Chaos, Bifurcations and Fractals around Us: A Brief Introduction

CHAOS Bifurcations ond Fractals flround Us

A brief introduction

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series A vol. 47

NONLINEAR SCIENCE Series Editor: Leon 0. Chua

CHAOS

Bifurcations and Fractals firound Us

A brief introduction

Wanda szemplinska-stupnicka Institute of Fundamental Technological Research, Polish Academy of sciences

World Scientific NEWJERSEY

• LONDON • SINGAPORE • S H A N G H A I • H O N G K O N G • TAIPEI • BANGALORE

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CHAOS, BIFURCATIONS AND FRACTALS AROUND US Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic ormechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-238-689-0

This book is printed on acid-free paper. Printed in Singapore by Mainland Press

Contents

1. Introduction

1

2. Ueda's "Strange Attractors"

5

3. Pendulum 3.1. Equation of motion, linear and weakly nonlinear oscillations 3.2. Method of Poincare map 3.3. Stable and unstable periodic solutions 3.4. Bifurcation diagrams 3.5. Basins of attraction of coexisting attractors 3.6. Global homoclinic bifurcation 3.7. Persistent chaotic motion — chaotic attractor 3.8. Cantor set — an example of a fractal geometric object

11 11 18 20 24 28 33 39 46

4. Vibrating System with Two Minima of Potential Energy 4.1. Physical and mathematical model of the system 4.2. The single potential well motion 4.3. Melnikov criterion 4.4. Fractal boundaries of basins of attraction and transient chaos in the region of principal resonance 4.5. Oscillating chaos and unpredictability of the final state after destruction of the resonant attractor 4.6. Boundary crisis of the oscillating chaotic attractor 4.7. Persistent cross-well chaos 4.8. Lyapunov exponents 4.9. Intermittent transition to chaos 4.10. Large Orbit and the boundary crisis of the cross-well chaotic attractor

49 50 53 57

4.11. Various types of attractors of the two-well potential system

94

5. Closing Remarks

62 71 75 79 82 84 87

98

Bibliography

101

Index

105 v

Chapter 1

Introduction

When we observe evolution in time of various phenomena in the macroscopic world that surrounds us, we often use the terms "chaos", or "chaotic", meaning that the changes in time are without pattern and out of control, and hence are unpredictable. The most frustrating phenomena are those, which concern long-term weather forecasting. We can never be sure about the change of weather patterns. The temperature, barometric pressure, wind direction, amount of precipitation and other important factors come as a surprise contradicting predictions made a few days ago. Sometimes we are caught in a storm, sometimes in a heat-wave. The world stock market prices are also an example of a system that fluctuates in time in a random-like, irregular way, and the long-term prognosis does not often come true. The two examples mentioned belong to the category of huge and complicated dynamical systems, with a huge number of variables. The unpredictability of the evolution in time of these interesting events is intuitively natural. Simultaneously, it also seems natural that evolution of physical processes in simple systems, the systems governed by simple mathematical rules, should be predictable far into the future. Suppose we consider a small heavy ball, which can move along a definite track, so that the position of the ball is determined by a single coordinate. Due to Newton's Second Law, the motion of the ball is governed by the second order differential equation. The well known physical system the pendulum belongs to this class of oscillators. We were told that if the forces acting on the ball, as well as its initial position and velocity are given, one could predict the motion, i.e. the 1

2

Chaos, Bifurcations and Fractals Around Us

history of the system forever into the future, at least if the powers of our computers are big enough. The scientific researchers were taken by surprise, some of them were unable to agree with the idea that even this type of system may exhibit an irregular motion, sensitive to initial conditions and though unpredictable in time, the motion is labeled as chaotic. This book is aimed at presenting and exploring the chaotic phenomena in the single-degree-of-freedom, nonlinear driven oscillators. The oscillators considered belong to the class of dissipative deterministic dynamical systems. The term "dissipative" means that drag forces act on the ball during motion (aerodynamic forces, friction forces and others), so that the free oscillations always decay in time, and the undriven system tends to its equilibrium position. The other essential feature is that all the forces acting on the ball are determined in time. Such systems are labeled deterministic. For a long time, researchers were deeply convinced that deterministic systems always give a deterministic output. Early discovery of chaotic output in deterministic systems came into view in the field of mathematical iteration equations of the type xn+\=f(xn), n = 0,1,2, The formula states that the quantity x at the "instant of time" denoted n+\ can be calculated, if the previous quantity xn is known. Interpretation of the parameter n as "instant of time" is useful in applications. One of the fundamental models of this type has its roots in ecology. Ecologists wanted to know the population growth of a given species in a controlled environment, and to predict the long-term behavior of the population. One of the simple rules used by ecologists is the logistic equation xn+i = kxn(l -xn),

n = 0,1,2,

Here, the "instants" n = 0, 1, 2,.... correspond to the end of each generation. Using this formula one can deduce the population in the succeeding generation xn+i from the knowledge of only the population in the preceding generation xn and the constant k. The results obtained for a wide range of values of the constant k were surprising. As long as k did not exceed the value of about 3.5, the behavior of the population changed in a regular way. But at higher ^-values, in particular within the interval ~3.6 < k < 4.0, strange results were obtained. Namely, the consecutive

Introduction

3

values xo, xj, x2, , xn+i looked like an irregular, random-like process, whose essential property was that the fluctuations were sensitive to the initial value x0. Dynamical systems generated by the iterative formulae belong to the category of dynamical systems with discrete time. In contrast, the physical systems governed by differential equations are labeled as dynamical systems with continuous time. In the latter case, the sought changes in time of the values of position and velocity can be found by numerical integration of the equation of motion. Indeed, we can apply a numerical procedure that enables us to obtain discrete values of position and velocity. For instance, we may record the sought values in selected instants of time, say, at intervals equal to the period of excitation T. Thus, a series of sought quantities in the discrete time 0, T, 2T, 3T, , nT, .... would be obtained. Yet, we are not able to find an analytical iterative formula for the relation between the position and velocity values at the instant n+l as function of the previous values. That is why the analytical results obtained by mathematicians for dynamical systems with discrete time are not always applicable in the continuous time systems. Yet, the fundamental new concepts of nonlinear dynamics are common for both types of systems. The book is addressed to general Readers, also to those who, although are interested in the fascinating chaotic phenomena encountered in our every day life, do not have a solid mathematical background. To make the book easily accessible, we try to reduce the mathematical approach to minimum, and to apply a simplified version of presentation of the very complex chaotic phenomena. The Reader may even skip the portions of material where equations of motion are derived, and confine his/her attention to the presented physical model. Instead of a mathematical approach, the book is based on geometric interpretation of numerical results. The effort is focused on an explanation of both the theoretical concepts and the physical phenomena, with the aid of carefully selected examples of computer graphics.

4

Chaos, Bifurcations and Fractals Around Us

Some portions of the material, written in small fonts, give additional remarks and refer the Reader to the literature on the problem considered. These portions might be skipped by those looking for an overview of the field. The same simplified approach is applied to the fundamental concepts, as well as to the advanced problems recently published by the Author and her associates in international scientific journals (see references [19-24]). Application of the simplified way of presentation of very complex problems is rather risky. The difficulty is in finding a compromise between strict mathematical accuracy and accessibility of the material. It is difficult to explain the chaotic phenomena in clear and simple language while avoiding simplifications that may lead to incorrect interpretation. In the search of compromise I asked for help from two types of Readers. I sent some parts of the manuscript to the Readers who are interested in chaotic phenomena but do not want to go deeply into mathematics of chaos, and to those who are involved in the research on chaotic problems and also give lectures for undergraduate students. The question addressed to the first group was whether the material was easy to read and understand, whereas the second group were asked whether the work was clear and exact. I owe special thanks to Prof. J. Zebrowski and Prof. A. Okninski for their critical comments and remarks. They helped me a lot with revision of the manuscript. I deeply appreciate the contribution of Dr. E. Tyrkiel for her carefully worked out color computer graphics that have added an aesthetic dimension to the study of chaotic dynamics.

Chapter 2

Ueda's "Strange Attractors'

At the International Symposium "The Impact of Chaos on Science and Society" organized by the United Nations University and the University of Tokyo in 1991, Prof. Yoshisuke Ueda presented a paper entitled Strange Attractors and Origin of Chaos. "At present, people say that the data I was collecting with my analog computer on 27 November 1961, is the oldest example of chaos discovered in a second-order nonautonomous periodic system. Around the same time, it was Lorenz who made the discovery of chaos in a thirdorder autonomous system" — this is how Prof. Ueda began his talk. Ueda considered a nonlinear oscillator governed by the second-order ordinary differential equation subjected to periodic excitation. In fact, both researchers, Ueda and Lorenz, studied a three-dimensional dynamical system with continuous time, and their interest was focused on the evolution of the solution in time. In that time, about 1961, Ueda worked as a postgraduate student at the University of Tokyo under supervision of Prof. C. Hayashi. The question he tried to answer was: what types of steady-state oscillations can occur in nonlinear driven oscillators? The expected "steady-states" were first calculated by means of the analytical approximate methods. Consequently, student Ueda was supposed to obtain simulation results that confirm the theoretical ones. It happened that, just on that day in November 1961, the oscillation phenomena portrayed by the analog computer did not agree with the expected, regular results. The approximate theoretical calculations predicted that the results should be mapped in the form of smooth, closed curves, whereas the obtained simulation portrait looked more like a "shattered egg with jagged edges". 5

6

Chaos, Bifurcations and Fractals Around Us

"My first concern was that my analog computer had gone bad" - said Prof. Ueda in his report in the year 1991. "But soon I recognized that the 'shattered egg' appears more frequently than the smooth closed curves. As I watched my professor preparing the report without any mention of this 'shattered egg' phenomenon, but rather replacing it with the smooth closed curves of the quasi-periodic oscillations, I was quite impressed by his technique of report writing". At the beginning of the year 1962 Prof. Hayashi changed the topic of Ueda's research study. Now it was the so-called "Duffing system", and the main point of interest concerned steady-state oscillations executed by this oscillator. The Duffing system, governed by the second-order differential equation in the form — Y + h— + Qlx + fJx3 = F cos cot, h>0 (2.1) dt dt was pretty well known in that time, because it was regarded as a mathematical model of a wide class of physical systems. One such system can be reduced to a ball moving along a definite track. The model is sketched in Figure 2.1: the y axis is vertical, and the ball remains in the x-y plane. The damping force resulting from friction and aerodynamic drag forces is assumed to be proportional to the velocity, and it is represented by the term hdx/dt. It is assumed that the ball is subjected to a horizontal harmonic force governed by the trigonometric formula F cos cot. The potential force, due to gravity, is represented by the term Q2ox + inx3;

this relation is due to a properly selected shape of the track. If the external harmonic force is not applied to the ball, it will exhibit free damped oscillations, and will finally settle on the rest position at the bottom of the oval track, that is at x = 0. Thus we deal with the dissipative and deterministic oscillator with periodic excitation. In the times of Ueda's simulation, the researchers were deeply convinced that the steady-state oscillations of the system were periodic. In the linear system, that is for [i = 0 in Equation (2.1),

