Hence in this case the system has changed into (15.5) of class (TII)a=O.· Thus, in the half-plane > 0 (or < 0), there exists at most one limit cycle, and if the system has a fine focus, it does not have a limit cycle.(9 ) Similar results were obtained later independently in [239]. REMARK. The results of [237] and [238] mentioned before for system (15.28) are proved by the method of proof of Theorem 15.12. In addition, in §14 we mentioned that the nonexistence of a limit cycle of system (14.5) under condition (14.17) is also proved by the same method [216]. Theorem 15.13 is only proved for equation (15.32). But Cherkas and Zhilevich in [222] carried out a similar study for (15.34). For this, they first used a transformation of variables
e
(15.53) to change (15.34) into
dfi dx
=
- f(x)jj- g(x) y
(15.54)
where
f(x)
= -P2(x)e-ao2x = -(ao1 + (2 + au)x- 2ao2x2)e-ao 2 x,
g(x)
= -P4(x)e-ao2x = -(aoo +
a10X
+ (a20- aol)x 2 - aux 3 + Bo2X4je- 2a 02 x.
(15.55)
--:----(9)Note that from (15.32) we can see that the vector field determined by this system indeed has a center of symmetry (0, 0).
THEORY OF LIMIT CYCLES
356
Then they used a similar method to show that Theorem 15.13 also holds for (15.54), and hence for (15.34). Finally we note that (15.31}, when boo 0, belongs to the class (Ill)o=o, and so Theorem 15.4 also holds for (15.31). Hence we obtain
=
THEOREM 15 .14. If the quadratic system (15.29) has two fine foci or two fine saddle points or one fine focus and one fine saddle point, then it does not
have a limit cycle. REMARK. According to the above discussion, all quadratic systems can be transformed into systems of Lienard type by an appropriate transformation of variables. But this method of transformation is not unique. Liu Jun [228] gave a transformation of variables suitable for systems more general than quadratic systems. THEOREM 15 .15. Equations of the form
dxjdt = fo(x) -!l(x)y, dyjdt
= go(x) + g1(x)y + g2(x)y2
(!l(x) #: 0)
(15.56)
can be transformed into equations of Lienard type by a change of variables. PROOF. First let x
dxjdt
= e,
= x and e= fo(x) -!l(x)y. de/dt
where
t/Jo(x)
Then
= -1/lo(x) -1/ll(x)e -11'2(x)e,
= !l(x)go(x) + fo(x)gl(x) + ~:~=~f~(x), 1/12(x) = g2(x) _ fi(x) !l(x) !l(x)'
t/J1(x) Again let
= ~:~:~fo(x) -fMx)- g1(x)- 2~:~=~fo(x).
e= uexp(- J; t/J2(z) dz). Then (15.57) becomes ~~ = uexp (- foz 11'2(z) dz), ~~ exp (- foz 11'2(z) dz) - u 2t/J2 exp ( -2 foz 11'2(z) dz) = -t/Jo(x) -1/11 (x )u exp (- foz tP2(z) dz) - 1P2(x)u2exp ( -2 foz t/J2(z) dz) .
(15.57)
357
§15. EQUATIONS OF CLASS III
The above formulas are the same as
~: = uexp (-los ~2 dz) ,
~; = -~ exp (los~ dz) - 1P1 u.
Next let
u loz 1P1 exp (lo' ~ dz) ds,
V= + Then we get
~: =V- los ~~(z)exp(Lz ~(s)ds)
dz, {15.58)
~ = -~(z)exp (2 los ~(z)dz),
which are the required forms of the equations. The theorem is completely proved. It is clear that (ill)n=O has already possessed the form of (15.56). In order to apply thi15 transformation to (ill)n~o, we first assume n 1, and apply the transformation z = i/(1 + y), y = fi/(1 + fi),
=
to the system
dz/dt = -y(1- y) + 6z + lz 2 + mzy,
dy/dt = z(1 + az +by)
which maps the line y = 1 containing the singular point (0, 1) to infinity (correspondingly, the equator becomes fi = -1), and thus we obtain a system of equations of the form (15.56). Finally, for singular closed tr9Jectories of quadratic systems, in addition to [14], we also mention the Soviet papers [192], [240], and [241]. In [192] the classification of singular closed tr9Jectories was tabulated, and examples were given to illustrate the possibility of being realized. In [240], L. H. Cherkas concentrated on the existence and stability of singular cycles of equation (15.31) when boo = 0, and in [241] he proved that when the coefficients of (15.31) satisfy the conditions
boo= 0, ag 1 - 4aooao2 < 0, (-1 + ao2)/ao2 < 0,
a~ 1 - 4a2o(ao2 -1)
< 0,
aooa2o < 0,
the equation has two critical singular cycles formed by the upper (lower) halfequator and the z-axis, but whose stability cannot be determined from the characteristic values of two saddle points in the singular cycle, and he obtained the sufficient conditions to distinguish its stability. But Zhou Kong-rong (242]
THEORY OF LIMIT CYCLES
358
pointed out mistakes in the above calculations, and corrected them, and also gave a simpler sufficient condition to distinguish its stability. Exercises
1. Prove Theorem 15.3. 2. Prove formula (15.16) and the formula for P2(x) after it. 3. Verify Figure 15.1, and construct under condition (15.11) the global phase-portrait of equation (15.15). 4. Use the method of computing the integral of the divergence around a closed trajectory once to prove that the equation
dyjdx = ( -x + ax 2 + bxy + cy 2 )Jy does not have a limit cycle provided one of the following three groups of conditions holds: (i) a < 0, c = 0; (ii) a = 0; (iii) a > 0, c = 0. 5. Prove (15.36) and (15.37). 6. Prove (15.40) and (15.44). 7. When ao2 = -1, 0 or 1, prove Theorem 15.12. 8. Prove (15.54) and (15.55). 9. Give details on the derivation of the formulas in the proof of Theorem 15.15. 10. Prove that the system of equations dx/dt = -y +ox+ lx 2 + mxy, dyjdt can be transformed into the Lienard equation dx dr
= -vdv dr
= x(1 +ax+ by)
[b b ox+ lx 2 ] m x + m 2 1n( 1 - mx)- 1- mx '
-=
x + ax 2 1-mx
bx(ox + lx (1-mx)2 '
2) +~'-----~
by the transformation
v
= y- !:x- _!_2 ln(1 m
m
mx) '
dt 1 = -:---dr 1-mx·
11. For the system of equations
dyjdt = x(1 +ax- y), prove that when o ~ m > 0, a(2l + 1) ~ 0 or o~ m < 0, a(2l + 1) ~ 0 it does not have a limit cycle, and if 0 ~ a - o ~ m and l + ma ~ 0, then it has one and only one limit cycle. 12. Study under the transformation X= 1/x, Y = (1 + xy)/x2 the corresponding relation between the regions in the projective plane (x, y, z) and (X, Y, Z), and point out in what regions they do not have a one-to-one correspondence. dxjdt = -y +ox+ lx 2 + mxy,
§16. The Method of Dulac Functions in the Qualitative Study of Quadratic Systems
In previous sections, in studying the nonexistence of a limit cycle for equations of classes I, II, and ill, we have already seen that the method of Dulac functions seems to have special advantages. For example in the proof of nonexistence of a limit cycle of (1) 6 = 0 (Theorem 12.4), we simply use two different Dulac functions (12.22) and (12.23), and then compute their divergences and immediately solve the problem. But [201] and [202] use three or four pages to prove the satne result. Similarly, for nonexistence of cycles of class (ill)a=6=0 in the previous section we presented the proof of [203] and also :used several pages. In the following we shall see the proof of Chen Lan-sun [243] and Ju Nai-dan: THEOREM 16 .1. The system of equations
~: =-y+lx2 +mxy+ny2, ~~ =x(1+by) center when m(l + n) = 0, and does not have a limit
has a m(l + n)
(16.1) cycle when
-1 0.
PROOF. 1) The case m 2 + 4n(n +b) ~ 0. Here n(n + b)02 - mO -1 = 0 has two real roots. Take any root of this equation, and let
Bl(x, y)
= (1 + by)(mn9-b-2l)fb(n0y- X- O)m9.
(16.2)
We compute 8(B1P)
ax
+ 8(B1Q) 8y
= -m(l + n)Ox2(1
+ by)(mnB-b-21)/b(nOy- X- o)m9-1;
(16.3)
also, we note that nOy- x- 0 = 0 is an integral line passing to the saddle point or nodal point (0, 1/n) and touching tangentially to one exceptional direction at this point, or all the trajectories intersecting it all cross through it in the 359
THEORY OF LIMIT CYCLES
360
same direction. Hence a limit cycle, if it exists, can only lie on one side of this line, and so from (16.3) we obtain the proof of this theorem. 2) The case m 2 + 4n(n + b) < 0. Here (16.2) cannot be used, sj.nce n(n + b)6 2 - m6- 1 = 0 has only a pair of conjugate complex roots. Now we follow [243) and take B 2(x,y)=(1+by)- 21 /b- 1 exp{ 2:tan- 1 [ where u = J-m 2
-
~in_+:;)- ;]}•
(16.4)
4n(n +b). Then we compute
:x (B2P) + :y (B2Q) = m(l +
n)~2(1 ;- ny) (1 +
x exp (2m tan_ 1 u
[-
by)-21/b-1
2n(n + b)x _ m]) u(1- ny) u '
(16.5)
where L = (1- ny) 2 - n(n + b)x 2 - mx( 1 - ny) is a positive definite quadratic form, and now 1 +by = 0 separates (0, 0) from (0, 1/n); hence any limit cycle in the vicinity of the origin cannot intersect 1- ny = 0. Thus from (16.5) we obtain the proof of this theorem. REMARK 1. Let b- 0 in (16.2). Then the limit function B(x,y) is the Dulac function in (12.23). REMARK 2. L can be factored as [(ny- 1)61 - x][(ny- 1)62 - x), where 61 and 62 are a pair of conjugate complex roots of n(n + b)6 2 - m6- 1 = 0. REMARK 3. H we use the well-known formula for the inverse trigonometric function in (16.4): tan- 1 z =
~ ln ~ = ln G~ :) 112i,
and rewrite -iu = Jm 2 + 4n(n +b)=
VIS., then we get
B' (x y) = (-1)mf.../X(1 + by)-21/b-1 2
'
m- ..{K
(16.6)
x [ x+2n(n+b)(1-ny)
[x + 2n(n m + VfS. (1- ny)]m/.../K +b) ] -m/.../K.
(16.7) If we drop the factor (-1)m/.../K. in front, then we can prove that this B~(x,y) can also be used in the case m 2 + 4n(n +b) ~ 0, replacing Bt(z,y) as the Dulac function there. We look at another example. In [201), the proof of nonexistence of cycles starts from the study of the properties of the system
dyfdt = z,
dxfdt = -y + lx 2 + mxy -ly2 + f(y).
(16.8)
§16.
THE METHOD OF DULAC FUNCTIONS
361
The author (G. S. Makar-Limanov) proved that if J(y) does not change sign, it is not identically zero in any neighborhood of y = 0, and the uniqueness of the initial value problem of (16.8) can be guaranteed, then (16.8) does not have a limit cycle. After choosing a suitable J(y), he also proved that (I).s=o does not have a limit cycle. Now we use the method of Dulac functions to prove the following theorem, which contains more information than the nonexistence of cycles for (I)o=O· THEOREM 16. 2. Suppose in the system dy (16.9) - =x=Q, dt where(} is a constant, J(y), in addition to satisfying the above conditions, also satisfies (l + n)6J(y) 2: 0. Then (16.9) does not have a limit cycle.
: = -y + lx 2 + mxy + ny2 + 6J(y) = P,
PROOF. Take a Dulac function (still the function in (12.23)), (16.10)
where a = (m + v'4n2 We can compute
a
a positive root of n 2 a2- ma- 1
= 0.
a
+ oy (B3Q) am[(l + n)x 2 + 6f(y)](x- any+ a)am-le(amn- 21 )11.
ox (B3P)
=
+ m2)/2n 2 is
(16.11)
Again we note that under the conditions of the theorem, the line L: x-any+ a = 0 is a line without contact, since
~~ IL=O = (l + n)x
2
+ 6/(y).
