Journal of Dynamics and Differential Equations, Vol. 15, Nos. 2/3, July 2003 (© 2004)
A Biographical Sketch of Victor A. Pliss Sergei Yu. Pilyugin 1 and George R. Sell 2
The 1960s saw many major changes in the global mathematical community. In this post-Sputnik decade, we witnessed the beginnings of an important exchange program between mathematicians (and other scientists) in the Soviet Union and the United States. Among the first Russian mathematicians to make an extended stay in the United States was Victor Pliss from Leningrad State University, now St. Petersburg State University. Researchers in dynamical systems and differential equations were delighted that Pliss was able to accept the invitation from the United States for this visit. At that time, he was well known for his theory of the Reduction Principle, which was later labelled the Center Manifold Theorem. Today, many experts consider this work to be one of the most prominent discoveries in dynamical systems in the last sixty years. Since its inception, this theory has grown in importance and significance. It has now become a central topic in hundreds, if not thousands, of works, and it is often used without citation. It is with pleasure that we dedicate this issue of the journal to a uniquely gifted scientist and mathematician, Victor Aleksandrovich Pliss. Pliss was born on February 10, 1932, in Syktyvkar, a small industrial city in the north of Russia. The 1930s were awful times for the Russian people, and especially for educated Russians. Pliss’ father was a prominent Leningrad chemist. He was also a political prisoner and was made to work at one of the new giant factories during this period. After World War II, Pliss returned to Leningrad, were he continued his studies at the Leningrad High School No. 181. He also met the charming Lilya Benois, whom he married in 1954. This marriage was destined to be long and happy, with two children and six grandchildren. 1 2
St. Petersburg State University, St. Petersburg, Russia. University of Minnesota, Minneapolis, U.S.A. E-mail:
[email protected] 225 1040-7294/03/0700-0225/0 © 2004 Plenum Publishing Corporation
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In his university studies, Pliss was a student of mechanics at Leningrad State University. Although his main scientific contributions are in mathematics, his training in mechanics had undoubtedly influenced the choice of his fields of research. Pliss received his Ph.D. in 1957 under the supervision of Professor Nikolai P. Erugin, a well-known specialist in differential equations. In 1959, Pliss earned his Doctor of Science degree, the highest degree available in Russia. It is equivalent to the Habilitation Degree in Western Europe. Pliss became a member of the Faculty of Mathematics and Mechanics at Leningrad in 1956, and in 1960, at age 28, he succeeded Erugin as the Chair of Differential Equations. The position of Chair is an important position of academic leadership within the Russian university system. In 1990, Pliss was elected a corresponding member of the Academy of Sciences of USSR, now the Russian Academy of Sciences. In his research, Pliss has demonstrated an outstanding ability to focus on the central problems in the theory of dynamical systems and differential equations and to develop new methods and new approaches. Let us now turn to a brief description of his mathematical contributions. The Reduction Principle. The basic issue behind the Reduction Principle (Center Manifold Theorem) arises with the system x˙=Ax+X(x, y),
y˙=By+Y(x, y),
(1)
where x and y are vectors, the eigenvalues of the matrix A have zero real parts, while the eigenvalues of the matrix B have nonzero real parts. Also the functions X and Y are assumed to be 0 at the origin (x, y)=(0, 0) and to have small Lipschitz constants near the origin. The problem is to find a function y=f(x) with two properties: • the dynamics of the ‘‘reduced problem’’ x˙=Ax+X(x, f(x))
(2)
in the vicinity of x=0 are the same as the dynamics of the original problem (1) near the origin; and • x(t) is a solution of (2) near x=0 if and only if (x(t), y(t)) is a solution of (1) near the origin, where y(t)=f(x(t)). In the language of the 1960s, the dynamical properties were typically described in terms of various Lyapunov stability concepts. The related theory of Pliss appears in his 1964 paper [1].