Veda's "Strange Attractors"

1

Fig. 2.1. Mechanical model of the Duffing system.

the period of oscillation is always equal to the period of excitation, that is to

T=W . /CO

The nonlinearity may affect the steady-state oscillation, and lead to the so-called subharmonic oscillations with periods 2T, 3T, , riT. The research problem concerning the steady-state oscillations was formulated as follows: in the Equation (2.1), assume £2 0 =0 and \i = CO = 1, and determine ranges of various subharmonic outputs in the plane of parameters F versus h. Since only the period of the final outcome was to be determined, student Ueda applied a mapping technique in his analog computer simulations. The mapping method relied on recording the quantities of the position x and velocity dx/ /dt at discrete instants of time, separated by the period of excitation T, that is at time t equal to 0, T, 27, 3T, , nT, Interpretation of results thus obtained was very simple. If the period of oscillations was equal to the excitation period T, only one point appeared on the screen of the analog computer; the two points on the screen indicated that the period of oscillation is 2T, three points - 3T etc. Therefore, it was enough to count the number of points displayed on the computer screen to answer the question of the period of the output considered. It happened accidentally that at certain values of the parameters F and h, student Ueda obtained a strange portrait that did not fit the theoretical prediction; his result looked like nonperiodic

8

Chaos, Bifurcations and Fractals Around Us

oscillations. The number of points on the screen was continuously growing as the time of computations went on, and this huge number of points formed a strange looking structure. Prof. Hayashi was sure that these results presented a transient state and, consequently, that after some time the system would settle on a subharmonic output. But the stubborn student Ueda repeated his simulations many times and watched the computer screen for long hours. Soon he became more and more convinced that the portrait he found represented the steady-state oscillations, rather than transient motion. It was a very strange steadystate, nonperiodic, irregular, random-like oscillation. Thus the term "strange attractor" was born. Yet, for a long time Ueda's results did not appear in any reports, were not presented at any conferences, and his papers were rejected by reviewers. It took many years before Ueda succeeded in publishing his results, first in Japanese and then in English, in a widely known American journal. At last, in about 1979, news about "strange attractor" found in the Duffing system spread across the world, and this stimulated research in the field of nonlinear oscillations (see references [29, 30]). When his simulations were verified by other researchers, and when the mathematical concept of "Lyapunov exponent" was applied to this irregular solution, it became clear that the Ueda's "strange attractor" indeed portrayed the solution unpredictable in time and sensitive to initial conditions, the solution labeled as "chaotic". The discovery opened a new chapter in the research of dynamical systems with continuous time, the chapter of exploration of chaotic phenomena in the nonlinear, dissipative oscillators, driven by periodic force. Thereafter, the early strange attractors found by Ueda with the aid of the analog computer became famous in scientific literature all over the world. One of them has been called "Japanese attractor", another became known as "Ueda's strange attractor". The attractors are presented in Figures 2.2 and 2.3.

Veda's "Strange Attractors"

9

Xp

Xp Fig. 2.2. J a p a n e s e attractor,

F= 12.0,

/7 = 0.1.

Chaos, Bifurcations and Fractals Around Us

10

Xp

Xp Fig. 2.3. Ueda's strange attractor, F= 7.50,

h =0.05.

Chapter 3

Pendulum

3.1. Equation of motion, linear and weakly nonlinear oscillations Consider a physical model exemplified by the plane pendulum depicted in Figure 3.1. The pendulum consists of a heavy, small-diameter ball with mass m suspended on a rigid and very light rod of length /. The rod can rotate around the horizontal axis O. It follows that the ball can move along a circle in a vertical plane, and its position is determined by a single coordinate, for instance, by the angular displacement denoted as x in Figure 3.1. The motion of the ball is ruled by the gravity force mg, the damping force Pt, and the moment of external periodic forces applied to the axis of rotation, M (T) . The considered physical model is often regarded as a satisfactory approximation of many technical devices.

M(T)

m

Fig. 3.1. Mechanical model of the forced pendulum.

11

12

Chaos, Bifurcations and Fractals Around Us

A physical experimental investigation of motion of the pendulum and, in particular, measurements of the sought position x and the velocity v = dx/dt of the ball, is not a convenient tool in the study of chaotic phenomena. Instead, we may apply a numerical approach, as the computer simulation enables us to find the output of the system. Numerical procedures also allow us to obtain "unstable solutions", that is the solutions that, although unrealizable in any physical experiment, play an important role in the analysis of system behavior and the related concepts. This will lead us to the discussion on the underlying structure of chaotic dynamics, namely, the geometric properties called fractal structure. To apply numerical procedures for the study of dynamic properties of the pendulum, one has to know the mathematical rule that governs its motion. That is, we need to know the equation of motion of the pendulum. The equation can be derived by the straightforward application of Newton's Second Law. The Law states that the product of the mass m of the ball (treated as a mass-point) and its acceleration is equal to the sum of all forces acting on the ball ma = F,

(3.1)

where a denotes vector of acceleration, and F is the vector sum of the acting forces. For the physical model of the pendulum shown in Figure 3.1, the equation of motion is obtained in the form mlf±

= -mgsinx-Pi+M(ll.

( 3.2)

dt I We assume that the moment of the external forces M (T) is a harmonic function of time M ( T ) = M 0 COS57T ,

and the damping force (resistance to motion) is proportional to the velocity dx where MQ and h denote constant coefficients.

Pendulum

13

In order to reduce the number of independent parameters, we introduce nondimensional time t and nondimensional driving frequency co , in the form t = rQ,0,

<« = = - >

where the variable

is the natural frequency of small amplitude oscillations of the pendulum. Next, we can also introduce nondimensional damping coefficient h and nondimensional amplitude of the forcing parameter F

A a s _L, mlQ,Q

F=^. mgl

In the nondimensional equation of motion obtained this way, the natural frequency of small amplitude oscillations is reduced to the value £lo = 1.

After transformations, equation of motion of the forced pendulum (3.1) takes the form of the ordinary second order differential equation x + kx + Q.2Qsmx = F cos cot,

(3.3)

where . = dx

.._d2x

x~Tt'x~dt2'

2i

°" '

h represents damping coefficient and F, CO denote amplitude and frequency of external excitation, respectively. Let us recall now that, at very low values of the angular displacement x, the trigonometric function sinx can be replaced by the first term in the Taylor series sin x = x, thus the nonlinear Equation (3.3) is reduced to the linear one

14

Chaos, Bifurcations and Fractals Around Us

x + hx + Q.20x = Fcos(ot.

(3.4)

It is worth noting that linear ordinary differential equations with constant coefficients possess analytical solutions in a closed form. Therefore, analytical analysis provides us with a full knowledge of the system behaviors, such as: • the free, conservative system, that is at F = h = 0, can swing around the lowest (hanging) position x = x = 0, and the motion is described by the trigonometric function of time x(t) = A c o s ( Q 0 t + 6 ) ,

(3.5)

where the constants A and 6 depend on the initial conditions; • in the free but dissipative system, that is at F = 0 and h > 0, the free oscillations decay in time, and finally the system settles on its rest (hanging) position, i.e. at x = x = 0. • in the dissipative, forced system, that is at F ^ 0 and h>0, the free oscillations decay in time, and the system tends to oscillate with the period of the excitation force T = 2n/<X). The final forced oscillations obey the trigonometric rule x(t) = acos(cot + (p),

(3.6)

where the constant values of the amplitude a and the phase angle (p are defined by the system parameters a = a(h,F,co), (p = (p{h,F,a). The amplitude a of the forced oscillations reaches its maximum value in the region of the resonance, that is when the excitation frequency 0) is close to the natural frequency £2o - Figure 3.2, the curve denoted 1. The linear oscillator obeys the principle of (linear) superposition. It means that the general solution of Equation (3.4) is a sum of the solution of the autonomous system (F = 0) and the particular solution of the nonautonomous system (F # 0). At sufficiently low values of the damping coefficient h, the general solution can be expressed in the form x(t) = Ae 2' cos(Q.dt + @) + acos(cot + (p),

Pendulum

15

where Qj stands for the natural frequency of the damped system, "
4

-

the constants A and 0 depend on initial conditions, and the constants a and (p are defined as follows a = .—

F

;

-hco

tg(D = —=

T-.

The first term in the general solution stands for "free oscillations", while the second one represents forced oscillations. At higher but still moderate values of the displacement x, the trigonometric function sinx in Equation (3.3) can be approximated by the first two terms in its Taylor series. Consequently, equation of motion of the pendulum takes the form x + hx + Q,20x—QQX3 6

= Fcoscot.

Thus, motion of the system becomes governed by the "weakly nonlinear" equation. The nonlinear equation does not possess an analytical solution in a closed form, and the superposition principle is no longer valid. The nonlinearity generates a variety of nonlinear phenomena, such as sub, ultra or sub-ultra resonances. Yet, in the neighborhood of the principal resonance, the steady-state oscillations remain close to the harmonic function of time. Theoretical analysis of the weakly nonlinear oscillations is based on approximate analytical methods. There are a variety of approximate techniques, each of the procedures being based on some approximate assumptions. The methods, and the obtained approximate solutions, are presented in many textbooks on nonlinear oscillations. The references [4, 11, 17] can be useful in the study of approximate methods, as well as on nonlinear regular phenomena that occur in the weakly nonlinear oscillators.

16

Chaos, Bifurcations and Fractals Around Us

1 2

snBc—£.

e

/

\

Dn i

snAjL/

\

PS.

\^

Sn

i— CO]

i <X>2

J

>

1

frequency

Fig. 3.2. Amplitude-frequency curves of the linear system nonlinear system 2.

1

and weakly

An essential effect of the nonlinear term proportional to -x3 in the weakly-nonlinear Equation (3.9) in the region of the principal resonance is that the resonance amplitude-frequency curves become bowed to the left, towards lower frequencies of excitation — see Figure 3.2, curve denoted 2. The essential new phenomenon relies on the existence of three solutions in the frequency interval between 0)x and co2, that is for wi < co < co2.