From this we immediately obtain the proof of this theorem. In particular, if J(y) 0, then this theorem reduces to Theorem 12.4; if l + n = 0, it reduces to the theorem in [201]. After reading about these Dulac functions, more complicated and more useful than those in the preceding sections, the reader will definitely ask some questions; How are these Dulac functions obtained? Does there exist a general method for constructing them? Indeed, at the beginning of the sixties, when we wanted to prove nonexistence of a limit cycle for some simple quadratic system, we depended completely on luck. At that time, the functions used were mainly exponential functions whose exponents were polynomials of lower degrees, or rational functions such as the function (12.22) of §12, (1-mx )- 1 in the proof of Theorem 14.1, and the function (15.3) of §15, etc. The function B3(x, y) was also obtained by luck. But once we obtain B3(x, y), we have obtained the inspiration: x - any + a = 0 is a line tangent to the separatrix
=
362
THEORY OF LIMIT CYCLES
through a saddle point at the saddle point, and is also a line without contact of the system. Moreover, when l + n = 0, on the one hand, x - o.ny + o. = 0 is an integral line; and on the other hand, B3 (x, y) is an integrating factor of (I).s=O· Hence, we realize that in Dulac functions we should extend fractional functions to the power functions as (ax+ by+ c)k where ax+ by+ c = 0 is either a line without contact or an integral line or a line passing a singular point in an exceptional direction-of course, this singular point can be a nodal point since a nodal point also has an exceptional direction. At the same time, to find a Dulac function, we can first find an integrating factor with center. Moreover, the passage from B 1 (x,y) to B2(x,y) shows that if a Dulac function can be suitable in even one case, then from the relation between the elementary functions of complex variables we can obtain another Dulac function suitable to other cases. (I) Finally we note the divergence in the above examples is either of constant sign or identically zero in the region where a limit cycle may exist. However, if the divergence has a linear function Ax + By + C of variable sign in addition to the part of constant sign, then the line Ax + By + C = 0 can have at most two points of contact with the trajectory of the quadratic system. H we have known this line passes through a singular point of nonfocal type of the quadratic system, then it has at most one point of contact on it; from this we can prove that limit cycles of this sytem, if they exist, must be centrally distributed. The above discussion is the main line of thought in the proof of Theorem 15.6 in §15. The Dulac function to be used will be given later. In the following we attempt to use the method of an integrating factor in the case of finding a center to obtain the Dulac function B 1 (x, y) and B2(x, y). Consider (III)a=cS=O:
dxfdt = -y + lx2 + mxy + ny2 ,
dyfdt
= x(1 +by),
(16.12)
which, when l + n = 0, has an integrating factor
JLt(x,y) =
v1- 1 v2- 1 ,
(16.13)
where Vt = 1- 2ny- mx + n 2 y 2 + mnxy- n(n + b)x2 = L (see (16.5)) and v2 = 1 +by. In order to make the formula for the divergence as simple as possible when l + n =I 0, we take the Dulac function of (16.12) as 71
= _1 _
2(l + n). b
(16.14)
( 1 )It is worthwhile to study the meaning of this fact in the qualitative theory in the complex region.
§16. THE METHOD OF DULAC FUNCTIONS
363
Thus we compute the divergence
a
a
ax (B4P) + By (B4Q)
= m(l + n)x2 [1- ny + 2n(:+ b) x] V1- 2 V2",
{16.15)
where V1
= L = -n(n + b)(ny- 191 -
x)(ny- 102 - x),
and 81 and 02 are the two roots ofn(n+b)92 -m0-1 then 81 and 02 are real, and the two lines
01(ny- 1)- x
=0
= 0.
Hm 2 +4(n+b) ~ 0,
and 02 (ny- 1) - x
=0
{16.16)
all pass through the singular point (0, 1/n), which is either a saddle point or a nodal point. The points of intersection of {16.16) with the x-axis are (-81,0) and (-82,0) respectively. Moreover 1- ny + 2n(n + b)x/m = 0 also passes through (0, 1/n), and its point of intersection with the x-axis ( -m/2n(n +b), 0) then lies between ( -Ot. 0) and (-02 , 0). From this we can see that if (/t8 2 < 0, i.e. n(n +b) > 0 (here (0, 1/n) is a saddle point), then 1- ny + 2n(n + b)x/m = 0 may intersect a limit cycle and does not lead to a contradiction. Hence (16.15) cannot be used to prove that (16.12) does not have a limit cycle when m( l + n) # 0. But if 8182 > 0, i.e. n(n + b) < 0 (here (0, 1/n) is a nodal point (or a focus)), then the two lines (16.16) intersect the x-a.xis on the same side of the origin (or they are both imaginary lines). When (0, 1/n) is a nodal point, the point of intersection of
1 -ny+ 2n(n +b) x= 0 m
with the x-a.xis lies between (Ot.O) and (82,0), and, since (16.16) is a line without contact, we know it is not possible to have a limit cycle. When (0, 1/n) is a focus but (16.16) and the x-axis do not have a real point of intersection, (16.15) still cannot be used to prove that (16.12) does not have a limit cycle when m(l + n) '::/:- 0. In the following we shall attempt to use
= ln [v,1/2n(n+b)(1 + by)1fb(n+b)] + 1 =C,
m
2u1n(n +b)
ln L+ L_
(16.17)
364
where
THEORY OF LIMIT CYCLES
= vrn2 + 4n(n +b) and L_ = -2n(n + b)x- (m- u1)(1- ny),
0'1
L+ = -2n(n + b)x- (m + ut)(1- ny),
(16.18)
v;
L+L1 =- 4n(n+b)"
Formula(16.17) isonlysuitableform2+4n(n+b) ~ 0. Ifm 2 +4n(n+b) < 0, then complex coefficients will appear in L+ and L_. Still using (16.6), we can get another general integral of (16.12) from (16.17): U:.~(x, y)
= ln[V11/2n(n+b){1 + by)l/b(n+b)]
m
+
u2n(n +b)
tan-l [-2n{n + b)x- m(1- ny)] u:~(1- ny)
( 16.19)
=0, where 0'2 = v-m2- 4n(n +b). Now when m 2 + 4n(n +b) ~ 0, we instead take the Dulac function as
B5 (x, y)
b)t
= exp ( 2n(n + u1- rn =
L~a1(1
Ut(x, y)) B"(x, y)
+ by)(mMl-21-b)/b 1
where a 1 = (m + ut)f2n(n +b) is a root of n(n + b)a 2 - ma- 1 = 0. Then clearly B 5 (x,y) is the B 1 (x,y) in (16.2); hence it can be used. Similarly, for m 2 + 4n(n +b) < 0, we instead take the Dulac function as B6(x, y) = e 2n(n+b)U2 (:c,y) • B4(x, y)
= {1 + by)-21/b-1 exp (2m tan-• [- 2n(n + b)x - .!!!.] ) i u2(1- ny)
0'2
{16.20)
0'2
then it is just B2{x, y) in (16.4) and therefore it can be used. From this we can see that the Dulac functions B 1 (x,y) and B2(x,y) can be constructed in a definite way, and are not completely obtained by chance. Now we give an example. Consider the system of equations [219]:
dxfdt 11-
= -y + lx 2 + mxy,
dyfdt = x(1 +ax).
It is easy to see that when l = 0 the system = 1/(1- mx) and a general integral U3
2 + _!_ (1 + .!_) X- !y2 + = ~X 2m m m 2
(1 +
(16.21)
has an integrating factor
~) ...!__ ln(1 m m2
mz).
Let (16.22)
§16. THE METHOD OF DULAC FUNCTIONS
365
It is obvious that
u1J.I. = (1- mx)2a/m +2/m-lea:z: -m11 +2(1+a/m)x 2
2
2
is still an integrating factor of (16.21), but this formula does not contain l. Similar to the previous example, in order to obtain the Dulac function at l ~ 0, we should take B(x,y)
= u1 J.I.e-21y = (1 -
mx)2a/m 2 +2/m-leax 2 -m11 2 +2(a/m+l)x-21 11 • (16.23) Here the new factor is just the integrating factor of {16.21) when a= m = 0. Then for system (16.21) we can compute
a
ax (BP)
a
+ ay (BQ) = -l[2a 2 mx:~ + 2a- m]x2B(1- mx)
_1 •
(16.24)
From (16.24) we can see that when l = 0 or a = m = 0, (16.21) has a center. When l ~ 0 and m ~ 2a, the origin is a focus. If m = 2a, then the origin is a fine focus, ii3 = 0, and the sign of iis can be determined from the sign of l (see (12.21) in §12); hence it does not have a limit cycle. Sometimes, for some equations, we can obtain at the same time several Dulac functions of different types. For example, the system (I)c=n=O dxfdt = -y + lx:l
has a center when l
+ mxy,
dyfdt = x
{16.25)
= 0, and has integrating factors
J.l.l;::: emz-m 2 y2 / 2 and J.l.2 = 1/(1- mx) and a general integral U = (1- mx)emx-m,y,/2.
Moreover, when m e- 21 11. The functions Bl
= 0, the system has a center and an integrating factor
= emx-my,/:l-2ly
and B2
= (1- mx)-le-21y
can also be used as Dulac functions of {16.25):
! :x
(BlP)
+ :y (Bl Q) = mlx2emz-21y-m,y, /2,
(B2 P) + :Y (B2Q)
= mlx 2(1- mx)-le- 2111 .
The above formulas show that for ml ~ 0, (16.25) does not have a limit cycle. For the above equations of class III, in fact we discuss only one case which may satisfy conditions of a center, i.e. {16.1). For quadratic systems which
THEORY OF LIMIT CYCLES
366
may satisfy other conditions of center, we say nothing here, but readers can refer to [244]. In the following we use the method of the Dulac function to prove Theorem 15.6 in §15. 1. The case m 2 + 4n(n +b) 2:: 0. Take L(al +m)b/2nat-1L(at-m)b/2nat-1 _ B 1_- eb(n+b)U1 /1-1+ , where U1 and 11- are given by {16.17) and {16.13) respectively, and L+ and L_ are given by (16.18). For the system of equations
dx dt
dy dt =x(1+ax+by)
2 = -y+lx2 +mxy+ny,
{16.26)
we compute its divergence, and obtain
a-
a 115
ax(B1P) + ay\BtQ = [Cx + A(1-
)
{16.27)
ny)]x2 L~t+m)b/2nat-2 L~t-m)b/2na.1 -2
where
A= m(l + n)- a(2l +b),
C = am(2l + b + n)- (2l + b)(n + b)(n + l). (16.28) The line l: Cx + A(1- ny) = 0 passes through (0, 1/n), which is in this case a saddle point or a nodal point. Hence there exists at most one more point of contact with the trajectory of (16.26). This shows that if (16.26) has a limit cycle (it must intersect with l, and hence its interior contains at least one point of contact on l), the limit cycles must be centrally distributed in the vicinity of one singular point of focal type of index +1. It may be (0,0), but it may be a singular point on 1 + ax + by = 0. 2. The case m 2 + 4n(n +b)< 0. Take
(2m(l+n)+mb t _ 1 [-2n(n+b)x-m(1-ny)]) B 2_TT(b+21)/2n - v1 ·exp · an ( ) • tw2 u2 1 - ny and compute its divergence. We get
a-
a-
ax (B2P) + ay (B2Q) = [Cx
+ (1- ny)AJVP'+b)/2n-t
·exp (
2m(l+n)+mb no-2
tan
_ 1 [-2n(n+b)x-m(1-ny)]) . u2(1- ny)
Note now that (0, 1/n) is a focus. Hence, if in the vicinity of (0, 1/n) there is no limit cycle, then as before we can prove the limit cycle, then as before we can prove that limit cycles of (16.26) must be centrally distributed. But if the
367
§16. THE METHOD OF DULAC FUNCTIONS
vicinity of (0, 1/n) does contain a limit cycle, then the vicinity of (0, 0) may still have a limit cycle and does not contradict the fact that on Cx+(1-ny)A = 0 there exist at most two points of contact with the trajectory of (16.26). In this case limit cycles need not be centrally distributed. An example was seen in system (11.10) of §11. But if there is some finite singula'r point (x11 yt) in addition to the two foci, then it must have an exceptional direction. We can then use this singular point to replace (0, 1/n) and use [19]:(2)
(i = 1,2) to replace L+ and L_ respectively, where fh and fh are two roots of (-1 + mx1
+ 2yt)0 2 + [(21- b)x1 + my1]0- ax1 = 0;
in fact, they are the slopes of two exceptional directions of (x1, Y1)· Moreover, we use B3 = [y- Y1- 01(x- xl)]k 1 [y- Y1- 02(x- xl)]k 2 to replace the above B1 and B2, where
k1 _ (01x1- YtHm- (b + 2l)02] 02-01 I k2
(02x1- YtHm- (b + 2l)01J. 01-02
=
Thus we can compute
:x
(B3P) + :y (B3Q)
= KL~'- 1 L~2 - 1 (ytz- z 1 y) 2 (A'x + B'y + C'),
where
A' = al(b + 2l) x 1 +am+ am(b + 2l), Y1
,
my1
B = -
2
+ mb + a(b + 2l) -
x1 C' = m(l + 1) - a(b + 2l).
m 2
x1
m2
+ -, x1
(16.29)
It is easy to verify that the line A' x + B' y + C' = 0 must pass through the singular point (xt. yt); thus, as before, we can show that in this case limit cycles cannot coexist in the vicinity of (0, 0) or (0, 1). It is worthwhile to consider: Can one use the method of Dulac functions to prove Theorem 15.12 of §15? We would guess that it is possible. (2)In the following calulations we assume n = 1; for all n :f: 0 we can always make this assumption.
THEORY OF LIMIT CYCLES
368
Finally, we would like to prove an important property of fineness of a focus for the quadratic system mentioned at the end of §11. THEOREM 16.3. If a quadratic system has two fine foci, then they can only be first-order fine foci. This theorem was first proved in [191] by the method of Dulac functions. But Yu Ren-chang [245] found a more simple and convenient method of proof was given. Hence we adopt the method of proof of [245]. As for the proof by means of the Dulac function, it is similar to the above approach and is left as an exercise. PROOF. We adopt the standard form of
= .X1x- y- .Xax 2 + (2.X2 + .Xs)xy + .Xay2, dyjdt = x + AlY + .X2x2 + (2-Xa + .X4)xy- .X2y2.
dxfdt
(16.30)
Suppose 0(0, 0) is at least a second-order fine focus. We would like to prove that if there exists a singular point N of focal type, then either N is a coarse focus, or both 0 and N are centers. From the formulas for tia and iis in (9.40) of §9, we know that if 0(0, 0) is at least a second-order fine focus, then we must have .X 1 = .X 5 = 0. At this time (16.30) becomes
dxjdt
= -y- .Xax2 + 2.X2xy + .Xay2 = P,
dyjdt
= x + .X2x2 + (2-Xa + .X4)xy- .X2y2 = Q,
(16.31)
and the divergence of the system is
D=
aP
aQ
+8y - =A4X. ax
H A4 = 0, then (16.31} is an exact equation, whose elementary singular point can only be a center or a saddle point. Hence we may as well assume .X 4 =# 0. At the same time, if another singular point N of focal type does not lie on the y-axis, then the value of D at N is not zero, and N must be a coarse focus. Now we suppose N is on they-axis, and we would like to obtain the coordinates of the singular point on the y-axis. Let x = 0 be substituted in P = 0, and obtain y = 0 and y = 1/ .Xa. Substituting Q = 0, we obtain .X2y2 = 0. Hence the possible coordinates of N are (0, 0} or (0, 1/.Xa}. In the first case, N = 0 and does not have a second fine focus. In the second case we must have .X2 = 0. But when .X 2 = 0, (16.31} can be integrated, and N can still be a center.