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In a recent lecture, Pliss related that his motivation for developing the mathematics that led to the Reduction Principle came in 1963, with the discovery of a major unpublished manuscript of A. M. Lyapunov, see [2]. In this manuscript, Lyapunov had made an in depth study of Eq. (1), under the assumption that the matrix A is a nonzero, 2 × 2, nilpotent matrix. As it happened, Lyapunov was unable to complete this work. A very special case escaped his analysis. In the paper [3], the ‘‘missing link of Lyapunov has been successfully completed.’’ (This quotation is from the Mathematical Review by S. Lefschetz.) The paper [3] contains the first ‘‘successful’’ application of the Reduction Principle. As we now know, it was to be followed by many, many other applications. Control Theory. Even before Pliss did his seminal work on the Reduction Principle, he had become well-known for his solution of the Aizerman problem in control theory. This problem concerns the equation x˙=Ax+F(x),
(3)
where A is an n × n matrix, and F is an n-vector with nonzero component, F1 , and F1 =f(xk ), for some k with 1 [ k [ n. It is assumed that there exist constants a < b such that the zero solution of system (3) with f(x)=hx is globally asymptotically stable for any h ¥ (a, b). The problem is: • to describe the region W in the parameter space {A=(aij )}, with the property that for any function f satisfying the conditions ax 2 < xf(x) < bx 2,
for x ] 0,
(4)
the null solution of (3) is globally asymptotically stable. By using new topological methods, Pliss gave a complete solution of the Aizerman problem for n=3. The parameter space {A=(aij )} is divided into two nonempty regions, W and its complement W c. If A ¥ W c, then there is a function f satisfying (4) where the origin fails to be globally asymptotically stable. Furthermore, W c is divided into two nonempty parts W c1 and W c2 . If A ¥ W c1 , there exist nonlinearities, where (3) has solutions that grow to infinity. If A ¥ W c2 , there exist nonlinearities, where (3) has nontrivial periodic solutions. These results are described in the 1958 book of Pliss [4]. Theory of Structural Stability. Pliss was one of the first to understand the importance of results of Cartwright and Littlewood [5, 6] and of Levinson [7] concerning the dynamical significance of the complicated
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behavior of solutions of the periodically forced Van der Pol equation, for example, d 2x +k(x 2 − 1) x˙+x=blk cos(lt). dt 2
(5)
In particular, Pliss included a detailed analysis of the Levinson equation, which is a simplified variant of (5), in his 1964 book [8]. At the end of the 1960s, one of the main problems in the theory of dynamical systems was to determine necessary and sufficient conditions for structural stability. J. W. Robbin and C. Robinson had shown that the Axiom A property with the strong transversality condition are sufficient conditions for structural stability. It remained to determine whether these conditions are also necessary. The first nontrivial step in the proof of necessity appears in the 1972 paper of Pliss [9]. In particular, he confirmed the Smale conjecture that any structurally stable system has only a finite set of stable periodic solutions. This paper of Pliss proved to be very important, because the methodology introduced in that work became widely used in the subsequent studies of the necessary conditions for structural stability. Pliss also constructed an example of a system of differential equations having an infinite set of stable periodic solutions such that these solutions belong to a bounded set and their Lyapunov exponents are separated from zero, see [10]. Of course, such a system cannot be structurally stable. The main achievement of Pliss in the study of structural stability appeared in [11, 12], where he studied the Poincaré transformation T generated by a two-dimensional time-periodic system of differential systems. Pliss maked the assumption that the nonwandering set W of T satisfies m(W)=0, where m is the two-dimensional Lebesgue measure. Under this assumption, he proved that the Smale conditions are necessary for structural stability. In order to do this, Pliss developed a method of structurally stable sequences of linear periodic systems of differential equations [13] and a variant of the closing lemma [14]. These results are also treated in his 1977 book [15]. 3 In this book, Pliss considered an important class of systems satisfying the following basic condition: there exists a C 1-neighborhood U of the given system such that any periodic solution of any system in U has only nonzero Lyapunov exponents. It is easily seen that any structurally stable system satisfies this basic condition. For a two-dimensional time-periodic system of differential 3
Unfortunately, the American Mathematical Society did not include this very important and significant book in its Translation Series.
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equations satisfying this basic condition, a problem, concerning the complexity of an attractor of the Poincaré map T, was solved. In particular, it was shown that an attractor A of T with m(A)=0 contains an indecomposable continuum if and only if T has infinitely many periodic points. The methods developed by Pliss allowed him to find necessary and sufficient conditions of structural stability for nonautonomous problems x˙=F(t, x)
(6)
x˙=G(t, x).