Only two of the three solutions, denoted by solid curves labeled Sr, Sn in Figure 3.2, can be realized in the system, i.e. we can observe them in both physical and numerical experiments. The third curve, marked by a dotted line labeled Dn, represents the so-called unstable solution. The solution satisfies the equation of motion, but every disturbance, even very small ones, leads the system to the resonant branch Sr or Sn. That is why the unstable solution Dn cannot-be realized in any experiment or numerical simulation. The stable solutions Sr and Sn are called attractors.

Pendulum

17

Thus, in the weakly nonlinear oscillator, there is a region of system parameters where two attractors, namely the resonant attractor Sn and the nonresonant one Sn, coexist. In the further study we will see that the initial conditions will decide which attractor will be realized in the system. Notice two particular points, labeled snA and snB, depicted on the resonance curve 2 in Figure 3.2. The points define the interval of (0 where two attractors coexist. At the related values of frequency, 2 with increasing frequency, are accompanied by a transient motion until the system settles finally on either Sn or Sr steadystate oscillations. It is worth recalling that in the nonlinear system the (linear) superposition principle is not valid. It follows that the general solution cannot be interpreted as a sum of solutions of the autonomous and nonautonomous system (see Equation (3.7)). That is why we would rather use the term transient motion, as well as the term motion on the attractor, to define the nonsteadystate and steady-state oscillations, respectively. Sudden changes of the amplitude of oscillations that take place at the frequency values to, and (O2 (see Figure 3.2), belong to the category of local bifurcations, and are called saddle-node bifurcations. The bifurcational phenomena play an essential role in nonlinear dynamics. The other examples of bifurcations will appear in the latter study. The concept of local bifurcation is closely related to the concept of unstable solution. For presenting both notions clearly it is useful to make use of the mapping method. This method reduces periodic in time solutions to constant solutions. In fact, this is the same method that was used by Ueda in his study of steady-state oscillations in nonlinear oscillators. Presently the method is referred to as Poincare method, and it presents a simple application of the mathematical theory that has a much wider scope.

18

Chaos, Bifurcations and Fractals Around Us

3.2. Method of Poincare map There exist a variety of graphical representations of periodic motion. The most commonly used is that sketched in Figure 3.3(a); it illustrates the evolution of the displacement x versus time t, and is called the time history of motion. We may also observe the periodic in time solution in the three-dimensional state space, with the coordinates x, x, t. Yet, it is more convenient to observe a projection of the three-dimensional trajectory on the two-dimensional phase plane, i.e. the displacement x versus velocity x (see Figure 3.3(b)). This form of graphical representation is referred to as the phase portrait of the solution. If the solution X = x(t) is a trigonometric function of time (as that in Figure 3.3(a)), the phase portrait is an ellipse (Figure 3.3(b)).

A A A /1 x

\

~

X

(b)

P

J

X

Fig. 3.3. Illustration of the periodic motion: (a) time history; (b) phase portrait.

Let us now focus our attention on the period of the considered solution. Instead of looking at the motion continuously, we may look at the same motion at discrete times, so the trajectory will appear as a

Pendulum

19

sequence of dots in the phase plane x - x. When the system is excited by a periodic force, a natural rule for the mapping is to choose the period of excitation, T = 2n/co, as the sampling time. Thus, we obtain and record the time-sampled sequence of data x(0), x(T), x{2T),

,x(nT)

=

7jt_

x(0),x(T),x(2T),

,x(nT),

" CO '

If the period of the solution is equal to the sampling period T, the solution will appear as one point in the phase plane, the point that is called the Poincare map of the T-periodic solution. In Figure 3.3(b), this point is defined by the Poincare coordinates xp and xp.

Consequently, the solution of period 2T will be represented by two points, of period 3T - by three points etc. Generally, the number of discrete points in the Poincare phase plane xp-xp, determines the period of the motion. Poincare maps of T-periodic and 2T-periodic solutions are presented in Figures 3.4(a) and 3.4(b).

*/>!

(a)

-M

•

xp

(b)

*•

i

xp

Fig. 3.4. Poincare map of: (a) T-periodic motion; (b) 2 T-periodic motion.

By applying the method of Poincare map, we reduce the periodic in time solution to the solution constant in time. We should remember, however, that this form of representation does not provide any information on the time history of motion.

20

Chaos, Bifurcations and Fractals Around Us

At this point the question arises: is that possible for the oscillator as simple as the pendulum excited by a periodic force to execute oscillations with the Poincare map consisting of points whose number is still growing with the time of computation? Theoretically it would mean that the number of points would tend to infinity as the time of calculation approaches infinity. In the further section it will be shown that the answer is "yes, it is possible". The Poincare map that possesses this property is called "strange attractor", and the motion on the strange attractor of the system belongs to the category of chaotic motion. The method of Poincare map, which reduces the periodic solution to the constant solution in the Poincare plane

appears to be very useful in clarifying the concept of stability of the solutions. 3.3. Stable and unstable periodic solutions To explain the physical sense of the notion "unstable solution", it is useful to consider firstly the stability of the constant in time solution. In the case of the pendulum considered in this chapter, the constant solution corresponds to equilibrium position of the unforced (i.e. autonomous) system, governed by equation of motion obtained from Equation (3.3) by putting F = 0 x + hx + sinx = 0.

(3.7)

The constant in time solution is the one that satisfies the condition sin x = 0. The condition is fulfilled by the two positions of the pendulum: • •

x = 0 which represents the hanging position and corresponds to minimum value of the potential energy;

x = ±n which represents the inverted position and corresponds to maximum value of the potential energy. Our everyday experience tells us that the inverted position of the pendulum cannot be realized in any kind of experiment. Every, even very

Pendulum

21

small, deviation from the x = ± 71 position, would grow with time and lead finally to the lowest, hanging position. Of course, there exist mechanical laws and the laws of mathematical theory of stability of motion, which prove that the solution corresponding to the inverted position satisfies the conditions of the unstable solution. Then we turn attention to the periodic in time solution of the driven pendulum, governed by the full Equation (3.3). The periodic solution is, thanks to the Poincare mapping technique, reduced to constant solution in the Poincare phase plane, x versus xp. If we add an initial disturbance to the considered solution and look at the time behavior of the disturbance, we would be able to answer the question whether the solution is stable or not. In the geometrical representation of the problem we look at the behavior of trajectories of motion, the trajectories that start in a close neighborhood of a point S, the point that represents T-periodic solution (see Figures 3.5 and 3.6). In Figures 3.5(a) and 3.5(b) all trajectories, which start from the vicinity of the point S, approach the point with time. Thus the point S seems to "attract" all the neighboring trajectories.

xD

(a)

xp Fig. 3.5(a). Example of stable Aperiodic solutions: focus.

22

Chaos, Bifurcations and Fractals Around Us

(b)

•

Fig. 3.5(b). Example of stable /"-periodic solutions: node.

Since the point 5 represents a periodic solution, this attraction property can be defined as follows: if we add an initial, even very small, disturbance S (to) to the periodic solution represented by the point S, the disturbance will decrease with time 5(0 ->0 with t-^°o. The point S in Figures 3.5(a) and 3.5(b) satisfies the condition and thus belongs to the category of asymptotically stable periodic solutions, labeled attractors. Trajectories around the point D in Figure 3.6 look quite different. They move away from the point, and the point D represents an unstable periodic solution. This particular unstable point is called saddle, or saddle point. One also notices that there are two pairs of the trajectories (drawn as bold lines in Figure 3.6), which look slightly strange. One pair, denoted Wsw,Wsm, tends directly to the point D, while the other pair W^,W™ moves away, and they all seem to "intersect" at the saddle D. In fact, these particular trajectories neither intersect nor approach or escape the saddle, because this would take an infinite time. In computer calculations, however, one can obtain the picture like that in Figure 3.6, because of some approximation techniques involved in the numerical procedure.

Pendulum

23

Xp Wu

Ws

Ws Wu •

Xp Fig. 3.6. An example of the unstable Aperiodic solution, D - saddle point; Ws^2),Wu^n represent stable and unstable manifolds of the saddle, respectively.

Thus we come to the new concept, the concept of invariant manifolds. The particular lines Wfl),Wsm that seem to approach the point D are called stable manifolds, while the others, which seem to escape D, Wum,Wu{2), are labeled as unstable manifolds of the saddle. A complete classification of the variety of stable and unstable solutions, as well as a more detailed discussion on the concept of invariant manifolds accompanied by mathematical analysis, can be found in many books on nonlinear oscillations and nonlinear dynamics. See for example the references [1, 3, 4, 11, 13, 17, 31].

24

Chaos, Bifurcations and Fractals Around Us

Let us notice that the stable manifolds Wsw,Wsi2)

mark a boundary between the trajectories, which tend to the left, and the ones, which tend to the right in Figure 3.6. This might suggest that the trajectories approach two different attractors. At this point it is worth looking back at the resonance curve denoted 2 in Figure 3.2. Here we face the situation that, inside the frequency range between ft), and ft)2, there exist two different attractors Sr and Sn, separated by the unstable solution Dn. Before we proceed to the discussion on a role the stable manifolds of a saddle play in the situation of coexistence of two or more attractors, we need to know what attractors exist in particular zones of the system control parameter plane, i.e. of the plane F versus CO, with fixed value of the damping coefficient h. We still confine our interest to the region of the principal resonance. Figure 3.7 covers the frequency interval from CO = 0.40 to CO = 0.95. The Figure indicates that the triangle-like region of coexistence of the periodic oscillating attractors Sn and Sr is bounded by the saddle-node bifurcation curves snA and snB (see Figure 3.2), as well as by the new types of bifurcation curves, denoted sb and cr. The new scenario of destruction of the Sr attractor, determined by the curves sb and cr, occurs at the forcing values F > Fi (F, ~ 0.15). 3.4. Bifurcation diagrams Let us now look closely at the bifurcational scenario of disappearance of the resonant attractor Sr in the forcing region where F > F,, but still F
Pendulum

25

«2 J\ °OR 0.8

I

P

-

'

^~

sn A

T'ITT

\

CHAOS

\

s;)u"

o.4

I

!

I

/ I CHAOS I

>y

//

ox//

0-3 ~ JF

- r

A/

£¥

0.2 -

I

'

\ // 7

SnJ

o.5 -

FT1

\

|Sr|

[Sn]

\sn A \

J ^

0.1 i

0.4

i

0.5

i

i

0.6

i

i

0.7

i

i

0.8

i

i

0.9

(0

Fig. 3.7. Regions of existence of various attractors (denoted as 5) in the control parameter plane F-u>, at /7 = 0.1.

26

Chaos, Bifurcations and Fractals Around Us

w

IF T 75 -

(a)

/

^

<]

-

sn

Dn 70

f

/Sr

/

65 -

J'

/

I /

\

"I

I]

I

__

v/»

cr

60

If 0.55 L -K

.