369
§16. THE METHOD OF DULAC FUNCTIONS
Of course, it ig possible for a quadratic system to have two first-order fine foci, such as the system (13.3) of §13, when m > -a > 1/a-a and 27a < 4m3 ; its global structure was shown in Figure 13.10; here 0(0, 0) and R( -1/a, 0) are both first-order fine foci. The theorem is completely proved. In [245], it was further proved that when a quadratic system has two firstorder fine foci, they must possess different stability. The proof is omitted, and left as an exercise for the reader.
Exercises 1. 2. 3. 4.
Prove (16.3) and (16.5). Prove Remark 1 after Theorem 16.1 and (16.7). Verify (16.27) and (16.29). Suppose that in the system
dx/dt =ax+ by+ x(x 2 + y 2 ),
dy/dt =ex+ dy- y(x 2
+ y2 )
we have a+ d > 0 and (a- d) 2 + 4bc < 0. Use the Dulac function [by 2 + (a- d)xy- cx 2 ]- 1 to prove that the system has a unique limit cycle and a stable cycle [170]. 5. Use the Dulac function 1/x(x + y) to prove that the system
dx/dt
= 3xy,
dy/dt
= y2 -
2x 2
-
4xy + 2x
does not have a limit cycle. 6. Prove that in the system of equations
dx/dt
= -y(1 + y- mx) + 6x,
dyfdt = x(1 +ax)
ifO < -1/a < m, then for 6 > m(1-y'1 + 1/ma)/2 the system does not have a limit cycle [246]. (Hint. Use the Dulac function [m 2 x- my+ d- m]- 1 .) 7. Prove that the system dx/dt = x(y- k), dy/dt = ""fX + (3y + gy 2 does not have a limit cycle [170]. 8. Prove that the system
dx/dt = y 2
-
(x + 1)[(x- 1) 2
+ ..\],
dy/dt
= -xy
does not have a closed trajectory when .,\ ;::: 1 [170]. 9. Prove that the system dx/dt = 2xy, dy/dt = 1 + y- x 2 have a closed trajectory [170]. 10. Prove that the system
dx/dt = 2x(1 + x2
-
2y 2 ),
does not have a closed trajectory [170}.
dyjdt = -y(1- 4x 2
+ y2
+ 3y2 )
does not
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THEORY OF LIMIT CYCLES
11. Take the Dulac function B = dx _ 1 - f3 dt - y- 1 + {3x'
e2e:z:,
and prove that the system
dy 2 2) dt = -E(x + Y
(where E > 0 and 0 < {3 < 1/3) does not have a closed trajectory in the half-plane 1 + {3x > 0 [170]. 12. Prove that the system dx
~
= xy
'
dy
&
= _!(x -1)(x + 2) + !y2 + !xy + !y 3
2
3
3
does not have a closed trajectory [170]. 13. Use the method of Dulac functions to prove Theorem 16.3.
§17. Limit Cycles in Bounded Quadratic Systems
The s~called bounded quadratic systems are quadratic systems whose trajectories remain bounded for t ~ 0. The research work in this area began with Dickson and Perko (247], who studied autonomous quadratic bounded differential systems in n-dimensional space, and then turned to a deep and detailed study of plane quadratic systems; the main part of the latter work was published in (248]. The authors made use of the results of [165] mentioned in §10 to give a detailed classification of bounded quadratic systems, and drew their global phase-portraits. However, there was no discussion at all on the existence and the number of limit cycles. Independently of (248], Levakov and Shpigel'man (249] obtained many sufficient conditions for a quadratic system to be bounded, but they too did not consider the problem of limit cycles. In this section we present the work of [250] and (253], which were the first papers to study existence and uniqueness of a limit cycle for a bounded quadratic system, and in which m;my results were obtained although the problems were not yet completely solved. In this area we also mention the recent papers (300] and [301], which among other things corrected a mistake in [250]; moreover, [300] added a new phase-portrait and the corresponding conditions to [248]. It is easy to see that any bounded quadratic system has at least one singular point; changing the origin to the singular point, we can rewrite the system as (17.1) where Pto,Pol!Qto, and Qol are constants, and X2(x,y) and Y2(x,y) are homogeneous quadratic polynomials. We study the boundedness of the quadratic system (17.1), to which the properties of the corresponding homogeneous quadratic system dyfdt = Y2(x, y)
dx/dt = X2(x, y), 371
(17.2)
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372
have a definite relationship. Changing (17.2) into polar coordinates, we get
~~
= r 2[X2(cos0, sinO) cosO+ Y2(cosO, sinO) sinO]
= r 2 F(O),
~: = r[Y2 (cos 0, sin 9) cos 0 = rG(O).
{17.3) X 2 (cos 0, sin 0) sin 0]
Let 9 = 00 be a root of G(O) = 0, and F(9o) # 0. Since G(IJ0 + 1r) =0, we may as well assume F(Oo) > 0. Then the solution of (17.3) which satisfies the initial conditions r(to) = ro and O(to) = 9o is 1
r(t) = F(Oo)
1
to+ (1/roF(Bo))- t'
9 = 90 '
(17.4)
It is a ray in the (z, y)-plane, and r(t) -+ +oo as t-+ to+ 1/roF(9o). DEFINITION 17.1. Suppose r = r(t) is a solution of (17.1) such that t approaches a finite value as r-+ +oo. Then r(t) is a solution possessing finite escape time. For (17.3), if there exists a Oo such that G(90 ) = 0 and F(90 ) # 0, and (17.4) is a solution possessing finite escape time, then {17.4) is called a ray solution. THEOREM 17 .1. If (17.2) has a ray solution, then (17.1) has an unbounded solution possessing finite escape time. PROOF. Apply the transformation z = rcos9, y (17.1) becomes
= rsinO, dr/dt = r. Then
~; = PlO cos2 9 +(Pot+ QtO) cos0sin9 + Qol sin2 0 + rF(O) = Pt(O) + rF(9), dO = ![Qto cos2 0 + (flot- PlO) cos8sin9- Pot sin2 9] + G(9) d r r
(17.5)
= !Qt(O) +G(O). r
From the hypothesis, we know there exists a 9 = 90 such that G(Oo) = 0 and F(Oo) # 0. We may assume F(Oo) = >. > 0. Let u = r- 1 • Then (17.5) becomes iJ = uGt(O) + G(O), which has a singular point (u = 0, 0 = 90 ) and the first equation has a negative characteristic root->.. Hence from the classical theorem ofLya.punov we know that the system of equations has a trajectory approaching to (u = 0, 9 = 00 ); correspondingly, system (17.5) has a trajectory (r(r),O(x))-+ (+oo,Oo).
§17.
LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
373
Applying the formula of variation of constants to the first equation of (17.5), we see that as r( r) -+ +oo, we must have r -+ +oo. Finally, from
r
ds t=Jo r(s) we can see that as r -+ +oo we have t < oo .. The proof is complete. From the necessary and sufficient conditions for a quadratic system to be bounded in [248], and the supplement from [300], we obtain the following: THEOREM 17.2. All the trajectories of quadratic system (17.1) are bounded when t ~ 0 if and only if there exists a linear transformation which changes (17.1) into one of the following types: (A) • 2 • 2 x = aux + a12Y + y , y = a21x + ~2Y- xy + cy ,
where lei < 2 and the other coefficients satisfy one of the following groups of conditions: (A1) au < 0,
(B)
(A2)
au = a21 = 0,
(A3)
au
= 0, a21 = -a12 =F 0, ca21 + a22 ~ 0;
x = aux,
where au < 0 and a22 ~ 0; or (C) . . x = aux + a12Y + y2 , Y• = a22Y, where au ~ 0, a22 ~ 0 and au + a22 < 0, or au < 0, a22 < 0, and 2a22 - au > 0. The proof depends on the results of [165] and Theorem 17.1, which includes many pages, and is therefore omitted. In the following, we plan to prove a rather special case of Theorem 17.3, and draw the global structural figures corresponding to several possible cases, in addition to the existence and number of limit cycles. REMARK. It is clear that under conditions (A2), (B), and (C) we can write the equation of the family of trajectories of (A), and using this we can prove it is a bounded system and does not have a limit cycle. Hence, we are only interested in the case when (A) satisfies (A1) or (A3). Now we change (A) and the corresponding conditions (A1) or (A3) into our familiar form of equations of class III. Without loss of generality, we can assume (0, 0) is an elementary singular point of (A) of index +1; hence aua22 - a21a12 = .!l > 0.
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374
Now in (A) we apply the change of variables
v = aux- a2 1y,
u = x or y = (auu- v)/a21,
x = u.
Then we get
u= -v +(au+ a22)u- u(auu- v)/a21 + cu 2, v=
u[(aua22- a21a12)- au(auu- v)/a21
(17.6)
+ (cau- a21)u].
Again let
Then the above system of equations is changed to
dxfdr = -jj +a -l/ 2(au + a22)x + xya - 112/a21 + (ca21- au)x2a - 1/a21. djjfdr = x + [a21(cau- a21)- a~ 1 ]a - 312x 2/a2 1 + auxya - 1/a21· 1t already possesses the form
~!
= -jj + ox + lx 2 + mxjj + ny 2,
djj dr
= x(1 +ax+ by).
(17.7)
Here we have
o=
(au + a22)a - 1/ 2, l = (ca2 1 - au)a - 1/a211 m=a- 112/a21, n=O,
a= [a2t(cau- a2t)- a~1]a- 3 1 2 /a21.
b = aua- 1/a21·
Note now that 4ma + (b -l) 2 = (c 2 - 4)a - 2,
mb = aua - 3 / 2 fa~1·
Hence condition (A1) of Theorem 17.2 and the inequality rewritten as
n = 0, and condition (A3) and n = 0,
(b- l) 2 + 4ma < 0,
lei <
2 can be
mb < 0,
lei < 2 together can be rewritten as
(b- l) 2 + 4ma < 0,
b = m +a = 0,
m(l + mo) ~ 0.
In the above two conditions we haven= 0. Hence we may rewrite (17.7)
as dx 2 dt = -y + ox + lx
+ mxy,
~~ = x(1 +ax+ by).
(17.8)
§17. LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
375
Thus we have THEOREM 17.3. A necessary and sufficient condition for system (17.8) to be bounded is that one of the following conditions holds:
+ 4ma < 0 and mb < 0.
(1)
(b -l) 2
(2)
(b- l) 2 + 4ma < 0, b = m +a = 0, m(l + mc5) :5 0, and m
# 0.
(17.9)
PROOF. Sufficiency. Examine the state of the trajectories of system (17.8) near the infinite singular point. Apply the Poincare transformation
x
= ujz,
y = 1/z,
= z.
dtjdr
Then (17.8) becomes
du/dr = mu- z+ (l- b)u2
+ c5uz- au3 -
u2 z,
dz/dr = -uz(b +au+ z).
(17.10)
From (17.9) we know that (17.8) can have a unique infinite singular point
(0, ±1,0). Since m # 0, we may as well assume m > 0 (otherwise we change the sign of x and y to achieve this). Let
x = -z,
jj
= mu- z,
dr/dr'
= 1/m.
Then (17.10) becomes
dx = 2x 1 _(.:x- y_) [b m
-d1 T
djj
1(
dr' = jj + m =
x- jj
a -] -1 (a+ m )x_ + -y m
m
= .rn 3 ( x,_ y_) ,
) [ mc5 + l_
l - b _ m + a__ a _2] ----,:n-x- m y - ---:;ri2xy- m2 y
ii + Q3(x, y).
Suppose from jj + Q 3 (x, jj) = 0 we solve for jj = cp(x) which satisfies cp(O) = = 0. Substituting in P3(f, jj), we get
cp'(O) 1/J(x)
= P3(x, cp(x)) =
-;x +m m 2
-4lb(mc5 + l)- m(a + m)]x 3
+ mc5 ~ l [b(mc5 + 2l- b) m
m(m + 2a)]x4
(17.11)
+ o(x4 ).
From (17.11) we know that if b # 0, orb= m +a= 0 and mc5 + l # 0, then (0, ±1, 0) is a semisaddle nodal point, and when b < 0, or b = m +a = 0, mc5 + l < 0, the state of the trajectory of (17.8) near the infinite singular
THEORY OF LIMIT CYCLES
376
o>O:
a-o
m>o, bO, ~+l
b-m+a-~+l-0
FIGURE 17.1
FIGURE 17.2(a)
b=m+a-~+l-0
o>0,-2
b-m+a-~+l-0
o>O, 0<
~
< 2
FIGURE 17.2(c)
FIGURE 17.2(b)
point (0,±1,0) is as shown in Figure 17.1; hence all the solutions of (17.8) are bounded when t ~ 0.( 1 ) Also, if b = a+ m = m6 + l = 0, then from (17.9) we must have lol < 2; then (17.8) can be written as
dxjdt
= (1 + ax)(ox- y),
dyjdt
= x(1 +ax).