(7)
and
In his 1980 paper [16], Pliss described the necessary and sufficient conditions under which system (6) has the following shadowing property: given E > 0 there exists d > 0 such that if r1 (F, G) < d, where r1 is a suitable C 1-metric, then one has • for any solution x(t) of system (6) there exists a solution t(t) of system (7) such that |x(t) − t(t)| < E,
t ¥ R;
(8)
• and for any solution t(t) of system (7) there exists a solution x(t) of system (6) such that inequality (8) holds. It is shown in [17] that if system (6) is time-periodic, then the conditions of [16] coincide with the Smale conditions. Pliss has analyzed the underlying structure of the Smale conditions. The first of the conditions, the Axiom A property, consists of the following two requirements: (1) the nonwandering set W is hyperbolic and (2) the periodic points are dense in W. It is shown [18] that if the second Smale condition (the strong transversality condition) holds, then the density of periodic points in W follows from the hyperbolicity of W. We also note an important uniformity property of hyperbolic structures discovered by Pliss. Let T be the Poincaré diffeomorphism of a structurally stable time-periodic system of differential equations. Let W s(p) and W u(p) be the stable and unstable manifolds of a nonwandering point p. It is shown in [19] that there exists a positive constant a having the following property: given E > 0 one can find d > 0 such that if for two points p, q ¥ W, there exist points x ¥ W s(p) and y ¥ W u(q) satisfying the inequality dist(x, y) < d, then W s(p) and W u(q) contain disks D s and D u of size a, respectively, such that D s and D u intersect at a point z satisfying the inequalities dist(x, z) < E and dist(y, z) < E and the angle between the
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tangent spaces of D s and D u at z is greater than a. This result is used in many applications. Unfortunately, there are very few known systems of equations which are both (1) models of real-world problems and (2) structurally stable. In this connection, Pliss analyzed a class of models that arise in radioelectronics, mx˙=F(x)+f(t), where x is a vector, m is a small parameter, and the function f is periodic. (This equation is closely related to (5), when the parameter b is large.) Pliss has shown that, under reasonable assumptions, this system is structurally stable with quasi-chaotic dynamics, see [20–22]. Systems with Invariant Measure. Let f: R × M Q M be a flow on a compact metric space M and let m is an ergodic invariant measure of f, with m(M)=1. Let f: R × M Q Gl(n) be a smooth (in t) cocycle over f. It follows from the multiplicative ergodic theorem that there exist numbers l 1 ,..., l n such that the Lyapunov exponents of the cocycle F equal l 1 ,..., l n for m-almost every point of M. In his paper [23], Pliss assumed that l i ] 0,
for each i=1,..., n.
(9)
He then showed that the cocycle F is hyperbolic on a set of measure arbitrarily close to 1 in the following sense: Let the Lyapunov exponents be ordered so that l 1 [ · · · [ l k < 0 < l k+1 [ · · · [ l n , and set l=12 min |l i |. It is shown in [23] that for any E > 0 there exists a measurable subset MŒ … M and a number a > 0 having the following properties: m(MŒ) > 1 − E, and for any point p ¥ MŒ there exists a k-dimensional linear subspace S(p) and an (n − k)-dimensional linear subspace U(p) such that |F(t, p) v| [ a |v| exp(−lt),
for v ¥ S(p), t \ 0,
|F(t, p) v| [ a |v| exp(lt),
for v ¥ U(p), t [ 0.