,

.

-2

-1

0

I

.

.

1

2

J Xp +71

Fig. 3.8(a). Bifurcation diagram at F - 0.35, h = 0.1 — bifurcations of the resonant attractor Sr at decreasing frequency u>, and of the nonresonant attractor Sn at increasing frequency.

We begin observation of the Sr attractor, starting at ft) =0.8 and then decreasing the frequency. The first sudden change occurs at the frequency value denoted sb. At this frequency, the symmetric Sr attractor loses its stability in favor of the two unsymmetric coexisting attractors Sr1 and Sr2, but only one of them, denoted Sr2, is depicted in the bifurcation diagram. In Figure 3.9(a), another form of graphical interpretation of this symmetry-breaking bifurcation is applied, namely the phase-portrait interpretation. Here we clearly see that the symmetric Sr attractor is replaced by two unsymmetric ones, Sr' and Sr2. The symmetry-breaking bifurcation is followed by the perioddoubling bifurcation, denoted pd (see Figures 3.8(b) and 3.9(b)). The T-periodic unsymmetric attractors S r; , Sr2 lose their stability and are

Pendulum

27

replaced by attractors with the period 2T. In the bifurcation diagram (Figure 3.8), the new 2r-periodic attractor is visualized by two lines. We can see this more clearly in the enlarged region of the diagram (marked by a rectangle in Figure 3.8(a)) - Figure 3.8(b). If the portion of the enlarged region of the diagram between pd and cr was enlarged again, we would see a cascade of period-doubling bifurcations. In our diagram we notice only the second period-doubling, and then the periodic attractor bifurcates into the oscillating chaotic attractor. This chaotic attractor looks like a black, horizontal strip covering the displacement range from xp = -2.5 to xp = - 2 . 3 , and exists in very small range of frequency. The chaotic attractor disappears suddenly in the crisis scenario at the frequency denoted cr. The phenomenon of oscillating chaos, as well as the scenario of boundary crisis, will be discussed in Chapter 4.

sr/

"I

(b) sb

^^_ 0.6Z

^*\.

\ s2 0.615

\

0.61

\

pd

^is-...n^ii'ii^ii-,..,.

ft

cr

0.605

0.6

_

.-2.7

.-2.6

.-2.5

.-2.4

.-2.3

.-2.2

.-2.1

t f

Xp

Fig. 3.8(b). Bifurcation diagram at F= 0.35, h = 0.1 — enlarged portion of the bifurcation diagram from Fig. 3.8(a).

28

Chaos, Bifurcations and Fractals Around Us

The "route to chaos" via a cascade of period-doubling bifurcations was discovered by M. Feigenbaum, and then mathematically analyzed for dynamical systems with discrete time. Apart from the theoretical study, this type of bifurcational scenario, leading from periodic to chaotic motion, was observed in numerous physical experiments. The problem is discussed in detail in a number of books devoted to nonlinear dynamics. See, for instance, the references [9, 13, 26, 32]. The bifurcational diagram in Figure 3.8(a) shows also the nonresonant attractor Sn when we observe it with increasing the frequency co (see blue arrows). We begin at CO = 0.55 and then see that the attractor loses its stability at the saddle-node bifurcation, denoted snA (an unstable branch Dn is also plotted). The frequency of the bifurcation cosnA is greater than that of the boundary crisis of the resonant attractor cocr, thus, in the frequency range cocr < co< cosnA, both attractors, Sn and Sr, coexist. Existence of two or more attractors at the same values of system parameters turns our attention to the question: what conditions have to be satisfied for the system to realize this or another type of motion? The question is closely related to the problem of basins of attraction. 3.5. Basins of attraction of coexisting attractors This way we come to the essential concept of basins of attraction of coexisting attractors. The basin of attraction of an attractor S is a set of initial conditions, i.e. initial positions and initial velocities, which lead the system to this attractor. This seems to be the simplest definition of the notion. Let us start with a discussion on the basins of attraction of the two periodic, resonant and nonresonant attractors, Sn and Sr, at the system parameter values F = 0.20 and (0 = 0.75. The basins obtained by means of computer simulation are recorded in the Poincare phase-plane xp versus xp inFigure3.10(a).

Pendulum

29

(a) x\

'

I Sr

f \T

\ symmetry ~~J breaking \ ^ ^ ^ ^ /

/

^

sb X P

'

'

'

Sr

Sr

X

(b) X I

'

I

Sr f

T

Vy

\

perforf )

^ - - ^ _ _ ^ - ^

doubling

pd

i r

'

•

I

'

'

Sr 2T

X

Fig. 3.9. Phase-portrait of the attractor Sr prior to and after: (a) symmetrybreaking bifurcation; (b) period-doubling bifurcation.

30

Chaos, Bifurcations and Fractals Around Us

VW,

^

2

s,, 0

•

-1

-2

x/0) —it,

Xpp

-*-

-

-i

.

u

•

•

/

L^ —7C

, -2

*-

i

^

^

/—\^n

^

Jr

_Z^^

,

,

-1

0

1

.

-rj[

^

ytf'i" ~^\S^'

—-r2

r d Xy, +71

Fig. 3.10. F= 0.20, w= 0.75, A =0.1, F
Pendulum

31

The two attractors are marked by dots, and the saddle Dn — by a cross. The basin of attraction of the nonresonant attractor Sn is represented by blue area, whereas the white region belongs to the basin of attraction of the resonant attractor Sr. It follows that all the trajectories with initial conditions belonging to the blue region tend to Sn attractor, while those ones, which start in the white region, settle finally on Sr attractor. One can see that the two basins of attraction are separated by a smooth, one-dimensional curve, the curve that seems to "cross" the saddle Dn. It is obvious that the boundary of the basins of attraction is a central point of the investigation. Then we compute also the manifolds of the saddle Dn (shown in Figure 3.10(b) in the same scale) and compare both Figures, 3.10(a) and 3.10(b). We observe that the stable manifolds, Wsm and W/2), coincide with the line, which separates blue and white regions, that is, with the boundary of the basins of attraction. Further study allows us to state that the stable manifolds of a saddle define the boundary of basins of attraction of the relevant coexisting attractors. The example in Figure 3.10 shows the basin boundary being a smooth, one-dimensional line. Let us now consider the same problem at slightly higher value of the forcing parameter F. Basins of attraction obtained at F = 0.27 and CO = 0.73 are drawn in Figure 3.11 (a), while the manifolds of the saddle Dn — in Figure 3.11(b). We immediately notice that the structure of the basin boundary, as well as of the related stable and unstable manifolds, appear to be essentially different from those presented in Figure 3.10. Now, the two basins of attraction (blue and white regions) are no longer separated by a smooth, one-dimensional line. In contrast, a detailed study would show that the blue irregular "fingers" which invade the white basin of attraction consist of infinitely many points, thus the basin boundary ceases to be a one-dimensional line. Such structures of the basin boundary are referred to as fractal. For the time being, we approach this concept only intuitively. Now let us look at the related structure of the manifolds of the saddle Dn (Figure 3.1 l(b)). Indeed, the complexity of the basin boundary is associated with the complex structure of the manifolds. One can notice that the stable and unstable manifolds intersect transversely many times.

32

Chaos, Bifurcations and Fractals Around Us

(a) Xp(0) 2

Sr

1

0

-1

-2

-3 -7C

Xp(0) -2

-1

0

2

11

+7U

(b)

XP 2 Wu

Sr

1

Ws

Dn

Sn 0

Wu Ws

-1

-2

-3

Xp —7C

-2

-1

0

1

2

+JI

Fig. 3.11. F= 0.27, w=0.73, /7=0.1, F> FM\{a) basins of attraction of the Sn and Sz-attractors; (b) stable and unstable manifolds of the saddle Dn.

Pendulum

33

Moreover, the number of intersections grows with the increase of time of computations. Theoretically, the number of intersections approaches infinity as the time of computations infinitely grows. The conclusion is that the intersections of the stable and unstable manifolds are responsible for the fractal structure of the boundary of basins of attraction. 3.6. Global homoclinic bifurcation The above analysis of the manifolds of the saddle Dn, and the related fractal structure of the basin boundary, brings us to the concept of the global homoclinic bifurcation of a saddle. The sudden changes depicted in Figures 3.10 and 3.11 were implied by the change of the value of the forcing amplitude F. It is useful to define the global bifurcation in terms of a general bifurcational parameter, say p. If we assume that the parameter pc is the critical value for the global bifurcation to occur, then at p < pc the stable and unstable manifolds of the saddle do not intersect, at p = pc become tangent, and at p > pc intersect transversally. One intersection implies infinitely many intersections as the time of computations approaches infinity. Referring the Reader to extensive literature on the global bifurcations, for instance, references [3, 14, 19-23, 27, 32], we confine our attention to their phenomenological effects. We have already noticed that the global homoclinic bifurcation results in fractal structure of the basins of attraction of the related attractors. This, in turn, implies sensitivity of the system to initial conditions. Therefore, the global bifurcation is responsible for the phenomenon of transient chaos. When this condition is satisfied, the system is labeled as "chaotic". To illustrate and explain the phenomenon of transient chaos, and the related unpredictability of the final system behavior (final outcome), we select the values of the control parameters F, (0 for which as many as three attractors coexist. This occurs in the narrow strip of parameters F, (0, marked by dots in Figure 3.7. The three attractors are denoted Sn, SQR and SQR . The attractor Sn is the familiar oscillating T-periodic nonresonant attractor, while the two attractors SQR and SQR are also

34

Chaos, Bifurcations and Fractals Around Us

T-periodic, but represent a new type of motion, the motion that consists in a regular combination of alternately rotating and oscillating motion.

xF

'

'

! si

'

'

n

!

^OR J

/

1 SOR

/

j

N,

n

1

\

»-----^---
,

,

!

,

,

-2.0

-1.0

0.0

1.0

2.0

JX 3.0 •

Fig. 3.12(a). Phase portraits of the three coexisting attractors: Sn, S'0R and SQR at F= 0.50, w=0.58, h =0.1.

The phase portraits of the three attractors are depicted in Figure 3.12(a), and the time histories of the oscillation-rotation attractors Sl0R , SQR are shown in Figure 3.12(b). Figure 3.13(a) presents basins of attraction of the three attractors. The latter Figure was obtained by applying the method of Poincare map, thus the r-periodic solutions are mapped as single points. The T-periodic attractors are marked by solid circles, while the related saddles Dn, DXOR, DQR - by crosses. The basin

Pendulum

35

of Sn is filled with white color, basin of Sl0R - with a red one, and the basin of SQR - with a green color. Figure 3.13(a) indicates that the boundary of the basins of attraction has fractal structure. In a large region of the Poincare phase-plane, the three colors seem to be "perfectly mixed". The word "perfectly" is used here to underline the essential property of the fractal structure: whatever fine scale we apply in the computation, we would always see that the colors are mixed. X

-'A A-A/A/A/A A: :.\ 1 HI 1 \:. 0

10

20

30

40

50

60

Fig. 3.12(b). Time-histories of the two oscillation-rotation attractors SQR , SQR .