The line 1 +ax= 0 is filled with singular points. The global phase-portrait of the system is as shown in Figure 17.2; hence all the solutions of (17.8) are bounded. do not assume m > 0, then when m < 0, b > 0 or m < 0, b = m +a= 0, 0, the figure of the trajectory near the infinite singular point can be obtained from Figure 17.1 by symmetry with respect to (0,0). Here the arrow does not reverse its direction; hence all solutions are still bounded when t ~ 0. ( 1 )If we
mo + I >
§17. LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
377
Necessity. We change the corresponding homogeneous quadratic system of (17.8) by the transformation used in the proof of Theorem 17.1 into polar coordinates:
= r[l cos 2 9 + (m +a) cos9sin9 + bsin 2 9] cos9 = rF(9), d9fdr = [acos2 9 + (b -l) cos9sin9- msin 2 9] cos9 = G(9). drfdr
If (b-l) 2 +4ma ~ 0, then G(9) = 0 has one or two real roots in [0, 1r). Since the eliminant of the two trigonometric polynomials G(9) = 0 and F(9) = 0 is
R = (bl- ma)[(a + m) 2 + (b -1) 2 ], to have a common root, we need R = 0. When bl- am = 0, we first assume b/m = afl = k ;f: oo. Thus G(9) and F(9) become cos 9(kcos9- sin9)(l cos 9 + msin9) and
cos9(cos9 + ksin9)(lcos9 + msin9), respectively. Take 90 = tan- 1 k. Then G(9o) = 0 and F(9o) '1: 0, i.e. (17.8) still has an unbounded solution. If k = oo, then m = l = 0; the proof of existence of unbounded solutions will be seen in the following Theorem 17.4. When (a+ m) = (b -l) = 0, (b -l) 2 + 4ma ~ 0 becomes -4m2 ~ 0; hence we also have m = 0. From the following Theorem 17.4, we know (17.8) still has an unbounded solution. Hence (b -l) 2 + 4ma < 0 is a necessary condition for (17.8) to be a bounded system. At the same time, we must have m ;f: 0. If mb > 0, or b = m + a = 0 and m(m6 + l) > 0, then from (17.11) we know that (0, ±1,0) is still a semisaddle nodal point, but the state of the trajectory of (17.8) near the singular point is as shown in Figure 17.3(a); hence (17.8) has an unbounded solution. If m + a = 0 and b '1: 0, then when b # l, to make R = 0, we need bl = ma. As before, if m # 0, then it has an unbounded solution. If m = 0, then, by the following Theorem 17.4, we know that (17.8) still has an unbounded solution. Also, if b = 0 and m +a ;f: 0, then from (17.11) we know that (0, ±1, 0) is a nodal point or a saddle point; hence as t -+ +oo, the trajectory must enter the singular point from the finite plane, i.e. (17.8) has an unbounded solution. The theorem is completely proved. THEOREM 1 7. 4.
The quadratic system
x = -y + ox + lx2 , has at least an unbounded solution.
1i = x(1 +ax+ by)
(17.12)
THEORY OF LIMIT CYCLES
378
m>o,li>O or m>O, tna+Z>O FIGURE 17.3(a) u
0
U=O
FIGURE 17.3(b) PROOF. When l =f. 0 and b =f. l, we can take 80 to satisfy acosOo+ (b - l) sin Oo = 0. Thus G(Oo) = 0 and F(Oo) :j.: 0. Hence (17.12) has an
unbounded solution. H b = l =f. 0, we may as well assume l > 0. Apply the change of variables v = -y- !bx2 , x = x; then (17.12) becomes
:i: = v + 6x + 3lx2 /2, Let V
= v + 3lx2 f4.
Then along the trajectory of (17.13) we have
(17.13)
§17. LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
379
Now we study the region D in the fourth quadrant in Figure 17.3(b). It is bounded by the parabola U = 0 and the horizontal line L. The point of intersection of L and U = 0 is P, whose abscissa is xo. It is easy to see, as long as xo is sufficiently large, that for x ~ 0 we have
dUfdtlu=o > 0,
viL <0,
i.e. the trajectory of (17.13), when entering D, must be unbounded. When l = 0 and b '# 0, we may as well assume b > 0. The proof is similar as before. The choice of v is unchanged, but we take U = v + kbx 2 , the sign of k depends on the sign of a+M, and the magnitude of lkl is chosen in order to make sure the coefficient of x 3 in the representation formula dU fdtlu=o has the same sign as a + bt5. The details are omitted. When l = b = 0, it is easy to see that there exists a region similar to D in the (x, y)-plane. The theorem is completely proved. In the following we begin to discuss the limit cycles of bounded quadratic systems. When b = m +a = m6 + l = 0, m '# 0, (17.8) does not have a limit cycle, even if 161 < 2, because in this case the right sides of the system have a common factor 1 - mx, and every point on the line 1 - mx = 0 is a singular point. Hence we only have to discuss the case when (17.8) has only a finite number of finite singular points. Clearly (17.8) has at most three finite singular points. In the following we carry out our discussion separately according to the number of singular points. I. The case of three finite singular points. We solve 1 +ax+ by= 0 for y, and substitute in the first equation of (17.8); we get
Dx 2 + Bx + 1 = 0,
(17.14)
where D = lb - ma and B = a + bt5 - m. A necessary and sufficient condition for (17.8) to have three finite singular points is B 2 > 4D. When this condition is satisfied the three singular points are (0,0), (x1,yl), and (x2,y2) where Xl
=
-B+JB2-4D 2D
a 1 Yl = -bx1- b'
- B - v,..B.,....2----:-4D=
a 1 Y2 = -bX2- b' 2D Note that when b = 0, by Theorem 17.3, we know that the condition B 2 > 4D does not hold; hence in this case we have only a finite singular point (0, 0). X2
=
LEMMA 17.1. When (b -l) 2 + 4ma < 0, mb < 0, and l + m6 '# 0, the structure of separatrices is homeomorphic to Figure 17.4 or 17.5; when (b-l) 2 +4ma < 0, mb < 0, and 1+m6 = 0, then the structure of separatrices
THEORY OF LIMIT CYCLES
380
(a)
(b)
(c)
(d) FIGURE 17.4
of (17.8) is homeomorphic to Figure 17.4(a) or 17.6. The symbols- 0 and 0 - in the figures represent the direction of the trajectory in the vicinity of a singular point of index +1, and whether there exists any limit cycle can only be determined after further analysis. PROOF. We may as well assume b > 0, for otherwise we can change the sign of x andy. Since when m =I 0 we have m < 0 and a> 0, from (17.9) we can get (b + l) 2 < 4D; hence x1x2 = 1/ D > 0. When B > 0, we have x2 < x1 < 0. Move the origin to (x1, Yl) and still use x, y to represent the new coordinates of (17.8). We have
dx 1 2 dt = b[B- a+ (lb+ D)x1]x + (mx1 -1)y + lx + mxy,
~~ = ax1x + bx1y + ax2 + bxy.
(17.15)
It is easy to calculate the product of the two characteristic roots of its linear approximate system to be
§17. LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
381
hence, the new origin (original singular point (Z1, Yl)) is a saddle point and the indices of the other two singular points 01 (-z1, -yl) and 02 (z2 - z1, Y2 - Yl) are + 1. Similarly, when B < 0 and z1 > z2 > 0, as before we can prove that (z2, Y2) is a saddle point, and the other two singular points are singular points of index +1. We first consider the case when (b-l) 2 +4ma < 0, mb < 0, and l+m6 '=f. 0. If B > 0, let v2 = y + (ajb)x, (17.16) and differentiate
vl and v2 along the trajectory of (17.15) respectively; then
we-
.
where
Note that
+ 6z1 + lx21 + mZ1Y1 = 0,
1 + az1 + by1
~
-yl
= 0,
and it is not difficult to simplify A11 A2, B1, and B2 as
A1
= __!_(myl + (l- b)x1Y1- axn Z1
= --A(myl + (l- b)x1Y1- ax~], zl B1 = (aDjb 2)(x1- z2), B2 = aD/b 2. A2
Hence (17.17) (17.18) and
dxl dt
= m z=O
(xl- ..!.) y, m
dyl dt
= ax(x + xt). 11=0
(17.19)
THEORY OF LIMIT CYCLES
382
(b)
(a)
(c)
(d) FIGURE 17.5
Note that (l- b) 2 +4ma < 0, m < 0; my~+ (l- b)x 1 y 1 - ax~< 0. 02 should fall into the region V1 > 0; hence we can assume y1 < 0. When l + mS #- 0, we have x1 - 1/m #- 0. Hence when x1 - 1/m > 0, the structure of separatrices of (17.15), and hence (17.8}, is homeomorphic to Figure 17.4; when x 1 -1/m < 0 the structure of separatrices of (17.8) is homeomorphic to Figure 17.5. H B < 0, we translate the origin to (x2, Y2) and still use x andy to represent the dependent variables of (17.8) after the translation; we then get dx 1 dt = b [B- a+ (lb + D)x2]x + (mx2- 1)y + lx 2 + mxy,
dy dt
= ax2x + bx2y + ax2 + bxy.
Apply the transformation
Y2
u1 =y- -x, X2
(17.20)
§17.
LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
383
FIGURE 17.6 and differentiate along the trajectory of (17.20). We get ddu 1 1
t u1=0
= - 12 [my~+ (l- b)x2Y2 X2
-du21 dt
U2=0
ay~]x(x + x2),
aD = -x[x-x1 +x2].
b2
The singular point (x1 - X2, Yt - Y2) always falls in the region Ut !xl t
=(mx2-1)y, z=O
> 0. Also,
!~I v=O =ax(x+x2)i
since mx2 -1 < 0, the structure of (17.20), and then (17.8), is homeomorphic to Figure 17.4 or 17.5. (The case of B < 0 can be included in the case of B > 0 by changing the signs of x andy.) Similarly, when B < 0, the structure of the separatrices of (17.8) is homeomorphic to Figure 17.4. Finally, consider the case when (b-l) 2+4ma < 0, mb < 0, and l+m.5 = 0. Under the hypothesis b > 0, we must have m < 0 and D > 0. Note that D = -m(a+bO); hence a+M > 0 and B = a+M -m > 0. H a+ M + m > 0, then Xt = m/ D, x2 = 1/m, and x = 1/m- m/ D are integral lines of (17.15), but the structure of the separatrices of (17.15), and hence (17.8), is still homeomorphic to Figure 17.4. H a+ M + m < 0, then Xt = 1/m, x2 = m/D, and x = 0 is an integral line of {17.15); hence the structure of separatrices of {17.8) is homeomorphic to Figure 17.6. The lemma is completely proved. Now we discuss the problem of limit cycles of bounded quadratic systems with three finite singular points. When l + mb = 0, from Lemma 17.1 we know the structure of the separatrices of (17.8) is homeomorphic to Figure 17.6 or 17.4. It has an integral line x = 1/m; hence, by the corollary to Theorem 15.4, (17.8) has at least one
THEORY OF LIMIT CYCLES
384
limit cycle. Also, the origin (0,0) is definitely of index +1, but the indices of the other two singular points
01 (;,
a~;D)
and 02
(!, -a~m)
have not yet been determined. We already know that the characteristic equation of (17.8) at (0, 0) is >. 2 - 6>. + 1 = 0; hence: 1) When 161 ~ 2, 01 is a nodal point and there is no limit cycle in its vicinity. 2) When -2 < 6 < 0, 01 is a stable focus, and if there are limit cycles in its vicinity, the number of limit cycles must be even; hence there are none. 3) When 6 = 0, 01 is a fine focus, and hence (17.8) does not have a limit cycle (Theorem 15.4). 4) When 0 < 6 « 1, 01 is an unstable focus, there is a unique limit cycle in its vicinity, and there is none in the vicinity of 0 2 • But the precise range of variation of 6 to guarantee existence of a limit cycle in the vicinity of 0 1 may be smaller than (0, 2), even though all the conditions which guarantee the system to be bounded and l + m6 = 0 remain the same. (See Example 1, below.) In order to study the properties of the other two singular points when l + m6 = 0, we can move the origin to (xi, Yi), and then we get the system of equations ~
dt
1 2 = b[Ba+ (lb + D)xi]x + (mxi- 1)y + lx + mxy,
~~ =
(17.21)
axix + bxiy +
ax 2
+ bxy.
First replacing (xi, Yi) by 02(1/m, (-a- m)/mb) we get ~
_ m(a+M)(a+M+m)
dtdy _ dt -
l
2
bD x+ x +mxy, a( a + M) _ b( a + M) 2 b x D y + ax + xy. D
From this we see that the two roots of the characteristic equation are (note = lb- rna= -m(a + M)):
D
>.1 =-(a+ lx5 + m)/b, Then replacing (xi,Yi) by 01(mjD, -(am+ D)/bD), we get
dx _ c5(a+M+m) _ m+a+M l 2+ dta+M x a+M y+ x mxy, dy a b 2 dt =- a+Mx- a+My+ax +bxy.