In the joint paper [24], S. Yu. Pilyugin and Pliss continued to study the behavior of a diffeomorphism with an ergodic invariant measure and nonzero Lyapunov exponents. It is shown that the phase space can be approximated with arbitrary accuracy by a finite number of hyperbolic periodic points connected by transverse homoclinic contours. Let f be a diffeomorphism of a closed manifold M such that the Lebesgue measure on M is ergodic and invariant. For fixed E > 0, the set S={pi : i=1,..., N; si, j : i, j=1,..., N} is an E-structure for f if • the pi are hyperbolic periodic points of f;
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• the set {pi : i=1,..., N} is an E-net of M; • any point si, j is a point in the (transverse) intersection of the stable manifold of pi and the unstable manifold of f nj (pj ) for some nj . It follows from the definition that the trajectories of the points si, j form a transverse homoclinic contour of f. Thus, there exists a closed topologically transitive f-invariant set containing the points pi (hence, E-dense in M). In addition, if S is an E-structure of f, then any diffeomorphism C 1-close to f has an E-structure close to S. It is shown in [24] that if condition (9) holds for the Lyapunov exponents l i of the diffeomorphism f, then f has an E-structure for any E > 0. Foliated Invariant Sets and Weak Hyperbolicity. In a series of papers [25–27], Pliss and G. R. Sell presented a study of small C 1-perturbations of a vector field in the neighborhood of a ‘‘foliated’’ invariant set K. An invariant set K for the system x˙=F(x),
x ¥ X,
(10)
is said to be foliated if it is a compact invariant set in X with the following properties: • K is the union of the family of leafs {S}, where each leaf is an invariant set for (10). Also each point x ¥ K is on a unique leaf S=S(x), and in the vicinity of the point x, S(x) is a smooth k-dimensional disk. Moreover, k is a constant, that is, it does not depend on x. We let Tx denote the k-dimensional tangent space to S at the point x. • The induced linearized skew-product flow has an exponential trichotomy over K, where the normal flow, at each point x in any leaf S, dominates the tangential, or neutral, flow along the leaf. (This property is a variation of normal hyperbolicity.) • The mapping x Q Tx is a Lipschitz continuous mapping on K. Examples of foliated invariant sets include: (1) an arbitary finite set product of Anosov flows, and (2) the set product of a normally hyperbolic compact invariant manifold with an Anosov flow. The finite dimensional problem, for example X=R n, is studied in [25, 26]. It is shown that, under reasonable assumptions, if G(x) is C 1-small in a neighborhood of K, then there is a homeomorhpism h: K Q X, such that h(K) is a foliated invariant set for the perturbed equation x˙=F(x)+G(x). The first step in extending this methodology to the infinite dimensional
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setting, where X is now a Banach space and the Eq. (10) is a suitable nonlinear evolutionary equation, is presented in [27]. In particular, it is shown that if K is a normally hyperbolic, compact invariant manifold, then there is a homeomorphism h, as described above. In this context, the theory applies, for example, to the study of bifurcation phenomena seen in the Navier–Stokes model of the Couette–Taylor flow. Other Topics. Together with Yu. N. Bibikov, Pliss has studied systems of second order differential equations without dissipation: d 2xi +l 2i xi +g(x1 ,..., xn )=0, dt 2
i=1,..., n,
(11)
in [28]. It is shown that if system (11) is time-reversible, then ‘‘most’’ points of a small neighborhood of the origin belong to invariant tori. Pliss also investigated planar diffeomorphisms having invariant sets of complicated structure. He has discovered deep relations between the arithmetical structure of the Birkhoff rotation numbers and the existence of periodic points of large periods, see [29] and [30]. Also, Pliss investigated the structure of the boundary of an attractor [31]. In a joint paper, Pilyugin and Pliss gave a complete description of the boundary of an attractor for a Morse–Smale system [32]. Pliss is not only a world-renowned researcher but also a gifted teacher and mentor. About 50 persons have received their Ph.D. degrees under his direction. Moreover, 9 of them (Yu. N. Bibikov, N. N. Petrov, S. Yu. Pilyugin, Zh. M. Myrzaliev, V. E. Chernyshev, Yu. V. Churin, I. M. Anan’evsky, O. A. Ivanov, and V. G. Romanovsky) have become Doctors of Science. Along with his marvelous sense of humor, his love of paintings and the arts are traits which are well known to the people closest to him. People who have had the experience of being with him at the Hermitage in St. Petersburg, for example, have marveled at his knowledge and his ability to comment in depth on works of the great masters, be it a landscape by van Ruisdael, the Prodigal Son by Rembrandt, or the Benois Madonna by Leonardo. He is indeed an avid student of the arts. S.Yu.P. My contact with V.A. began in 1966 when he was the lecturer in the basic course in differential equations at Leningrad University. I (Sergei) was deeply impressed by his personality combining the intellectual charm of a high-level mathematician and the physical strength of a trained sportsman. Later V.A. became my scientific advisor, and our cooperation has continued successfully over the last 35 years.