36

Chaos, Bifurcations and Fractals Around Us

C2

n2

u'«

Dn

Xp(0)

2.0

1.0

0.0

-1.0

-2.0

-3.0 -3.0

-n

-2.0

2.0

3.0

+n

Fig. 3.13(a). Basins of attraction of the three coexisting attractors Sn, SOR and SQR (white, red and green colors, respectively); F = 0.50, OJ = 0.58, /7= 0 . 1 .

Considering this example we do not search, however, for the related global bifurcation, but we focus on the phenomenon of sensitivity to initial conditions. To this end, we choose the initial positions and velocities x(0), i(0) from the strongly fractal region of the basins of attraction (see the rectangle in the neighborhood of the origin of coordinates in Figure 3.13(a)), and then record the time-histories of the response x = x(t) for three different, but very close sets of initial

Pendulum

37

conditions. Results are presented in Figure 3.14. In all three cases, the initial portion of the time-history is very irregular and looks like a random-like combination of the oscillating and rotating motion of the pendulum. Finally, the system settles on one of the three coexisting attractors, though, in each case, it could be a different one: (a) the oscillating nonresonant attractor Sn ; (b) the oscillation-rotation attractor SQR ; (c) the oscillation-rotation attractor SQR , with the opposite direction of rotations.

1

"

r

^

-

'

.

-

'

/

Fig. 3.13(b). Structure of the basins of attraction of the oscillation-rotation attractors SlOR, SQR in the "periodic window" —see Figs. 3.7 and 3.15.

38

Chaos, Bifurcations and Fractals Around Us

X

-Mci- x i

i m

n

ft! A

f'

1 I

n

'

•

'

:

\ f , I " If

T 11 1 7 Y T

r lr

0

irii i n

I

I. I i

100

1 I. I 1 I

^

TJ I, I . I I

,

200

,

.

300

/

400

X

- ^ r i i r i i f 111 i n i i r i n i i i i i r i i { H I (dry

ifllmlW llffln T —71 r

0

X

I- ^

I'

^ J

A

,

1 I, I

100

f A A i l ^ U L AJiAJk AikikA> , i A A A A ^ .A A ^ J f I I ,1 1 1 I ! ,

200

I 1 | M

| 1 M i l

300

I I

I I, 1 I I M

t

Fig. 3.14. Three time-histories at very close initial conditions; F= 0.50, co= 0.58, h= 0.1: (a) x(0) =-0.220, i(0) = 0.010 ; (b) x(0) = -0.218, i(0) = 0.010;

400

Pendulum

100

200

39

300

400

Fig. 3.14(cont.). (c) 4 0 ) = -0.220, x(0) = 0.000. The presented results, as well as the other numerical and theoretical investigations, allow to conclude that: • the time duration of transient motion, i.e. between the initial time of computations and the time the system settles finally on the attractor, is unpredictable, that is, is neither related to the value of damping coefficient nor to the initial conditions; • the transient motion looks like steady-state oscillations, and it possesses properties of chaotic motion. It does not show any decay over time and ends suddenly, settling on one of the regular attractors. 3.7. Persistent chaotic motion — chaotic attractor We proceed now to the phenomenon of persistent chaotic motion. In the considered forced pendulum, it occurs in the V-shaped region in the control parameter plane F - CO (see Figure 3.7), the region bounded by the saddle-node bifurcation curve snA, and the curve of crisis cr, at the forcing parameter values

40

Chaos, Bifurcations and Fractals Around Us F>FQ,

COsnA<(Q
In this region, the system does not possess any regular attractors. to n

'

1—'

'

'

'

<~

/s, 0.75 -

/

ci

X

I

C2

\

*>

sb

r

0 70 , * %

*

*$,**&^~Cr

°- 65 l% '

J"

mm

/J?s

.

v
-si

7 ' L

0.55

-

i

7

1

-

i

2

-

1

i

I

i

0

1

2

J

Xp

+Tt

Fig. 3.15. Bifurcation diagram at F= 0.60, h= 0.1.

We begin with constructing the bifurcation diagram at the forcing level F = 0.6 (see Figure 3.15). At the frequency co = 0.80 there exists the T-periodic resonant attractor Sr. With the decrease of the frequency, we face the symmetry-breaking bifurcation of the attractor (sb on the 0) axis), the same one as in Figure 3.8. This time we apply the numerical procedure of the program Dynamics [12] that allows us to graph all the coexisting attractors in the diagram. An enlarged portion of the diagram would be similar to that in Figure 3.8(b), except for one essential difference. Namely, after disappearance of the oscillating chaos, that is at frequency (0 below the critical (Ocr value, our bifurcation diagram shows chaotic motion as a shaded region that spreads over the whole

Pendulum

41

range of the displacement xp, from xp = -it to xp=+n . It means that this attractor is not only non periodical but it covers full rotations of the pendulum. Within the range of frequency from cocr to cosnA, one can see two "periodic windows" inside the chaotic region. One of them is very narrow and is not studied here, the other one is quite wide, and the two periodic attractors SQR and SQR that occur there are identical with those shown in Figure 3.12. At the frequency of the saddle-node bifurcation cosnA the shaded region disappears suddenly, and chaotic motion is replaced by the Tperiodic nonresonant oscillating attractor Sn. The same bifurcation diagram is obtained with the increase of the control parameter CO . Let us focus now on the shaded region of the bifurcation diagram in Figure 3.15. The next Figure (Figure 3.16) shows a sample of the timehistory of the motion at F = 0.60 and CO = 0.69. Similarly as in Figure 3.14, the motion looks like an irregular combination of the oscillating and rotating motion of the pendulum. But now this type of motion lasts "forever" as it represents motion on the attractor. In the next step of the investigation we apply the Poincare mapping technique to study the structure of the chaotic attractor thus obtained Figure 3.17(a). hi the course of drawing the picture one can easily notice that the number of points of the attractor keeps growing with the time of computation. Moreover, that huge number of points forms a highly organized structure; the points seem to be arranged in what appears to be parallel lines. We look more closely at the strange structure of the attractor by making two sequential enlargements of small selected portions of the attractor - Figures 3.17(b) and 3.17(c). It is worth emphasizing that the "enlargements" we have in mind are not of the type used in photography, where the number and the magnitude of grains of the original photographic material remain unchanged. In our case, we choose a small portion of the original attractor (defined by a rectangle in Figure 3.17(a)) and compute again the Poincare map of this region using the previous grid size on the computer screen, as long as we come to a clear picture of the structure, hi Figure 3.17(b) thus obtained, we see again a highly organized set of points whose number grows with the increasing time of computation.

42

Chaos, Bifurcations and Fractals Around Us

X +K t n—i

pi

]—i

-7C I H i

I .aoo 1 I

1—j

j—ri—1—i

1

,gso I

1 1 . 4 1 j t

Fig. 3.16. Sample of the time-history of the persistent chaotic oscillations; F=0.60, CO =0.69, /7= 0.1.

Xp

,

,

,

,

,

3 2 1

(a)

0 -1

-2 . -TZ

. -2

. -1

. 0

. 1

, 2

\XP +71

Fig. 3.17(a). Poincare map of the chaotic attractor.

Pendulum

-0.2 ^J%>,...

43

•"^^s^-^^r: (b)

. . . ^ . "'." U g 0.4

0.5

0.6

0.7

4I ' ' ' ' ' ' ' -0.264

' H ^ K M I T ^ " ' ^ v' "^"": •0.268-

(C)

^%^&%.

'

-0.272

'

^

I «*/> 0.61

0.62

0.63

0.64

0.65

Fig. 3.17(b) and (c). Poincare maps of the chaotic attractor — enlargements of the small rectangle regions.

44

Chaos, Bifurcations and Fractals Around Us

By repeating the same procedure for the small region defined by a rectangle in Figure 3.17(b) we obtain the picture shown in Figure 3.17(c). We see again that the region, which in Figure 3.17(b) seemed to cover some area of the phase-plane, reveals another highly organized structure consisting of non-connected points. An attractor whose Poincare map consists of infinite number of noncountable points organized in such a way that the "enlarged" pictures show similar "structure in the structure", belongs to the category of geometric self-similar objects with a non-integer dimension, labeled as "fractals". At this point it is worth noticing that sequential enlarged structures, although similar, do not look the same. It follows that the attractor does not satisfy the condition of exact self-similarity, and the notion "statistical self-similarity" is used instead. Due to the strange geometry, this type of attractors is called strange attractors. In the literature we often read the remarks that the strange attractors are closely related to the Cantor set, or that they have Cantorset like structure. To learn more about Cantor set - see Section 3.8. At this point we face the most important problem: is the fractal geometry of the attractor a sufficient condition to be a chaotic attractor? Extensive studies of nonlinear oscillators with periodic excitation enable us to give a positive answer; in the class of dynamical systems considered, the strange geometry of the attractor guarantees that the motion on the attractor is chaotic, i.e. is sensitive to initial conditions and is unpredictable over a long time. There is also a direct mathematical method for the estimation whether a given trajectory of motion is chaotic or not. The method relies on computation of Lyapunov exponents. To prove that the motion is chaotic, it is enough to show that the largest Lyapunov exponent is positive. More remarks on the concept of Lyapunov exponents are presented in Section 4.8. In this section, however, we turn to the illustration of the sensitivity to initial conditions by presenting the time-history of solutions, see Figures 3.18(a) and 3.18(b). Figure 3.18(a) shows two time histories of the two solutions which start from very close initial conditions, Figure 3.18(b) demonstrates effects of different size of the step of integration

Pendulum

45

used in the numerical procedure. Comparison of the time-histories in both figures shows clearly that in each case the solutions diverge over time. X

120

140

160

180

200

t

220

X i~7n „

120

.

,

140

1—,

160

.

p

,———,—|—|

180

200

_

t

220

Fig. 3.18. Illustration of sensitivity of chaotic motion to initial conditions: (a) x(0) = 0.606, i ( 0 ) = -0.242 (black line), x(0) = 0.610, i ( 0 ) = -0.242 (red line); (b) spc = 100 (black line), spc = 300 (blue line); spc - number of steps per cycle in numerical integration procedure.