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385
\
\
1+~+2y-O
FIGURE 17.7
From this we can get the characteristic equation ,\ 2 _
c(a+M+m)-b,\- (a+M)(a+M+m) _ 0 a+M (a+b6)2 - ·
(17.22)
According to the previous hypotheses, b > 0. Hence, by condition 1) of Theorem 17.3, m < 0 and a > 0; but the sign of a + M + m cannot be determined yet. H a+ M + m > 0, then from (17.21) we know that the two characteristic roots at 02(1/m, (-a- m)/mb) are negative, i.e., it is a stable nodal point. On the other hand, from (17.22) we can see that the two characteristic roots at 01(mjD, -(am+ D)/D) have opposite signs; hence 01 is a saddle point. Conversely, if a+M +m < 0, then 0 2 is a saddle point. Also from (b-l)2+4ma < 0, we can deduce that (b+l) 2 < 4(bl-ma) = -4m(a+b6); hence a+ M > 0, and so by (17.22) 01 has index +1. In short, as far as the abscissa is concerned, the saddle point must lie between two singular points of index +1, but under the above conditions, 0 1 may be a focus or a nodal point. When 0 1 is a focus, we can discuss as before the existence of a limit cycle. H it exists, then it must be unique, and cannot coexist in the vicinity ofO. EXAMPLE 1. Study the system of equations £ £ 2 -dx = -y + vX + vX dt
-
xy,
dy
dt
= x(l + 6x + 2y),
(17.23)
386
THEORY OF LIMIT CYCLES
FIGURE 17.8 where 0 ~ 6 ~ 2, l We have
FIGURE 17.9
FIGURE 17.10
= 6, m = -1, a= 6, b = 2, B = 7 + 26, and D = 26 + 6.
l + m6 = 0,
(b -1) 2
+ 4ma < 0,
mb < 0,
B 2 > 4D.
System (17.23) always has three finite singular points. When 6 = 0, they are (0, 0), (-1/6, 0), and (-1, 5/2); when 6 = 2, they are (0, 0), (-1/10, -1/5), and ( -1, 5/2). As 6 increases from 0, 0 changes from stable to unstable, and there exists a unique stable cycle in its vicinity. When 6 varies to 61 ~ 2, there exists a separatrix cycle in the vicinity of 0. It is formed by a separatrix h starting from the saddle point ( -1/10, -1/5) advancing to its upper left side, and finally entering ( -1, 5/2), and a separatrix h starting from (-1/10, -1/5), advancing to the lower right side, and finally entering (-1,5/2), as shown in Figure 17.7. When 8 > 2, 0 becomes an unstable nodal point, and there is no limit cycle in its vicinity. Note that in this example x = -1 is always an integral line, (0,0) and (-1,5/2) remain fixed, but the saddle point moves as 6 varies. Also, it is not known whether 61 equals 2. (In the original manuscript there were some errors in this example, which were corrected according to the suggestion of Han Mao-an.) When l + m6 :/:0, we can start from (17.15) and study similar problems of limit cycles as before. But since now the system does not necessarily have an integral line, even though under certain conditions we can prove there exists a limit cycle, we are not sure of its uniqueness. The details are omitted. II. The case of two finite singular points. From (17.14) we know that the necessary and sufficient condition for system (17.8) to have two finite singular points is B 2 = 4D. At the same time, 0(0, 0) is an elementary singular point of index +1, and O*(xo, Yo) is a higher-order singular point, where Yo=
2a-B bB .
§17. LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
(a)
(b)
(c)
(d)
387
FIGURE 17.11
FIGURE 17.12
Let
x = x- xo andy= y- Yo·
Then (17.8) becomes
~: =-b~(a+M+m)x- !
2a _
2b _
_2
dt = - B x - By + ax
b- _
+ xy.
Denote u =
-(1/bB)[a(a + M + m) + 2b2 ].
.(17.24)
388
THEORY OF LIMIT CYCLES
(a)
(b)
(c)
(d)
FIGURE 17.13 We have the following lemma. LEMMA 17 .2. If
B 2 = 4D, and (b -1) 2 + 4ma < 0, mb < 0, then the
structure of the separatrices of (17.8) is homeomorphic to 1) Figure 17.12 when a( a+ M + m) + 2b2 = 0; 2) Figure 17.8 when B[a(a + M + m) + 2b2 ] < 0; 3) Figure 17.9 when B[a(a + M + m) + 2b2 ] > 0 and a( a+ bt5 + m) 4) Figure 17.11 when a( a+ M + m) > 0; and 5) Figure 17.10 or 17.13 when a+ M + m = 0. PROOF. We first consider the case u
= 0.
Then a(a + bt5
and system (17.24) can be written as dx 2b_ 2b2 _ 1_2 __ dt = Bx+ Bay+ x +mxy, • djj 2a _ 2b _ _2 b- _ dt = - B x - By + ax + xy. Let
x=u,
_ v lu2 + 2bu/B y = 2b2jBa+mu- 2b2jBa + mu·
< 0;
+ m) = -2b2 ,
§17. LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
389
We get
du dt =v, dv = 2D u2 [1- B u+o(u)] dt B 2
+
~+~+D b
2 [ +· maB 2b2 v 1 -
&mD ] uv 1- 2b(D+b2+lb)u+o(u) [
(17.25)
maB ] 2b2 u + o( u) .
Therefore 0* is a higher-order degenerate singular point; hence the structure of the separatrices of (17.25) (and so also (17.8)) is homeomorphic to Figure 17.12. Next, suppose u :f:. 0. Applying the transformation X
*
b_ =X+ ay,
y
*
a ( ~.~ ) = 2a_ B X- bB a+ + m 'f} 00
to (17.24), we get dx* a dt = ux*- 2bBu2 (a+ bt5 + m)[2u(b + l) + B]x* 2
bD 1 + """'22y* 2 + 2 ~._ 2 [a(a+ bt5 + m)B- 2bu(b 2 + ma)]x*y*,
aq auu dy* a3 dt = B 3b2u2 (a+ bt5 + m)[(a + bt5 + m)(Bu + 2l) + 4mb]x* 2
(17.26)
2bD 2 a ( 2 + Boo2 y* + B 2bu2 [a+ bt5 + m)(bBa + 2D + 2bl) +4mb ]x*y*. Again applying the transformation x* = v, y* = u, dtfdr = 1/u to (17.26), we get du 2bD 2 a 2 dr = Boo3 u + bB 2 u3 [(a+ bt5 + m)(bBu + 2D + 2bl) +4mb ]uv a3 + B 3b2q3 (a+ bt5 + m)[(a + bt5 + m)(uB + 2l) + 4mb]v 2 = P2(u,v), dv bD 1 dr = v + a2u3 u2 + 2abu3 [a(a+ bt5 + m)B- 2bu(b2 + ma)]uv
- 2b;u3 (a+ bt5 + m)[2u(b + l) + B]v 2 = v + Q2(u,v). Since v + Q2 (u, v) = 0, we can solve
v = cp(u) ==- bD u 2 + o(u 2 ), a2q3
(17.27)
390
THEORY OF LIMIT CYCLES
and substituting it in the first equation of (17.27), we get
.,P(u)
=P2(u, cp(u)) = Auu + o(u 2
2 ),
(17.28)
where Au= 2bD/Bau 3 ; hence 0* is a semisaddle nodal point. We may as well assume b > 0; if not we just change the sign of x and y in (17.8). Since B 2 = 4D, we get 1 mx1 -1 =- B(a + b8 + m). (17.29) Note that under the condition B 2 > 4D, when B > 0 we have x2 < x 1 < 0; and when B < 0 we have 0 < x2 < x1. Hence when B 2 > 4D changes to B 2 = 4D, if B > 0 (< 0), then the higher-order singular point corresponds to the coalescing of the saddle point and the elementary singular point on its left (right) of index + 1. Now we consider the case B[a(a+bt5+m)+2~] < 0. If a(a+bt5+m)+2b2 < 0, then B > 0 and a( a + b8 + m) < 0; hence, we must have m < 0. Thus a> 0 and a+ b8 + m < 0. From (17.29) we know that 0* compounded of the saddle point in Figure 17.5 and the elementary singular point of index +1 on its left. Also by (17.28) the hyperbolic region near 0* contains the negative u-axis (in this case Au > 0). Returning to the xO*y plane, we see that 0* is compounded of the saddle point in Figure 17.5(a) and the elementary singular point of index + 1 on its left. Similarly, we can prove that if a( a+ bC + m) + 2b2 > 0, then 0* is compounded of the saddle point in Figure 17.4(a) and the elementary singular point of index +1 on its right. In these two cases, the structure of separatrices of (17.8) is homeomorphic to Figure 17 .8. Next, we consider the case when B[a(a + b8 + m) + 2b2 ] > 0 and a( a+ M + m) < 0. If a( a+ b8 + m) + 2b2 > 0, then B > 0, and so m < 0. 0* is compounded of the saddle point in Figure 17.5 and the elementary singular point of index + 1 on its left. But since Au < 0, the hyperbolic region near 0* contains the positive u-axis. Thus in the xO*y plane, 0* is compounded of the saddle point in Figure 17.5(d) and the elementary singular point of index + 1 on its left. Similarly, if a( a+ bC + m) + 2b2 < 0, then 0* is compounded of the saddle point in Figure 17.4(d) and the elementary singular point of index +1 on its right. In these two cases, the structure of the separatrices of (17.8) is homeomorphic to Figure 17.9. Third, consider the case a( a + b8 + m) > 0. Then a( a + b8 + m) + 2b2 > 0, m < 0, B =a+ bC + m- 2m > 0, and from (17.29) we know that 0* is compounded of the saddle point in Figure 17.4 and the elementary singular point of index +1 on its left. Hence the structure of the separatrices of {17.8) is homeomorphic to Figure 17.11.
§17. LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
391
FIGURE 17.14
Finally, if a + b8 + m = 0, then m < 0, for otherwise we would have B = 0, which is impossible. Since B > 0, x = 1/m is an integral line. H a+ b8 + m changes from negative to 0, then 0* is compounded of the saddle point in Figure 17.6 and the elementary singular point of index +1 on its left; hence the structure of separatrices of (17.8) is homeomorphic to Figure 17.10. If a + b8 + m changes from positive to 0, then 0* is compounded of the saddle point in Figure 17.4 and the elementary singular point of index +1 on its left; hence the structure of the separatrices of (17.8) is homeomorphic to Figure 17.13. The lemma is completely proved. We now turn to the problem of limit cycles. First of all, when 161 ~ 2, the unique elementary singular point of index +1 of (17.8) is a nodal point; hence (17.8) does not have a limit cycle. Thus we can assume 161 < 2. Under this condition, if a( a + b8 + m) + 2b2 ::/: 0, then B > 0; otherwise, let
- YOV =y=-x, xo -2/ Band Yo= (2a- B)/bB. Differentiating V and xo along the
where xo = trajectory of (17.24), we get
~I
t V=O
= 8~ [(4- 62 )b2 +(a+ m) 2 ]x(x + x0 ),
=~-x=O =-!(a+b8+m)y,
:1_
y=O
=ax(x+xo),
when B < 0 and xo > 0. Thus the structure of part of the separatrix of (17.24) is as shown in Figure 17.24 (the figure corresponds to the case of a > 0); but this is impossible. Since the shaded sector in the figure does not have a singular point, the trajectory starting from the semisaddle nodal point (0, 0) cannot get out of the sector as t - +oo.
THEORY OF LIMIT CYCLES
392
Thus, under the condition J8J < 2, conditions 2) and 3) of Lemma 17.2 can in fact be written as 2') a( a+ b8 + m) + 2b2 < 0, B > 0; 3') a( a+ b8 + m) + 2b2 > 0, B > 0, a( a+ b8 + m) < 0, respectively. Now it is not hard to see that under conditions 2') and 3') plus conditions 1) and 5) of Lemma 17.2, we have m '=/: 0. Hence to discuss existence and number of limit cycles, we can assume m '=/: 0 in the above cases. In fact, for system (17.8), if m = 0, then the uniqueness of a limit cycle was proved earlier in [226], as we mentioned in §15. Since we can assume b > 0, we must have m < 0; for convenience, we take m = -1. Thus (17.8) can be written as dx 2 dt = -y + 8x + lx - xy,
~~ = x(1 +ax+ by).
(17.30)
The boundedness condition 1) of Theorem 17.2 and the condition B 2 = 4D for (17.8) to have two finite singular points can be written respectively as (b - l) 2
< 4a,
b > 0,
(17.31) (17.32)
(a+ b8 + 1) 2 = 4(lb +a),
and conditions 1), 2), 3), and 5) of Lemma 17.2 can be written respectively as a( a+ b8 -1)
+ 2b2 =
0, a(a + b8 -1) + < 0, a+ b8 + 1 > 0, 2 a( a+ b8- 1) + 2b > 0, a( a+ b8 -1) < 0, B > 0, a+b8 -1 = 0.
2b2
(17.33) (17.34) (17.35) (17.36)
THEOREM 17.5. Suppose (17.30) satisfies (17.31) and (17.32), and when 0 < 8 < 2 it satisfies one of the conditions (17.33), (17.34) or (17.36). Then · the system has exactly one limit cycle, and this cycle is stable.
PROOF. Let x
= x and e = lx2 + 8x- (1 + x)y.