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The influence of V.A. on my life has been deep and fruitful. I am grateful to him for the great opportunity to discuss various problems in differential equations and dynamical systems. I have been, and I continue to be, amazed by the speed at which he understands new ideas. Of equal importance, V.A. gave me a real appreciation of the great Chebyshev– Lyapunov tradition of mathematics in St. Petersburg. This is a tradition which holds that one should not do mathematics for the sake of mathematics, but instead to do mathematics arising from applications. In fact, V.A. was my ‘‘teacher of life,’’ and now I see with definiteness how much he gave me in the study of mathematics, as well as in Russian history, Dutch art, and European soccer, for example. G.R.S. I (George) met Victor for the first time during the 1967–1968 academic year when he visited the University of Southern California in Los Angeles. While he was there, he gave a beautiful and challenging series of lectures on his recent work in dynamical systems. All of us in the seminar, H. A. Antosiewicz, R. J. Sacker, W. T. Kyner, and I benefited greatly from his visit. It was there that I really began to appreciate Victor, the mathematician. I also began to know Victor, the man. He was very enthusiastic about his skills as a boxer. In his youth, he became a boxing champion of Leningrad for his age group. During his California visit, he went to a local (boxing) gymnasium and volunteered to be a sparring partner. Also, on one memorable occasion, after a party in his honor, he treated a small group of the mathematicians to a demonstration of shadow-boxing under the street lights of Santa Monica! During the 1970s and the 1980s, it was extremely difficult for Russian and American mathematicians to maintain scientific contacts. Fortunately, there was a very valuable series of Equadiff Conferences, alternating among Bratislava, Brno, and Prague in Czechoslovakia. We in the West found that it was easier for the Russian mathematicians to get permission to go to these conferences, than to meetings held elsewhere. I took advantage of these opportunities to go and meet with Victor and many others. These were great meetings, with good mathematics (of course), but also good food, good beer, good wine, and very good fellowship. These conferences also showed me Victor’s concern for other people. At the social events, one found persons of all backgrounds, including Russian speakers who did not know English, and folks like me who have great difficulty with the Russian language. Victor very graciously would interrupt the speaker and say, ‘‘please let me translate.’’ By doing this, back and forth, he kept everyone in the conversation, and he made me feel like a member of the family, and not a stranger.
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In the 1980s, I was able to participate on two occasions in the Academies of Sciences Exchange Program, with extended stays in Leningrad. This created an opportunity for Victor and me to have serious discussions about our current research interests. In 1989–1990 Victor came to the Institute of Mathematics and its Applications for a year-long visit. It was at this time that our joint work came to fruition. The academic year 1989–1990 also coincided with a period of great political change in Europe. The wall came down! This has benefited many. Since that time Victor and 1 have exchanged visits in St. Petersburg and Minneapolis. During these visits, I was privileged to meet many students and colleagues of Victor, including Erugin and Pilyugin. Also, I experienced the warmth of genuine Russian hospitality. Upon arriving in St. Petersburg, I was quickly invited to the home of Victor and Lilya for ‘‘dinner,’’ which is actually a great Russian feast. The visits begin and end with a feast, but there is much more. Over the years I have had the good fortune to be taken to many cultural events: operas, ballets, concerts, museums, palaces, etc. (I cannot even count them all.) I have learned to love the city of St. Petersburg, which is certainly one of the greatest centers of culture in the world. There were also memorable visits outside: to Petrovoretz, to Novgorod, and to the dacha. Spasibo bol’shoe za vsë, Victor. Happy Birthday to my very good Friend, Victor. May you be blessed with many, many pleasant returns of the day. REFERENCES 1. Pliss, V. A. (1964). A reduction principle in the theory of stability of motion. Izv. Akad. Nauk SSSR Ser. Mat. 28, 1297–1324. 2. Lyapunov, A. M. (1963). A Study of One of the Special Cases of the Problem of Stability of Motion [in Russian], Izadt. Leningrad Univ., Leningrad, 116 pp; (1966). Stability of Motion [in English], Academic Press, New York/London, 203 pp. 3. Pliss, V. A. (1964). Investigation of a transcendental case in the theory of stability of motion. Izv. Akad. Nauk SSSR Ser. Mat. 28, 911–924. Also see the English version in [2]. 4. Pliss, V. A. (1958). Some Problems in the Theory of Stability of Motion in the Large, Leningrad University. 5. Cartwright, M. L., and Littlewood, J. E. (1945). On nonlinear differential equations of the second order: I. The equation y¨ − k(1 − y 2 y˙) y˙+y=blk cos(lt+a), k large. J. London Math. Soc. 20, 180–189. 6. Cartwright, M. L., and Littlewood, J. E. (1947, 1949). On nonlinear differential equations of the second order: II. The equation y¨+kf(y) y˙+g(y, k)=p(t)=p1 (t)+kp2 (t). Ann. Math. 48, 474–494; 50, 504–505. 7. Levinson, N. (1949). A second order differential equation with singular solutions. Ann. Math. 50, 126–153. 8. Pliss, V. A. (1964). Nonlocal Problems in the Theory of Oscillations, Nauka, Moscow.