-,

46

Chaos, Bifurcations and Fractals Around Us

At this point it is essential to notice that all chaotic solutions that are obtained in both physical and numerical experiments lie on the same chaotic attractor. This also means that, in spite of different time histories, any one of these solutions is sufficient to obtain the chaotic attractor. The above remarks raise some questions about the stability of the chaotic solution. The classical definitions of stable and unstable solution are based on the evolution of the individual trajectory. Remember that the definitions were formulated by mathematicians in the time when the concept of chaotic solution was not known yet. If one considers stability of a chaotic solution in the classical way, he (or she) will conclude that the chaotic solution is unstable (see, for example, references [4, 11, 17]). The discovery of chaotic solutions and the concept of chaotic attractor required reconsideration of the concept of stability of motion. It is not very risky to state that, since all the chaotic solutions lie on the chaotic attractor and stay there "forever", they should be regarded as a stable bundle of solutions.

3.8. Cantor set — an example of a fractal geometric object When we discussed the basins of attraction and their geometric structure, we used the term "fractal". Our understanding of that notion was rather intuitive, with no mathematical definition. The concept of fractal appeared firstly in mathematics, and it referred to some geometric objects obtained by iteration procedures. Following Mandelbrot and his associates (see reference [6]), we define fractals as geometric objects, which satisfy two inseparable conditions, namely selfsimilar structure and fractal (i.e. non-integer) dimension. In the Euclidean geometry, the only existing objects are of integer dimensions. The dimension of a point is 0, of a line - 1, of an area - 2, of a volume - 3. The notion "dimension" can be generalized if we apply the following procedure. Consider the line segment of a length equal to 1 and divide it

Pendulum

47

into N identical parts, each of which is scaled down by the ratio r = l/N from the whole. The full length of the segment can be expressed as

An analogous relation for a square of a unit area is Nr2 = 1, while for a cube of a unit volume Nr3 = l. Applying the procedure to other geometric objects by dividing them into very small elements r (r —> 0 ) one can write NrD

=\.

Thus we arrived at a generalized concept of dimension D that does not need to be an integer

D = lim^4.

(3.8)

r-*o ln(l/r) The definition of the generalized dimension D requires the assumption that the object is self-similar, i.e. that it retains of the same structure, independent of the scale of observation. Let us consider in detail the construction of the fractal geometric object called Cantor set. We begin with a closed line segment of a unit length [0, 1] from which we remove the open middle third (1/3, 2/3), i.e. without its end points 1/3 and 2/3. Then we proceed in the same way with the two remaining closed segments [0, 1/3] and [2/3, 1]: we remove the open middle thirds of them and obtain four closed subsegments. If we continue this iteration procedure ad infinitum, the limiting set results as an infinite, non-countable set of limit points that are not connected (see Figure 3.19). Note that at each step of iteration we apply the same procedure. Therefore, at each step we see the same structure, independent of the magnitude of the segment r. Then we find out that, in this case, the values of N and r in the relation (3.8) are N = 2, r = 1/3, so the dimension D takes the value

48

Chaos, Bifurcations and Fractals Around Us

1

1

• I ••

1

•• ••

•• ••

•• ••

Fig. 3.19. Four sequential steps in the construction of a Cantor set.

D=**S0.63. In 3 Thus the Cantor set somehow fills more of space than a point, but less than the Euclidean one-dimensional line 0 < D < 1.

Chapter 4

Vibrating System with Two Minima of Potential Energy

In this section, we consider some problems of chaotic dynamics of a vibrating system that possesses two minima and one maximum of the potential energy. Further on, the system will be referred to as the "twowell system". Its mathematical model was originally derived in 1979 as a single-mode equation of a buckled beam [5]. First results of a related physical experiment were presented by F. Moon in 1980 [8]. In the following years, the model has found applications in several branches of physics. It soon appeared that the system was so rich in various nonlinear phenomena that it became an archetypal model being explored in many textbooks, also in books written by applied mathematicians [31]. Chapter 3 that presented a case of a pendulum familiarized the Reader with such concepts of nonlinear dynamics as: saddle-node bifurcation, period-doubling bifurcation, Poincare map, bifurcation diagram, basins of attraction, strange attractor, fractals and others. In the present chapter, we try to explain and illustrate new phenomena and concepts such as: • boundary crisis of the chaotic attractor; • unpredictability of the system behavior following the destruction of chaotic attractor; • intermittent transition to chaos; • Melnikov criterion; • Lyapunov exponents.

49

50

Chaos, Bifurcations and Fractals Around Us

4.1. Physical and mathematical model of the system In the physical model of the two-well system, a ball (mass-point) moves in a vertical plane along a strictly defined track, the track that possesses one "hill" (i.e. one maximum of potential energy of gravity force) and two "wells" (two minima of the potential energy) - Figure 4.1. On the bottom of each well, the ball is in the lowest position on its track, so the potential energy reaches its minimal values there. If no external force is applied, the ball is subjected to the gravity force, the reaction of the base, and to the resistance to motion. In this case, the ball will oscillate with decreasing amplitude, tending to the rest position on the bottom of one of the potential wells. Let us now apply to the ball the horizontal (along z axis) periodic force ACOSCOT . In the simplest experiment, instead of applying the force, we may put the base into horizontal oscillating motion Z 0 COS57T. We must, however, remember that our ball keeps sliding on the track surface without bouncing.

ZQCOSOOX

Fig. 4.1. Physical model of the two-well potential system.

Vibrating System with Two Minima of Potential Energy

51

No doubt that we deal here with the dissipative, deterministic oscillator, driven by periodic force. It is easy to predict that the ball subjected to low-amplitude periodic force will oscillate around its rest position, thus the motion will be confined to one of the two potential wells. However, with the increase of the driving force, the displacement of the oscillating ball will also increase. At this point we face the question: what happens when the ball reaches the top of the "hill" (z = 0) and starts to jump over it to the opposite well? As the system is symmetric with respect to z = 0, the ball has the same tendency to jump over the top of its track when it finds itself in the opposite well. It is therefore natural to ask whether the ball will now jump to and fro between both potential wells? And whether the motion will be regular and predictable, or not? In what follows we show that the motion, which consist in jumping from one well to the other (thus crossing the potential barrier - the "hill"), can be chaotic. Moreover, this irregular motion may occur in wide ranges of amplitude and frequency of the driving force. The considered oscillator exhibits a broad variety of strongly nonlinear phenomena. To explore some of them, detailed numerical computations are required. Hence, one has to formulate a mathematical model of the system, i.e. the equation of motion of the ball, and then to perform numerical analysis of it. This can be done, for instance, with the use of the software package for nonlinear dynamics, that accompanies the book Dynamics [12]. Our physical model (shown in Figure 4.1) is governed by the equation of motion in the form of ordinary, second order differential equation, the equation that finds a lot of applications in various branches of physics

-^4 + £—~a z + Pz3=Acoso)T. dx

(4.1)

dx

Here, cc,/3 > 0 are constant coefficients, k represents a resistance coefficient (viscous damping), A, 57 stand for amplitude and frequency of the driving force, respectively. d/,

- denotes differentiating with respect to time x.

52

Chaos, Bifurcations and Fractals Around Us

After suitable transformations and a change of variables, the above equation may be reduced to the following nondimensional (standard) form —z- + h dt dt2

2

x +—xi = Fcoscot, 2

(4.2)

where

/—-

(J

,

\a

_ A [p

k V2«

2a V a

to 42a

All numerical computations presented in this section have been performed for the equation of motion in the nondimensional form (4.2). Let us now consider how to adjust the shape of the track the ball is moving along to obtain the following form of potential forces, the form assumed in Equation (4.2) (

1

1 3^|

( 2

2J

Notice, that the shape of the track, i.e. the dependence of the vertical coordinate y upon the horizontal one x, y = y(x), is decisive for the change of the potential energy V as a function of x. From the course of Mechanics we know that potential force along x-axis equals to the first derivative of potential energy with respect to x, with a negative sign. We may write it down as follows ( \ 1 3\ dV - — x + -x3 = ,

( 2

2 J dx'

and thus we obtain V(x)= {[--X + -X3 \dx + C = --x2+-x* i{ 2 2 ) 4 8

+C,

(4.3)

where C is an arbitrary constant value. For the system coordinates assumed as in Figure 4.1, 8

Vibrating System with Two Minima of Potential Energy

53

Figure 4.2 presents the diagram of the potential energy V as function of x (and, consequently, an analytical form of the track's shape in Figure 4.1). We can see that the potential energy reaches its minimum value for x = +1 and x = - 1 .

V(x)k

X

-1

0

+1

Fig. 4.2. Diagram of the potential energy of the system.

4.2. The single potential well motion Let us start with Figure 4.3 where only the right half-plane of the potential diagram V = V (x), i.e. the neighborhood of the right minimum of potential energy, is displayed. We also introduce an additional coordinate x = x-l, which defines horizontal displacement from that minimum. Equation of motion, with respect to the new coordinate x, takes the form d 2x

dt

+h—+x—x2 dt

+—x3 = Fcoscot.

(4.4)

where the natural frequency of oscillations of the ball around the equilibrium position x = 0 equals 1.

54

Chaos, Bifurcations and Fractals Around Us

V(x)

x

j / « x

-y,L/ 0

i+l

(b)

0 -h

1—

i 0

•

x

+1

Fig. 4.3. The right potential well; (a) potential energy diagram; (b) phase portrait of the Aperiodic single-well oscillation.

Vibrating System with Two Minima of Potential Energy

55

Figure 4.3(a) shows clearly that the single potential well is asymmetric with respect to x — 0. Therefore, we may expect that the motion of the ball within the well will be asymmetric too. Indeed, the phase portrait of T-periodic single-well motion appears to be asymmetric (Figure 4.3(b)). Next, in Figure 4.4(a) we plot a schematic diagram of the maximal displacement xmax versus the driving frequency CO , in the vicinity of the principal resonance, i.e. close to to = 1. Likewise in the case of the pendulum, the diagram indicates the so-called "soft characteristic" of the system, when the resonance curve xmax = xmax(co) is skewed to the left, towards lower frequencies. We also see that, in the frequency range from cosnA to C0snB, the system possesses two stable solutions of a period T (T-periodic attractors) depicted as solid lines, namely the nonresonant attractor Sn, and the resonant one, Sr (Figure 4.4(a)). The attractors disappear at the saddle-node bifurcations points, snA and snB, respectively. Thus we notice full analogy to the phenomena relevant to vibrations of a pendulum, except that, in this case, both attractors Sr and Sn are asymmetric (S,(x) * S^-x), i = r,n). What one can see next is that, with the slight increase of the forcing amplitude F, at the top of the resonance curve Sr new irregular phenomena appear (Figure 4.4(b)). The saddle-node bifurcation snB ceases to exist, being replaced by a cascade of period-doubling bifurcations (the onset of the cascade is denoted pd in Figure 4.4(b)). The cascade is followed by a narrow strip of the oscillating chaos (chaotic attractor), which is finally annihilated by the mechanism of the boundary crisis crj. For the time being, this is only a schematic diagram; the outlined phenomena will be discussed in detail in the following sections. Upward and downward arrows placed in Figure 4.4(b) at the point of crisis cri draw attention to possible further motions of the system, after disappearance of the single-well chaotic attractor. The downward arrow indicates that, after the crisis of the chaotic attractor, the trajectory of motion tends to the nonresonant attractor Sn, so it remains confined to the same potential well. And what is the meaning of the symbolic upward arrow? In the next section we show that, after disappearance of the single-well chaotic attractor, the trajectory may also overcome the potential barrier Vmax (at x = 0) and settle onto the nonresonant attractor in the opposite well.