= e, de/dt = -
Then (17.30) becomes
dx/dt
x(1 + ax)(1 + x)- bx(lx 2 + 8x)
+ [o + (21 + b)x- x(~x++x6)] e + 1 ~ x'
§17. LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
Again applying the transformation x
393
= x, u = U(1 + x), we get
dx dt = u(1 + x), du dt
= - x(1 +ax)- bx2(lx + 6) + [6 + (2l + b)x-
Finally, let
x = x,
(17.37)
1+x
x(~x++x6 )] u.
r 6 + (2l +(1b)s+ s)2+ (l + b)s ds, 2
y = -u + lo
dt 1 dr = 1+x" Then we obtain an equation of Lienard type:
dyfdr = g(x),
dxfdr = -y- F(x),
(17.38)
· where
g(x)
= (1 +x x)2
F( ) - -
x-
(
1+
r 6 + (2l +(1+s)2 b)s + (l + b)s
lo
It is easy to see that xg(x) > 0 when x then /(0) = -6 < 0. At the same time,
~ [!(x)] dx g(x)
a+M+1 ) 2 x
# 0,
2
' 2
d
s.
and if we assume f(x)
= F'(x),
= _1_1 +(a+ b6 + 1)x/2W(x) g2(x)
(1 + x) 4
'
where
W(x)
= !O + b)(a + b6 + 1)x3 +
[l(a + b6 + 1) + (l + b)(a + M)]x 2
(
17.39)
+ ~6(a + b6 + 1)x + 6.
From (17.31) and (17.32) we can get a> 0 and (a+ b6- 1) 2 = 4b(l- 6); hence (17.40) l > 6 > 0. Since 6 > 0, (17.30) has at least one limit cycle. It is easy to prove that the line x = 1 is a line without contact; hence the limit cycle must lie in the region x > -1. Now suppose (17.33) holds. Then W(x) in (17.39) takes the form W(x) = ( 1 + a+M+1) x Z(x), 2
THEORY OF LIMIT CYCLES
394
where
Z(:o:)
= (l + b)x 2 + 2[(2l +b)~+ b:)- b] x
a+ + 2 [(2l + b)(a + M)- b] + (a+ M + 1)2 It is easy to compute the minimum point of Z = Z(x) to be xo=-
(2l+b)(a+M)-b a+M+1 '
and its minimum value is
Z(x ) = [(2l + b)(a + M)- b][-(2l + b)(a + M- 1) + 2b] 0 (l + b)(a + M + 1)2 Since a + M - 1 = - 2b2 /a
< 0, we have
-(2l + b)(a + M- 1) + 2b > 0. Also because
4(l +b) - 2(2l +b) 6= 0 (a+M+1)2 a+M+1 + ' or
2(2l + b)(a + M + 1}- 4(l +b)= 6(a + M + 1} 2 , we deduce that
(2l + b)(a + M) - b = !6(a + M + 1) 2 > 0, and then Z(x) ~ Z(xo)
.!!__ [f(x)] dx g(x)
> 0 when x > -1; thus
= _1_ (1 +(a+ M + 1)x/2} 2 Z(x) > 0 (x + 1} 4
g2(x)
'
when x > -1. According to Theorem 6.4 of §6, system (17.30} has exactly one limit cycle, and it is a stable cycle. Next, if (17.34} holds, then a+ M- 1 < 0, and then
-
a+
2 M
+ 1 < -1.
Since
[6 + (2l + b)x + (l + b}x2 ]x=-2/(a+bc5+1)
= -b\::~ ~ ~) [a(a+
M-1} + 2b < 0, e) 2]
it follows that
4(l +b)- 2(2l + b)(a + M + 1) + 6(a + M + 1) 2 < 0, 2This inequality does not hold under condition (17.35).
§17. LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
395
or
4(1 +b)- 2(21 + b)(a + M + 1) + 48(1b +a) < 0. Thus
b(a + M- 1) > -2a(l- 8).
(17.41)
From (17 139) we have
W'(x) = ~(1 + b)(a + M + 1)x2 + 2[l(a + M + 1) + (l + b)(a + M)]x
+ ~8(a + M + 1), and then the roots of equation W'(x) = 0 are =
-2[l(a + M + 1) + (l + b)(a + M)]- .fK 3(1 + b)(a + M + 1) '
x2 =
-2[l(a + M + 1) + (1 + b)(a + M)] + .fK 3(1 + b)(a + M + 1) '
Xt
where
~
= 4[1(a + M + 1) + (l + b)(a + M)] 2 - 98(1 + b)(a + M + 1) 2.
First we suppose~> 0. Hence Xt < x2 < 0 and W(x) takes its maximum value W(xt) and minimum value W(x2) at Xt and x2 respectively; moreover, W(x2) < W(xt). We have
W(x2)
=! [1(a + M + 1) + (l + b)(a + M)]x~ +8(a+M + 1)x2 + 8.
Consider
1/J(x)
= l[1(a + M + 1) + (l + b)(a + M)]x2 + o(a+ M + 1)x +8,
1/J'(x)
= i[l(a + M + 1) + (l + b)(a + M)]x +8(a+M+l).
The root of the equation
1/J' (x) = 0 is 38(a+M + 1}
xo =- 2[l(a +
M + 1) + (1 + b)(a + M))'
and 1/J(x) takes its minimum value at x = xo:
1/J(xo)
=
1
2o(a + M + 1)x0 + 8 + 1} + (l + b)(a + M) - 38(lb +a)] l(a + b8 + 1) + (l + b)(a + M)
_ 8[l(a + M
-
THEORY OF LIMIT CYCLES
396
Since a(l- o)
> 0, we have -o(lb +a) > -l(a + bO),
and then
l(a+ bO + 1) + (l + b)(a+ bO)- 3o(lb+ a)
> l +(a+ b0)(2l + b- 31) = b(a + M) -l(a + bO -1) >0; hence W(x2) = ,P(x2)
.,P(x2) > 0. Moreover, W(-1) = ~(l- o)(a + M) + !(l- o) +~(a+ bO -1). ~
According to (17.41),
W(-1)
> J(l- o)(a+ bO) + !(l- o)- a(l- o)
= !(l- o)(a +3M+ 1) > 0. Now, if~ < 0, then W(x) is a monotone increasing function of x (when x > -1), and W(-1) > 0. Hence, in either case, W(x) > 0 when x > -1. Thus
[!(x)] > 0
d dx g(x)
when x
> -1.
According to Theorem 6.4 in §6, system (17.30) has exactly one limit cycle, and it is a stable cycle. Finally, if (17.36) holds, then l = o > 0, a+ bO = 1, and a+ bO + 1 = 2. Then xg(x) > 0 when x :1: 0; /(0) = -o < 0, and (17.39) becomes
W(x)
= (b + 6)x3 + (36 + b)x2 + 3ox + o = (x + 1)[(b + o)x2 + 2ox + 6].
Since~= 46 2 -4o(b+o) = -4b0 < 0, it follows that W(x) > 0 when x > -1. Thus as before we can deduce the uniqueness of a limit cycle, and this cycle is stable. The theorem is completely proved. REMARK. This theorem requires that condition (17.22) hold. Hence for fixed a, b, and l, there exists only one value of 6. Hence the conclusion of this theorem cannot guarantee that system (17.30) has a limit cycle for all in (0,2). As mentioned before, when 161 ~ 2, (17.8) does not have a limit cycle; hence we only have to discuss the nonexistence of a limit cycle under the condition . 161 < 2.
o
THEOREM 17.6. Suppose that system (17.30) satisfies the conditions (17.31) and (17.32) when ~ 0, and one of (17.33), (17.34) or (17.36) holds.
o
Then it does not have a limit cycle. PROOF. Here we only prove the case when (17.34) holds, since the remaining two cases are similar. First we suppose o = 0; then
1Ja
= ~ [m(l + n) -
a(b + 21)]
= -~ [1 + a(b + 21)] < 0;
§17. LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
397
that is, the elementary singular point of (17.30) is a stable first order fine focus. Hence if limit cycles exist, the number of limit cycles must be even, and these cycles must lie in the region x > -1. But now W(x) in (17.39) becomes
2[l(a+1)+a(l+b)]] W( ) -!(l b)( 1) 2 [ X - 2 + a+ X X+ (l + b)(a + 1) ' and (17.41) becomes b(a- 1) > -2al; thus 1 2[l(a + 1) + a(l +b)] _ 1 = [l(a + 1) + 2al + b(a- 1)] (l + b)(a + 1) (l + b)(a + 1) l(a + 1) l > (l + b)(a + 1) = l + b > O. Hence
W(x) > !(l + b)(a + 1)x2 (x + 1) > 0, x > -1; that is, system (17.30) has at most one limit cycle. This contradiction shows that (17.30) does not have a limit cycle. Next we consider the case when -2 < 6 < 0. Here the elementary singular point (0, 0) o£{17.30) is a stable coarse focus. As before, if limit cycles exist, the number of limit cycles must be even and they must lie in the region x > -1. In the following we discuss the two cases l + b ~ 0 and l + b > 0. (1) The case l + b ~ 0. 1 Denote -y- F(x) P(x, y) and g(x) Q(x, y), where the meaning of F(x) and g(x) is given in (17.38). Now suppose (17.30), and thus (17.38), has limit cycles, and suppose r is any one of them; then
=
D=
rf,. (8P + 8Q)
1;
dr = rh_ (l
ax ay V(x) = ']{. (1 + x)2 dr,
'}{.
=
+ b)x2 + (2l + b)x + 6 dr (1 + x)2
rf...
where V(x) = (l + b)x2 + (2l + b)x + 6. H l + b = 0, then from (17.40) we know that V(x) = lx + 6 = l(x + 1)(l- 6) < 0 when x > -1. H l + b < 0, then it is easy to compute a maximum point of V(x): 2l +b b xo = 2(l+b) = - 1 + 2(l+b) < - 1. Since a + b6 - 1 < 0 and a + b6 + 1 > 0, we see that
-2
a+ lx5 + 1 < -1. Moreover, V(-1) = -(l- 6) < 0. This shows that V(x) < 0 when x < -1, and thus D < 0 when x > -1. That is, system (17.30) has at most one limit cycle, contradicting the statement that the number of limit cycles is even.
~
THEORY OF LIMIT CYCLES
398
{2) The case l + b > 0. First suppose {21 + b)(a + b8 + 1)- (1 +b) ~ 0, and suppose r is any limit cycle of {17.30) and f* is a limit cycle of the system corresponding to r. Then
D= 1 (aP + aQ) lr· ax ay aP + -a aQ) = -a
i( r·
1
= Tr·
x Y u(x) (x + 1)2 dr,
dr dr -
i
l+b b8 g( x) dr a+ + 1 r•
where
u(x)
l+b 3 = -4-(a + M + 1)x +
( 21 + b- a+1+b M + 1 ) x + 8.
In this case,
and
l-8 u{-1) =a+ M + 1 [-{a+ b8 + 1) + b(1 +b)] 1 a(a + M -1) + lb]
= - 41 (1- 8)(a- b8 + 3) < 0. Hence D < 0 when x > -1. Thus system {17.30) has at least one limit cycle, and so {17.30) in fact does not have a limit cycle. Next, if {21 + b)(a + b8 + 1) - (l +b) > 0, then 21 + b > 0, and
D= 1 (aP + aQ) lr· ax ay =
£. (~= + ~~) 1
=
dr dr- (21 +b)£. g(x)dr
K(x)
Tr• (x + 1)2 dr,
where
K(x) = -(21 + b)(lb + a)x 3 + [(1 +b)- (21 + b)(a + M + 1)]:t:1 + 8. Then Xl
=0
and
X2
=
2[(21 + b)(a + M + 1) - (1 +b)] -3(21 + b)(1b +a)
§17. LIMIT CYCLES IN BOUNDED QUADRATIC SYSTEMS
are maximum and minimum points of K K(x 1 ) = 6 < 0. It is easy to compute K( -1)
= K(x)
399
respectively, and K(x2)
<
= (l- 6)[b(2l +b) -1].
Since
0 < ((1- b6)/2)(a + b6 + 1) - b6
= 1- 2b6- !(a+ b6- 1)(b6- 1)
= 1- 2b6- !(a+ b6- 1) 2 + ia(a + b6- 1) = 1- 2b6- 2b(l- 6) +!a(a+ b6- 1) < 1- 2b6- 2bl + 2b6- b2 = 1- b(b + 21), we conclude that K( -1) < 0, and thus D < 0 when x > -1. AB before, (17.30) does not have a limit cycle. The theorem is completely proved. III. The case of one finite singular point. When b = 0, the necessary and sufficient conditions for (17.8) to have one finite singular point are m +a = 0 and l +m6 i- 0. Ubi- 0, then by (17.14) a necessary and sufficient condition for (17.8) to have a finite singular point is that B 2 < 4D or B = D = 0. Consider the boundedness condition of Theorem 17.3. We get LEMMA
1 7. 3. Suppose system ( 17.8) satisfies one of the following
conditions: (1) b = m +a= 0, 12 + 4ma < 0, and m(l + m6) < 0; (2) (b -1) 2 + 4ma < 0, mb < 0, and B 2 < 4D. Then it is a bounded system, and the structure of its separatrices possesses an infinite saddle poinl and a finite focus or nodal point. (The meaning of B and D is given in (17.14).) For case (1) of Lemma 17.3, we have the following result. THEOREM 17.5. Under condition (1) of Lemma 17.3,(3 ) system {17.8)
has exactly one limit cycle when 0 cycle when 6 ::; 0 or 6 ~ -lfm.