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9. Pliss, V. A. (1972). On a conjecture of Smale. Differ. Uravn. 8, 268–282. 10. Pliss, V. A. (1974). A system of differential equations having an infinite number of periodic solutions. Differ. Uravn. 10, 2179–2183. 11. Pliss, V. A. (1971). Position of separatrices of saddle periodic movements for systems of differential equations of second order. Differ. Uravn. 7, 1199–1225. 12. Pliss, V. A. (1972). Behavior of solutions of a periodic system of two differential equations having an integral set of zero measure. Differ. Uravn. 8, 553–555. 13. Pliss, V. A. (1971). Structural stability of a sequence of linear systems of differential equations of second order with periodic coefficients. Differ. Uravn. 7, 261–270. 14. Pliss, V. A. (1971). A variant of the closing lemma for differential equations. Differ. Uravn. 7, 840–850. 15. Pliss, V. A. (1977). Integral Sets of Periodic Systems of Differential Equations, Nauka, Moscow. 16. Pliss, V. A. (1980). Stability of an arbitrary system with respect to C 1-small perturbations. Differ. Uravn. 16, 1891–1892. 17. Pliss, V. A. (1981). Relation between various conditions of structural stability. Differ. Uravn. 17, 828–835. 18. Pliss, V. A. (1972). Analysis of neccesity of Smale’s and Robbin’s conditions of structural stability for periodic systems of differential equations. Differ. Uravn. 8, 972–983. 19. Pliss, V. A. (1984). Position of stable and unstable manifolds of hyperbolic systems. Differ. Uravn. 20, 779–785. 20. Pliss, V. A. (1990). Existence of a hyperbolic integral set of a special periodic system. Differ. Uravn. 26, 800–808. 21. Pliss, V. A. (1990). Nonwandering set of a special periodic system. Differ. Uravn. 26, 966–975. 22. Pliss, V. A. (1991). Structural stability of a periodic system of differential equations. Differ. Uravn. 27, 2077–2081. 23. Pliss, V. A. (1986). Hyperbolicity of smooth cocycles over flows with invariant ergodic ˇ asopis pro pestov. matem. 111, 146–155. measure. C 24. Pliss, V. A., and Pilyugin, S. Yu. (1999). Existence of persistent structures for diffeomorphisms with ergodic invariant measure. Differ. Uravn. 35, 116–120. 25. Pliss, V. A., and Sell, G. R. (1991). Perturbations of attractors of differential equations. J. Differential Equations 92, 100–124. 26. Pliss, V. A., and Sell, G. R. (1998). Approximation dynamics and the stability of invariant sets. J. Differential Equations 149, 1–51. 27. Pliss, V. A., and Sell, G. R. (2001). Perturbations of normally hyperbolic manifolds with applications to the Navier–Stokes equations. J. Differential Equations 169, 396–492. 28. Bibikov, Yu. N., and Pliss, V. A. (1967). Existence of invariant tori in a neighborhood of the zero solution. Differ. Uravn. 3, 1864–1881. 29. Pliss, V. A. (1969). To the theory of invariant sets in periodic systems of differential equations. Differ. Uravn. 5, 215–227. 30. Pliss, V. A., and Chernyshev, V. E. (1970). Some properties of homeomorphic transformations of the plane into itself. Vestn. Leningr. Univ. 13, 62–68. 31. Pliss, V. A. (1969). Structure of asymptotically stable invariant sets of structurally stable autonomous systems of differential equations. Differ. Uravn. 5, 979–991. 32. Pliss, V. A., and Pilyugin, S. Yu. (1978). Boundary of a stable invariant set of a Morse– Smale system. Differ. Uravn. 14, 1997–2001.