56

Chaos, Bifurcations and Fractals Around Us

*""j

(a)F0
Jf snA

i

^***+*^^

I

Stt

CO s n B

CG snA

1

f * crt


Pa

;••-.. \

z

/

tsnA

1 COcr

^<*^^^

i OOsnA

^oo 1

Fig. 4.4. Amplitude-frequency curves in the neighborhood of the principal resonance; (a) low value of the forcing parameter F, (b) moderate value of F.

Vibrating System with Two Minima of Potential Energy

57

The schematic diagram in Figure 4.4(b) signals that, at sufficiently high amplitude of the driving force F, the regular T-periodic motion within single potential well may be accompanied by quite irregular phenomena. Before proceeding to detailed exploration of the irregular phenomena, let us first examine the diagram of regions of existence of different attractors in the plane of the control parameters F, co , with fixed value of the damping coefficient h. The diagram is presented in Figure 4.5. We notice that the shape of the resonance curve shown in Figure 4.4(a), is observed only in a relatively narrow range of the forcing amplitude, from Fo to Fi, because only in this region can exist both saddle-node bifurcations, snA and snB. The range of frequency co , where both single-well attractors (S r and Sn) coexist, diminishes at the forcing level between F, and F2 and there appear the curves denoted pd (onset of the period-doubling cascade) and cri (crisis of the chaotic single-well attractor). This indicates that the form of the resonance curve corresponds to that presented in Figure 4.4b. In Figure 4.5 one can see an additional line M that is not related to the boundary of existence of any attractors. What is the meaning of the line M and what role does it play in the system behavior? The problem will be explored in the next section. 4.3. Melnikov criterion Let us now examine the manifolds of the hilltop saddle of the system. To this end, we apply the method of Poincare map, the method that transforms a T-periodic solution into a single point in the phase plane (see Section 3.2). In our case, the hilltop saddle represents the unstable T-periodic solution, which appears close to the top of the potential energy. For the sake of simplicity, we will concentrate our attention on the region of control parameters F, co where only one resonant attractor Sr exists in each potential well. In Figure 4.5, the defined region is situated to the right from the bifurcation line snA.

58

Chaos, Bifurcations and Fractals Around Us

0-095 f

fp*#T

W /Pd V/ •** V

^ 0.085

r«n 0.075

/ /

0.065

\

."V/

'

<

^

'

17

fC~1

/ / \ I£D

cr

0.055

/

\

^

M ^

\snA

^^^^-----[§i^-----V-------F1 0045 ~~~~^—-^^^ \ 0.035 :

0.55

^NA :*- r 0 0.60

D.65

0.70

0.75

0.80

0.85

0.90

CO Fig. 4.5. Regions of existence of different attractors in the control parameter plane F- co, at fixed value of the damping coefficient h = 0.1.

Figure 4.6(a) displays, in the Poincare plane xp-xp, the resonant attractors in both left and right potential wells (represented by points Sr and Sr, respectively), as well as the hilltop saddle (denoted DH), and its stable and unstable manifolds (depicted Ws and Wu, respectively). At this point, we recall that the stable manifolds separate the basins of attraction of different coexisting attractors, whereas the unstable manifolds tend directly to the attractors (in this case, to the points Sr, Sr).lt follows that all trajectories starting in the white region tend to the

Vibrating System with Two Minima of Potential Energy

59

left attractor (i.e. periodic oscillations of a ball in the left well), whereas all trajectories starting in the blue region tend to the right attractor S"r. The boundary of both basins of attraction is a smooth, regular curve that allows us to state, without any doubts, to which attractor the given trajectory tends.

-1.5

-1.0"

-05

0.0

0.5

1.0

1.5

Xp

Fig. 4.6(a). Basins of attraction of the resonant attractors Sr (white) and 5, (blue) together with the stable and unstable manifolds of the hilltop saddle DH , below Melnikov criterion; F= 0.05, at = 1.0, h= 0.1.

Such a clear situation takes place only for the values of F, 0) which lie below the line M, i.e. for F(co)
60

Chaos, Bifurcations and Fractals Around Us

Sri •y

~

J~/\

-1.5

rDH ^~l

-1.0

. ^ j - ^ —

-0.5

*

0.0

rsr |^pp

0.5

^£P

1

1.0

^ ^

f—r

1.5

.

Xp

Fig. 4.6(b). Stable and unstable manifolds of the hilltop saddle DH above Melnikovcriterion; F= 0.05, a; = 1.0, h=0A.

If we perform long time numerical computations of the manifolds, we shall find that the number of intersections increases with time (theoretically, it approaches infinity). As the stable manifolds separate basins of attraction in the phase plane, what we intuitively expect is that the manifold intersections must essentially affect the structure of the boundaries of the coexisting basins of attraction. And indeed, after crossing the line M, the basin boundaries cease to be smooth and regular and become fractal in nature. An example of the fractal boundaries of basins of attraction is shown in Figure 4.6(c). If the trajectory starts from the initial conditions x(0), i(0) which lie in the fractal region of the phase plane, the transient

Vibrating System with Two Minima of Potential Energy

0.5 £/ %

61

%

-0.5 <-

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Xp

Fig. 4.6(c). Basins of attraction of the resonant attractors S^ and S'r above Melnikov criterion; F- 0.05, co= 1.0, A=0.1.

motion will be chaotic, and the final state (i.e. whether the ball will finally oscillate in the left or in the right potential well) will be unpredictable. We have just mentioned this phenomenon in Section 3.6 where the concept of the global homoclinic bifurcation was introduced. In our present case, the related global bifurcation, which is referred to as the Melnikov criterion (M curve in Figure 4.5), appears to be the homoclinic bifurcation of the hilltop saddle. With crossing this boundary, i.e. for F(co) > FM (co), the hitherto regular system becomes chaotic. Let us note, however, that the system still possesses regular periodic attractors, and so, in the course of both physical and numerical experiments, this chaotic behavior may happen to be missed. If the trajectory of the ball starts close enough to the existing attractor, the

62

Chaos, Bifurcations and Fractals Around Us

transient motion will be regular, and the final state will be predictable as well. Critical parameters of the homoclinic bifurcation of the hilltop saddle (Melnikov bifurcation) may be predicted analytically in an approximate manner, by applying a perturbation method. Mathematical analysis of the problem gained a lot of fame, and has been quoted by a number of authors in many books and papers. The original derivation of an analytical formula governing the bifurcation was first presented in the book [3]. It follows from the analysis presented there that the dependence of critical values of the forcing amplitude F upon the driving frequency CD as well as the damping coefficient h takes the form „ h-Jl , nco cosh . Fu = In the present book, Melnikov criterion has been calculated by the use of the above formula, and then verified by numerical analysis of the stable and unstable manifolds of the hilltop saddle. The results obtained by both methods were so consistent that they might be visualized in the diagram as a single line M (Figure 4.5).

4.4. Fractal boundaries of basins of attraction and transient chaos in the region of principal resonance Let us now turn back to the region of the principal resonance, i.e. to the range of system parameters F, co where two T-periodic oscillating attractors, namely the resonant attractor Sr and the nonresonant one Sn , exist in both potential wells. In Figure 4.5, this triangle-like zone is bounded by the bifurcation curves snA, snB and pd. In the case when four attractors coexist, the diagram of their basins of attraction appears to be evidently more complex than that presented in Figure 4.6. To illustrate this fact, first we show basins of attraction for the region of control parameters F, CD where the system behaves regularly, i.e. below the Melnikov criterion, F
Vibrating System with Two Minima of Potential Energy

63

xp Fig. 4.7. Basins of attraction of the four coexisting attractors S'r,S'n,Sr,S"n (green, yellow, blue and red, respectively) prior to the Melnikov global bifurcation: F= 0.050, cu = 0.72, /?= 0.1 (point 1 in Fig. 4.9).

In Figure 4.7, the basins of attraction of the attractors existing in the left potential well are filled with green and yellow colors, while the basins of the attractors related to the right well - with blue and red colors. The attractors are denoted by the points Sr,Sn,Sr,Sn, and the hilltop saddle by the point DH . Then we make a crucial observation that, apart from a larger number of attractors, two additional, new saddle points have appeared, namely D and D . They constitute the mapping points of the unstable T-periodic solutions (denoted Dn in Figure 4.4), and the stable manifolds of the solutions define the boundaries of basins of attraction of the resonant and nonresonant attractor in each potential well. One can see it clearly by comparing the basin boundaries (Figure 4.7) with the manifolds of all three coexisting saddles (Figure 4.8).

64

Chaos, Bifurcations and Fractals Around Us

jrij

DH

|rf'

|

xp Fig. 4.8. Manifolds of the three coexisting saddle points DH ,D , D prior to the Melnikov global bifurcation: F= 0.050, co = 0.72, h= 0.1.

The global bifurcations that involve these additional saddle points result in the fractal structure of the basins of attraction within single potential wells, as well as in the appearance of different types of transient motions. It follows that, in addition to the homoclinic bifurcation of the hilltop saddle DH (line M in Figure 4.5), new types of global bifurcation occur. In Figure 4.9, the line horn represents the homoclinic bifurcation of the single-well saddles D and D , while the line het corresponds to the heteroclinic bifurcation of the single-well saddles D (D ) and the hilltop saddle DH . The latter one is a new type of global bifurcation that has not been discussed so far.

Vibrating System with Two Minima of Potential Energy

65

Next we will show the sequences of related changes in the basins of attraction, as well as the examples of time histories of the chaotic transient motions, the motions that result from the fact that the structure of the basins became fractal.