< 6 < -l/m, and does not have a limit
The proof is not difficult, and is left as an exercise. For case (2) of Lemma 17.3, we can only get partial results, i.e. when 0 < 161 ¢:: 1 with appropriate sign, there exists a unique limit cycle near (0, 0). Chen Guang-qing [252] and Yang Xin-an [253] used the original equations in [248] to study the problem of limit cycles for bounded quadratic systems. Chen [252] obtained some sufficient conditions to guarantee that system {A) has at most one limit cycle, but when system (A) is changed to the form ( 3 )We
may as well assume m
> 0 and l < 0, and thus tJs
= 3ml < 0 when 6 = 0.
THEORY OF LIMIT CYCLES
400
of (17.8), its results duplicated part of the results of [250] mentioned at the beginning of this section. [253] contains work of Yang Xin-an done before [250], in which (with the notation of [248]) he proved that a bounded quadratic system possessing two finite singular points (one of which is a degenerate higher-order singular point, which satisfies lei < 2, au < 0, a21 < 0, aua 22 = a12a211 and au + a22 = 0) has one and only one limit cycle when cau + a21 > 0, and does not have a limit cycle when cau + a21 ~ 0.
Exercises 1. Prove the remaining two cases of Theorem 17.4. 2. Give a detailed proof of Theorem 17.5. 3. Prove that the system dxjdt = -x - y- xy, dyjdt unbounded solution [249]. 4. Prove that the system
dxjdt
= -y + xy,
dy / dt
=x -
= x + xy has an
y + x2 + 2xy
has an unbounded solution [249]. 5. Prove that in the system
dxjdt
= aux + a12Y,
except for the case when a22 < 0 and a21 # 0, if along any trajectory t can be continued to +oo, then this trajectory must be bounded [249]. 6. Use (17.32) to deduce that (a+ lx5- 1) 2 = 4b(l- 6), and use these two equalities to prove the equality on the fourth line after (17.40):
[6 + (2l + b)x + (l + b)x2 ]x=-2/(a+b6+1) -(a+ lx5- 1) 2 = b(a + lx5 + 1) [a( a+ lx5 + 1) + 2b ]. 7. Change the several cases of the bounded quadratic system in this section to the several cases of [248]. 8. Use Bautin's method to prove that for the bounded quadratic system (17.8) there may exist two limit cycles in the vicinity of the origin, but (17.8) cannot have (0, 0) as a third-order fine focus.
§18. Appendix
The goal of this section is to give a brief presentation of some new results on the theory of limit cycles from recently received journals or preprints, and also to indicate the direction of research on limit cycles tried in some other papers, not necessarily recent ones, whose contents did not fit into the previous seventeen sections.
I. Plane polynomial systems. In [255], Cai Sui-lin and Wang Zhong-wei studied the sys.tem of equations dxjdt
= -y + lx 2 + mxy,
dyjdt
= x(1 +ax+ by),
(1)
assuming that a 'I 0, ml - a(b + 2l) = 0, m
(mlja 2 )(ma 2
-
'I 5a, and ml 2 + 2al 2 ) of 0.
Then (0, 0) is not a first- or third-order fine focus and is not a center, but is a genuine second-order fine focus, and when m varies the system can change its stability and generate a limit cycle; when ajm < 0 or ajm > 1/5, system (1) does not have a limit cycle in the whole plane; when 0 < 1/5- afm « 1 and 1-15(l/m) 2 of 0 there is at least one limit cycle in the vicinity of 0(0, 0) but there is no limit cycle surrounding any other singular point. But whether the precise range of variation of ajm which guarantees the existence of a limit cycle is 0 < afm < 1/5 or not is still unknown. Han Mao-an [304] proved that if there exists a limit cycle of (1) around its second-order fine focus (0, 0), then it must be unique. Moreover, Chen Wei-feng [305] also obtained many results for the existence and nonexistence of limit cycles around the secondorder fine focus (0, 0) of (1), when the right-hand side of the first equation in (1) contains an additional term ny2 • Ye Bai-ying [256] pointed out that [231] did not completely solve the pro~ lem of centralized distribution of limit cycles of equations of class II, and also gave some other sufficient conditions, and thus extended the range of variation 401
THEORY OF LIMIT CYCLES
402
of b to guarantee the centralized distribution of limit cycles of system {14.5) in §14,(1) but the problem has not yet been completely solved. [257], [293], and [294] all studied the problem of whether quadratic sy~ terns with parabolic solutions can have a limit cycle. The answer is affirmative. Thus the situation is different from the case of quadratic systems with hyperbolic solutions [122]. Wang Dong-da [259] proved that when the parameter A varies, the equation dy dx
-(2Ax + 1)(ax +by+ c)+ o:(Ax2 + y 2 + x + 1) 2y(ax+by+c)+,8(Ax 2 +y2+x+1)
(2)
can have several forms of orbits which are quadratic algebraic curves such as hyperbola, parabola, real ellipse, point ellipse, and imaginary ellipse; but if (2) has a limit cycle, it must be unique. Thus the results in {17] were extended. P. Holmes and D. Rand [260] studied the phase-portraits and bifurcation curves of the nonlinear oscillation equation
(3) In particular, they applied the theory of bifurcation to point out this equation can have @ distribution of limit cycles in the phase-plane. Li Ji-bin [261] continued to study, after [180] and [260], the general properties of a closed trajectory of the cubic system dxjdt = y,
dyjdt = Q3(x, y)
(4)
and pointed out that there are many properties of (4) which are not possessed by quadratic systems. He rigorously proved that system (4) has @ as its distribution of limit cycles. He also studied at the same time the. cubic system possessing nine singular points dyjdt
= Q3(x, y),
(5)
0
and proved that the system has ~ as its distribution of limit cycles, and gave concrete numerical examples. Also he gave sufficient conditions for a general system of Lienard equations dxjdt = y- F(x),
dyjdt = -g(x)
(6)
for the existence of a limit cycle whose interior contains three singular points. L. A. Cherkas [262] then got a sufficient condition for the equation yy' = -g(x)- f(x)y
{7)
(where f and g are rational functions) to have a family of closed trajectories containing three singular points, and proved that in this case the absolute (1 )See footnote 6 in §14.
403
§18. APPENDIX
values of the two characteristic roots of a saddle point in the interior of the closed trajectory must be equal. Li Ji-bin [263] first studied the perturbed system of a Hamiltonian system containing two parameters J.L and A, fJH dy dx fJH (8) dt = - ox - J.LY(q(x, y)- A), dt = fJy - J.LXp(x, y), and the question of generation of limit cycles when 0 < IIJ.LII « 1, and then discussed the question of limit cycles following the change of A and J.L in the cubic system
~: = dy dt
y(ax 2 + by 2 +c),
(9)
= -x (x 2
m 2 - A) • + ay - d - J.LY lx + '3y 2
)
(
2
His results are far better than those of [260] and [264], and the equations he considered are more general. We should point out that for the branching problem of cubic systems, there is work earlier than [260] and [264] on the interesting system x = y- (ax 3 + bx2 +ex), iJ = ( 1 - AX- y (10) in which the existence of S type distribution of limit cycles was pointed out and the branching curves in the (A, u)-parametric plane were sketched (see [265]). Huang Ke-cheng (266] studied the system of equations 2m
2m+l
i+j=O
i=l
dy
dx dt = y,
.. ~ . dt = -y L..... aiix'y3 - L..... aix', ~
(11)
and gave a very succinct sufficient condition to guarantee existence of limit cycles, and he also pointed out some errors in the proof of Theorem 2 in [267]. Note that system (11) is more general than the system
x=y, studied in [268]. Ye Yan-qian [269] extended several general properties of limit cycles of quadratic systems to the quadratic system
dxfdt
= P(x, y),
dyfdt
= Q(x, y)
(12)
with real coefficients, but with complex unknowns x and y, and the equivalent quadratic system in four-dimensional space
dxddt = Pr(x,y),
dx2/dt = Pj(x, y),
= Qr(x,y),
dy2jdt = Qj(X, y);
dytfdt
(13)
THEORY OF LIMIT CYCLES
here x = Xt +ix2 andy= Yl +iy2, while Pr, Pj and Qr, Qj represent the real and imaginary parts of P(x, y) and Q(x, y). Tsutomu Date [270] used the theory of nonvariables to classify homogeneous differential systems, and corrected some wrong figures in [166], and pointed out that their method of classification was not altogether independent of earlier methods. Shi Song-ling [271] obtained sufficient conditions for the existence of k limit cycles containing (0, 0) for the nth degree polynomial system dx dy dt = >.x- y + P(x, y), dt = x + >.y + Q(x, y) (14) and extended the results of [19] and [20]. Jorge Sotomayor [272] proved that if every singular point (including the singular point on the equator) of a plane polynomial system is an elementary singular point and the skeleton figure on the Poincare hemispherical surface is a simple figure, then the polynomial system can only have a finite number of closed trajectories. (A skeleton figure is formed by saddle points and separatrices, in which the w-limit set and the a-limit set of every separatrix consists of saddle points, and in the figure for every saddle point there exist at least one separatrix entering it and one separatrix leaving it; suppose the two characteristic roots at the saddle point Pi are JLi < 0 < Ai· If for all the saddle points P1, ... , Pk in the figure we have k
p
= II IJLil ~ 1,
i=l >.i then this figure is a simple figure.) Based on [272], Sotomayor and R. R. Paterlini [273] further proved that in a vector field x2 formed by all the vector fields of quadratic systems, there exists a nonobvious function R: X2 - R such that if X E X2 satisfies R(X) ~ 0, then X has only a finite number of closed trajectories. We should point out there are several lemmas in this paper which had been obtained earlier in [14]. Rodrigo Bam6n [274] proved that in a vector space X2 formed by the totality of vector fields of quadratic systems, there exists an analytic submanifold 8 of codimension 1 such that every vector field in 8 has one separatrix cycle surrounding one limit cycle. However, Bam6n's claim that this structural figure had not previously been discovered is wrong, since we saw the possibility of its appearance many times in §14. Liang Zhao-jun [275] studied several possibilities of phase-portraits, top~ logical structures, and branching curves of the systems of class II whose first equation contains two quadratic terms
dxjdt = -y + lx 2
+ mxy,
dyjdt = x- x 2 •
(15)
§18. APPENDIX
405
Note that, unlike system {13.3) of §13, {15) may have a limit cycle or a separatrix cycle. Cai Sui-lin and Zhong Ping-guang [302] proved that the limit cycles of quadratic systems possessing a fine saddle point, if they exist, must be centrally distributed in the vicinity of a focus. Also, if the quadratic system has a fine saddle point and a line solution not passing through this saddle point, then this system does not have a limit cycle. There are some other new results, different from the above results, which can be found in [303]. Moreover, we mention the papers [126], [214], and [295]-[297] on the study of quadratic systems.
II. Limit cycles and the typical integral and Darboux integral of differential equations. In the mid-fifties, N. F. Otrokov, K. S. Sibirski'i, and M. V. Dolov in the Soviet Union considered the system of equations
dxjdt = P(x, y),
dyjdt
= Q(x, y)
(16)
(P and Q are analytic functions) and its corresponding first-order partial differential equation
pal +Qaf =O.
(17) ax 8y They obtained a solution /o(x, y) of (17) which satisfies the initial condition [f(x, y)]., = c
{18)
in a neighborhood of a closed trajectory of {16) (where 8 is a segment of the line without contact x = xo +ac, y = Yo+bc, lei :::; co of lata point (xo, Yo) on it). Using the analytic function f(x, y) (generally speaking, it is multi-valued), obtained through all analytic continuations of the solution /o(x, y), as a tool, they studied the decomposition of multiple limit cycles and the existence of Darboux integral, and obtained some significant results, as follows. Suppose h (x, y) is a function obtained from fo(x, y) after continuation for one complete ~ound along a fixed direction in a neighborhood of l. Since on the segment 8 of the line without contact we have 00
[!l(x,y)].,
=w(c) =I:
"'fkCk
(19)
k=l
{here w(c) is a Poincare successor function), from the uniqueness of the solution we get It (x, y) = w(fo(x, y)), (20)
THEORY OF LIMIT CYCLES
406
and it is well known that 11 = exp h = exp
/,(P~ + Q~) dt = exp loT (P~ + Q~) dt.e)
(~1)
We call the analytic solution f(x, y) of (17) satisfying condition (18) a normal integral of system ( 16) along the closed trajectory l. It is well known that there exists a neighborhood S(e, l) of lin which either the equation
w(Jo)- fo
=0
(22)
has a unique solution (then lis a limit cycle of (16)), or the left side of (22) is identically equal to zero (then the neighborhood of l is completely filled with closed trajectories). Otrokov [277], [276] proved that l is an n-tuple limit cycle( 3 ) of (16) i( and only if there exists a first integral f(x, y) = c defined as above, where f(x,y) is single-valued along land has order n. Here "f(x,y) has order n on l" means that f and all its partial derivatives up to order n - 1 have period T with respect to t, and at least one nth order partial derivative is not periodic with respect to t. Hence it is easy to see that in this case any integrating factor J.L(x, y) of (16) on l is a single-valued analytic function of order n - 1. Moreover, Otrokov used the normal integral as a tool to solve, under given conditions, the problem of decomposition of multiple cycle of the original system (16) into single cycles with the aid of the perturbed system
dx dt
= P(x, y) + p(x, y),
dy
dt = Q(x, y)
+ q(x, y).
(23)
For example, in [276] he proved that if l is an n-tuple limit cycle of {16), and we take n-1
p(x, y)
=L
n-1
.\knk(x, y),
q(x, y)
=L
k=O
).kf3k(x, y),
k=O
where nk(x, y) and !3k(x, y) are analytic functions, and the ).k are small independent parameters, then no matter how p and q are chosen, l can at most be decomposed into n single cycles of (23). In [277) he further proved that we can take p and q to be rational functions such that the above conclusion remains the same. Dolov [278] first proved that equation (17) has a real analytic integral, whose first and second branches satisfy the simple relation:
F1(x,y) = Fo(x,y)exph.
(24)
(2) Here we suppose that the equations of l: x = cp(t), y = tl>(t) have period T with respect to t. ( 3 )I.e., in (19} we have '"'11 = l,-y2 = '"'13 = ·· · = '"'fn-1 = 0, and "'n ::/; 0.
§18. APPENDIX
407
This kind of integral is called a typical integral. He then proved: For a single limit cycle l of system (16), there exists a neighborhood S(l, c) such that the equations (25) aQ - (3P = P~ + Q~ and a~
+ (3~ = 0
(26) possess unique single-valued analytic solutions a( x, y) and (3( x, y). But if for every trajectory l we have h = 0, then these a and (3 exist if and only if all the trajectories of system (16) belonging to S(l, c) are closed. From this we can see that the existence of a and (3 is characteristic for a limit cycle to be single. Also, from (26) and the equality h=
i
a dy - (3 dx
we know that a and (3 in the region enclosed by l must have a singular point. For a and (3 which satisfy (25) but not (26), Dolov [278] proved that if in a simply connected region G there exist single-valued continuously differentiable solutions a art
THEORY OF LIMIT CYCLES
408
Dolov in [280] discussed the problem of whether there exists a typical integral in a neighborhood of a separatrix cycle h passing through a saddle point, and obtained several similar results as in [279]. Here we use the integral ht = /_:
[P~ (rp1 (t), 1/J1 (t)) + Q~(rpt (t), 1/Jt (t))] dt
to replace h in (21), where x = rp1(t) andy= 1/Jt(t) are the equations of l 1. In [281] Dolov studied the polynomial system
dxjdt
= y + p(x, y),
dyjdt = -x + q(x, y),
(27)
with (0, 0) as its center. Suppose it has a Darboux integral k
II 41j; = c
(28)
j=l
(where the 41; are polynomials of complex coefficients, the /3; are complex numbers, and 41; and 41, have no common factors when j '#l). He proved that if in this case (27) still has a limit cycle l, then l must be a closed branch of a real algebraic curve, this cycle must be structurally stable, and the irreducible polynomials which determine l must be contained in (28). If l is a single cycle, then in a neighborhood of l there exists a typical integral F(x, y, l), which is analytic in a neighborhood of the origin, and F(O, 0, l) ::1 0. Later, in [282] Dolov studied the formula for the typical integral corresponding to l in the case when a single limit cycle l of system (16) contains a unique coarse focus in its interior. He also proved that if in this case the system has a Darboux integral (28), and also has a limit cycle, then there are two algebraic curve solutions passing through the coarse focus, and they are complex conjugates. In [283] Dolov proved that if system (16) has a limit cycle l whose interior contains a unique singular point which is a nodal point 0, and the ratio of its two characteristic roots is not an integer, then (16) does not have a Darboux integral. Moreover, he gave some examples to show that Sibirskil's conjecture in [286], "there exists a Darboux integral (28) in which the set of the totality of polynomial systems in which 41;(xt, Yt) ::1 0 is dense in the space of all polynomial systems possessing elementary singular points of center-type" is incorrect. In [284] Dolov proved that for a quadratic system (16) with a fine focus, the Darboux integral cannot exist and the system cannot have more than two different irreducible algebraic curve solutions passing through this fine focus which are complex.
§18. APPENDIX
409
Finally, in [285] Dolov studied the polynomial system (16) whose Darboux integral has the form k
G=
m
II ~:; exp II w:· = c
j=l
(29)
s=l
(where the /3; and n, are in general complex-valued, the ~j and W, are in general polynomials with complex coefficients, and there is no functional relatiOn between u = n~l w:• and r = n~=l ~:; 1 there are nO COmmOn factors among the ~;, and there are no common factors among the W,); and the problems he solved are similar to the previous several papers. P. S. Atamanov and V. P. Zakharov in 1976 proved that the single-valued integrating factor M(x, y) of system (16) is unbounded near a limit cycle l. Dolov [287] further studied the relation between the order of increase of M(x, y) and the multiplicity of l. Since recently the study of polynomial systems, especially in the theory of limit cycles of quadratic systems, has become much deeper, we feel that the above work should be especially mentioned.
m. Equations of limit cycles.
For some artificially constructed systems
such as
(30) we can write down not only the equation of the limit cycle x2 + y 2 also its first integral
= 1, but (31)
and the formula for its solution. Many similar examples can be constructed. However, for the quadratic system studied in §11,
~: = -y(ax + {3y + 1)- (x 2 + y 2 -1), dy dt
(32)
= x(ax + /3y +"f),
we can only write down the equation of its limit cycle x 2 + y 2 = 1 (when > a 2 + {32 ) and the first integral of (32), but not the formula for its solution. As for other equations of class I studied in §12, although we have used qualitative methods to prove that its limit cycle, if it exists, must be unique, we cannot even write down the equation of the limit cycle. 12
410
THEORY OF LIMIT CYCLES
When the equation of the limit cycle of a dynamical system can be written down, not only can we determine the precise or approximate position of its limit cycle, but also we can study, when the limit cycle disappears owing to the variation of the parameter in the differential equations, whether it runs to the two-dimensional complex space. [269] gave examples to show that this situation is just like the case when a pair of real roots in an algebraic equation with real coefficients coincide and disappear (because of the variation of the coefficients): we can then obtain a pair of conjugate complex roots. When the limit cycle in the real plane shrinks towards the singular point in its interior, or coincides with another limit cycle and then disappears, we can often get it back in the two-dimensional plane or two-dimensional integral manifold of the fourdimensional vector space corresponding to two-dimensional complex space. Hence the study of equations of limit cycles is an important and interesting problem. For this problem, even for the limit cycle of the well-known van der Pol equation, in the past mathematicians have confined themselves to obtaining the first one or two terms of the approximate representation formula by approximation methods when the parameter is very small. Sun Shun-hua [55] first noted that if in the system
dxjdt
= y,
dyjdt
= -x + J.t(l- x 2)y
(J.t ::/: 0)
(33)
we write J.t as J.t 2 , and apply the transformation x' = J.tX, y' = J.tY, and get (we still denote x', y' as x, y) (34)
dxjdt = y,
then the system of equations forms a family of rotated vector fields with respect to JJ. 2 • As far as system (34) is concerned, when JJ. 2 changes from 0 to positive, the limit cycle is generated because of change of stability of the origin, not because of the circle of radius 2 as in (33). In the complete family of rotated vector fields, a closed trajectory monotonically expands or contracts according to the variation of its parameter and trajectories of the system
dx/dt
= P(x, y, JJ.),
dyjdt
= Q(x, y, J.t)
(35)
corresponding to different parameters do not intersect. Hence in the region G covered by the changing trajectories whose union and outer boundaries all contain singular points, we can define a function JJ.(x, y), whose value at the point (x, y) is the value of J1. of the system (35) which has a limit cycle passing through this point, and J.t(x, y) = JJ.(xo, Yo) = JJ.o is obviously the
§18. APPENDIX
411
equation of all the closed trajectories of {35) 110 , one of which passes through the point (xo, Yo). Sun [55] proved that when P and Q in {35) are continuously differentiable with respect to x, y, and J.t, this function is also corrfiiiuously differentiable in G, and satisfies the first order quasilinear partial differential equation a~-'
Q(x, y, J.t(x, y)) ay
a~-' + P(x, y, J.t(X, y)) ax = 0.
{36)
Finally, seeking the equation of the limit cycle of system (34) in the form 00
1-'2
= ~ Ok(cp)p2k k=l
in polar coordinates (where nk(cp) is a periodic function of period 211"), and returning to system (33), we obtain the equation of the unique limit cycle of {33) in power series form: 1-
~p2
+:,.
+ 16~4
(2sin2cp
+
sin4cp)p4
( 3 cos 6cp + 5 cos 4cp + 5 cos 2cp -
2!~4) + ... = 0
(37)
p6
(which converges when 11-'1 « 1). L. A. Cherkas [288] obtained system (36) again under the condition that P and Q are real analytic with respect to x, y, and J.t, and proved that in this case J.t(x, y) is also real analytic in G. Moreover, he sought the limit cycle 1-'2 = F(x,y) = !(x2 + y2) + F3
+ F4 + · · ·
in the form of a power series solution for the partial differential equation corresponding to system (34), and got F _ 1( 2 2) 1 3 1 a 1 2 4 - 4 x + y - Bx y- 6144 y - 2048 x y
+
+
143 x4 2 - -...!:.!._xs 1 3 5 385 5 3 2048 y 6144 + 4608 X y - 1152 X y terms of higher degrees.
(38)
Returning to (33), he got the equation of its unique limit cycle as 1( 2 2) 1 3 1-'2 0- 1- 4 x + Y + Bp,x Y + 6144(y6 + 3x2y4- 429x4y2 1-'3
+ 6144
(6160 5 3 4 3 5 ) -3-x y - 3x Y -+- .•.•
+ 17x6) (39)
412
THEORY OF LIMIT CYCLES
Ye Wei-lin [289], independently of [288], applied the transformation x'
=
J.I.X, y' = J.I.Y to the Rayleigh equation equivalent to (33)
dyfdt
= x,
dxfdt = -y + J.1. 2 (x- x3 /3)
(40)
and obtained (we still denote x' and y' by x and y) dyfdt
= x,
(41)
and then assumed the equation of the limit cycle of (41) to be J.1. 2 = F(x, y). From [55], we know that F(x, y) satisfies the partial differential equation
aF ( -y+xF-x ) -=0. aF x-+ By 3 ax 3
(42)
Suppose F(x, y) can be expanded into a power series F(x, y)
= a1x + b1y + a2x 2 + ~xy + c2y2 + · · ·,
and substitute it into (42). Then Ye [289] proved that all the coefficients in the expansion formula can be uniquely determined. All the coefficients up to the terms of 8th degree in x and y were computed, and the equation x2+y2 1 J.l. 2 = { + - - (29x6 4 18432 +
X
-
9x4 y2 - 153x2y4
3 { 1 ( -46 2 2 34 y 24 + 18432 X y - 18432 y
-
51y6 ) + · · ·
4) + .. •}
} (43)
= S1(x) + x 3 y82(x) was obtained. Since there was no way to obtain the formula for the general term of the coefficient, he used a computer to compute the approximate values of all the terms up to the homogeneous expression of 20th degree, and observed that S1(x) and S2(x) seem to have majorant series sinh(x2 +y 2) and cosh(x2 + y2). Hence (43) seems to be convergent in the whole plane. Ye Wei-lin [289] also discovered that the calculation of the fourth term on the left side of (37) was wrong: the correct answer should have been J.l.2 (7 1024 2 + cos 2
s
s]
413
§18. APPENDIX
From this we can see that the terms of 8th degree in (39) also have calculation errors. Finally, it is worth pointing out that the method of G. V. Kamenkov used in Chapter 14 of [290] to seek the equation of the limit cycle is not as simple and convenient as the method of undetermined coefficients introduced before in the case of the family of plane rotated vector fields. Moreover, from the theorems concerning equations of class I in §12 of this book, we know that the conclusion of §5 in Chapter 14 of [290] concerning the existence of a limit cycle for the system of equations dx dt = y,
at b = 0 is wrong.
dy [ 2 dt = -x + 1-' ax+ 2by + a2ox
+ a11 xy + ao2y2]
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=
I:
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=
I:
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qoo + q10x + Qo1Y + q2ox 2 + quxy + Qo2Y2
= Poo + P10X +PolY+ P2ox 2 + PuXY + Po2Y2 .
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=
y + aox2 + a1xy + a2y2 x+box 2 +b1xy+~y 2 '
(1)
Differentsial'nye Uravneniya 2 (1966), 353-360=Differential Equations 2 (1966), 174-178. 168. I. S. Kukles and M. Khasanova, On the distribution of singularities of the first and second group, lzv. Vyssh. Uchebn. Zaved. Mat. 1964, no. 6(43), 88-97. (Russian) 169. A. N. Berlinskil, On the behavior of the integral curves of a differential equation, lzv. Vyssh. Uchebn. Zaved. Mat. 1960, no. 2(15), 3-18. (Russian) 170. A. A. Andronov, E. A. Leontovich-Andronova, I. I. Gordon and A. G. Ma.ler, Qualitative theory of second-order dynamical systems, "Nauka", Moscow, 1966; English transl., Israel Program Sci. Transl., Jerusalem, and Wiley, New York, 1973. 171. [Same as [152].] 172. Cai Sui-lin, Remark on [152], Zhejiang Daxue Xuebao 1979, no. 4, 105-113. (Chinese) 173. Shi Song-ling, Topological classification for general quadratic vector fields without limit cycles, Report at Third National Conf. Differential Equations. (Chinese) 174. Liang Zhao-jun, Nonexistence and phase portrait of singular cycles of a quadratic system of type I (unpublished). (Chinese)
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I
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•+k=n
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-(2>.x + 1)(ax +by+ c)+ a(>.x 2 + y 2 + x + 1) 2y(ax +by+ c)+ .B(>.x2 + y2 + x + 1)
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