W/V

F 0.085

W

0.075

'

/

Cri/A\

}

//^f\

0065

Jl"

y''/"%*'^''hom ,.-Y """' 0.055

^ - ^ ^ ^ ' ' - " " ' " " "

^^E^ZT 0.045

""

#1 '

\

[iSrS^ \snA

^ - ^ ^ _ \ snB ^*********+. \

0.035 0.55

\ 0.60

0.65

0.70

0.75

0.80

0.85

\ 0.90

(0 Fig. 4.9. Regions of existence of various attractors close to the principal resonance, and the sequence of global bifurcations, defined by lines M, horn, het, at h= 0.1.

66

Chaos, Bifurcations and Fractals Around Us

After the bifurcation curve horn is crossed, the fractal structure of the boundary between basins of attraction of the attractors Sn and Sr in the left potential well, and of the attractors Sn and Sr in the right well, appear respectively (Figures 4.10(a) and 4.10(b), point 2 in Figure 4.9). In Figure 4.10(a) one can also notice the effects of the Melnikov criterion, but they are hardly discernible and involve only the initial conditions, which lie on the boundaries of the phase plane shown in the Figure. Fractal structure of the basin boundary between the resonant and nonresonant attractor in each potential well results in a single-well chaotic transient motion. The phenomenon is illustrated in Figure 4.11 by the two examples of time-history of motion. Initial conditions of both trajectories lie in the region of the fractal boundary of the basins, and are very close to each other. One notices that, for some time, the system exhibits quite irregular, chaotic transient motion until the trajectory settles on either the resonant or the nonresonant attractor (Figures 4.11 (a) and 4.11(b), respectively). Hence, a small uncertainty in the initial state of the system leads to the inability to forecast its future. Both the history of the transient motion and its time of duration are unpredictable; we are also unable to predict whether the system finally "decides" to exhibit resonant, or nonresonant periodic motion. After crossing the bifurcation curve het, one can notice qualitatively new effects. These are visible in Figure 4.12(a) (point 3 in Figure 4.9). In the regions of initial conditions that formerly involved fractal basin boundary between the attractors in the same well, now there appears a mixture of all four colors, and so we conclude that the boundaries of the basins of attraction of all four coexisting attractors become fractal. The fractal structure of the basins is intensified and, with the increase of forcing parameter F, it involves larger and larger areas. This essential feature is nicely illustrated in Figure 4.12(b) that corresponds to F = 0.075 and (0 = 0.79 (point 4 in Figure 4.9). Figure 4.13 presents examples of the cross-well chaotic transient motion that appears after crossing the curve of the global heteroclinic bifurcation (het).

Vibrating System with Two Minima of Potential Energy

67

Fig. 4.10. (a) Basins of attraction of the four attractors Sr,Sn,S"r,Sn after the global bifurcation horn, F= 0.058, cu= 0.72, h= 0.1 (point 2 in Fig. 4.9); (b) Enlargement of the region defined by a small rectangle in (a).

68

Chaos, Bifurcations and Fractals Around Us

s"

1 III11 O

JOB

ZOO

pOO

(4
5OQ

*_

O^

,1OQ

,200

,300

,400

,5OO

* _

Fig. 4.11. Examples of time-history of the single-well transient chaos; F= 0.058, co =0.72, /7= 0.1.

Vibrating System with Two Minima of Potential Energy

p Mlk I||fl,i|>

fr&

'

^

i^ow^i^HtK^I^^^^^^^^^^^^^^^^^Val

Pi

1.5

^

^o/

<

69

^B

\ *

• • ' ' ' ,

Fig. 4.12. Basins of attraction of the four attractors S'r,S'a,S"rJS"n after the global bifurcation het (a) F= 0.060, to = 0.72 (point 3 in Fig. 4.9); (b) F= 0.075, w = 0.79 (point 4 in Fig. 4.9).

70

Chaos, Bifurcations and Fractals Around Us

x

.. >Aoo=O.OO3

_1.B

^OO

x

, >Aoo=O.OO4

^OOO

.1500

*_


Fig. 4.13. Examples of time-history of the cross-well transient chaos: F= 0.075, cu =0.79, /?= 0.1.

In this case, the following numerical experiment has been performed: to the system exhibiting T-periodic nonresonant oscillations in the left potential well ( S n ) , two slightly different perturbations Aft) of driving frequency were added. As a result of each perturbation, the system behaves as follows: during the transient motion, the displacement x overcomes the potential barrier (for x = 0), exhibits some irregular oscillations in the opposite potential well, and then comes back to the previous well etc. This way, for a certain time interval, the system executes an irregular, unpredictable motion that has been called crosswell transient chaos.

Vibrating System with Two Minima of Potential Energy

71

The considered transient motion disappears suddenly, and after that the system returns to the r-periodic regular attractor. At the perturbation value Act) = 0.003, the final output is the resonant attractor Sr in the left (the same) potential well, while for Aft) = 0.004 the system finally settles on the resonant attractor in the right (opposite) well, Sr. Thus, the system behavior appears to be quite unpredictable again; we are unable to predict neither the time history, nor the time duration of transient motion. The final state, that is the steady-state oscillation after the decay of transients, is also very sensitive to small changes of the system parameters. We have just mentioned that the phenomena illustrated in Figures 4.12-4.13 take place after crossing the critical threshold of the global heteroclinic bifurcation. The bifurcation occurs when the stable and unstable manifolds of two different saddles begin to intersect. An essential difference between the homoclinic and heteroclinic bifurcation is outlined in the schematic diagram Figures 4.14(a) and 4.14(b). In Figure 4.14(a) one can see only one saddle D, and the intersection of its stable manifold W^ with the unstable one W^. In contrast, Figure 4.14(b) illustrates the heteroclinic bifurcation. There are two saddles, D[ and D2, and the intersection of the unstable manifold of the saddle D[

with the stable manifold of the saddle D 2

W2). More detailed considerations on the problem the Reader may find in the paper [20].

4.5. Oscillating chaos and unpredictability of the final state after destruction of the resonant attractor We still remain in the region of the principal resonance, i.e. the region where all four attractors, namely two resonant attractors (S r , Sr) and two

72

Chaos, Bifurcations and Fractals Around Us

nonresonant ones (Sn,Sn), coexist. Proceeding with our considerations into the range of forcing amplitude F> FM , we turn our attention to the

x

(a)

~sn

) x

i

(b)

/

V*

Fig. 4.14. Schematic diagrams illustrating: (a) global homoclinic bifurcation; (b) global heteroclinic bifurcation.

sequence of transformations of the resonant attractors Sr and Sr with decreasing frequency (O , the transformations which lead finally to disappearance of the attractors from the phase space. While considering Figure 4.4(b) and Figure 4.5 in Section 4.2, we have mentioned that the resonant attractor undergoes first the cascade of period doublings, then is transformed into an oscillating chaotic attractor, and finally is destroyed in the boundary crisis scenario. To perform a thorough study of the above transformations, we make use of the bifurcation diagram, i.e. the diagram where the Poincare

Vibrating System with Two Minima of Potential Energy

73

displacement xp versus the bifurcational parameter co is plotted (Figure 4.15). The diagram is constructed for the constant forcing amplitude F = 0.062, and for the initial value of frequency CD = 0.74. During numerical calculations, the parameter (O decreases. 00 L

'

0.73 -

!

i

: (a)

I

A.-

!

(*V

\

!

'

-

•**

r

• 1—.———^-.—, -1.0 -0.5

co f

•

»n " i

.—.—.—.—.—. 0.0 0.5

'

; i

0.73 -

(Her, oi»

i

I

'

'

.——J-i—^J XD 1.0 F

^

:

S"

!

(b)

i

0.72 -

\

'

"*" ,,

, S1

&pd

A

!

0.71 -

0.70

•*-

•

0.70 0.69 [

' '

»r

;

0.72 -

'

-

A

r

.'(Ocr,

i I

0.69 I

L

,

!

.

.

-1.0

-0.5

0.0

0.5

1.0

1 Xp

Fig. 4.15. Bifurcation diagram of the resonant attractor Sr and the unpredictability of the final outcome, /7 = 0.1: (a) F= 0.0620; (b) F= 0.0621.

y

74

Chaos, Bifurcations and Fractals Around Us

What one can see first in the diagram, is a single, nearly vertical line that represents a map of the T-periodic resonant attractor Sr in the right potential well. At the value 0) - copd, the attractor undergoes its first bifurcation of period doubling, and is replaced by the 2r-periodic attractor. In the bifurcation diagram, the new attractor is represented by two lines. As ft) further decreases, a whole cascade of successive period doubling bifurcations appear; as a result, the attractor ceases to be periodic, and becomes transformed into an oscillating chaotic attractor. In Figure 4.15, this chaotic attractor is represented by a narrow shaded region just above the value cocrl. The chaotic attractor also loses its stability and disappears from the phase space, being destroyed at ft) = cocrl by the so-called scenario of the boundary crisis. At this point, an essential question arises: what happens with the trajectory of motion after the crisis? Let us first have a look at the upper figure (Figure 4.15(a)) where a horizontal arrow pointed to the right, towards the coexisting nonresonant attractor Sn, is placed. The arrow indicates that the resulting trajectory of motion settles finally onto this attractor, thus being confined to the same potential well. Instead, in the bottom figure (Figure 4.15(b)) one can see an opposite situation: after disappearance of the oscillating chaotic attractor, the ball jumps over the potential barrier V^ and settles finally onto the r-periodic nonresonant attractor Sn in the opposite well. Both cases presented in Figure 4.15 differ only slightly in the value of forcing amplitude (AF =0.0001). Thus we may conclude that the final state of the system after destruction of the oscillating chaotic attractor is unpredictable. To clarify the reasons of this unpredictability, we shall make use of the diagram of basins of attraction on the XP-*P

plane (Figure 4.16). Moreover, this diagram will also help us to answer the question: what is the boundary crisis of the chaotic attractor?

Vibrating System with Two Minima of Potential Energy

75

4.6. Boundary crisis of the oscillating chaotic attractor The points S'n and S"n , shown in Figure 4.16(a), represent T-periodic nonresonant attractors in the left and right potential well, respectively. In the middle of the Figure, we also see the main saddle DH situated near the top of potential energy. For the assumed parameters F, (Q , the resonant attractors Sr and Sr are just transformed into chaotic oscillating attractors. One can see the chaotic attractors as white curves in both left and right potential well (virtually, Poincare maps of the attractors consist of a great number of points which are not discernible in the figure). Thus, we have four coexisting attractors and, consequently, four basins of attraction. The relative basins are depicted in yellow (S n ), red (S n ), green and blue (left and right chaotic attractor, respectively). The boundaries of the basins of attraction have a strongly fractal nature, which manifests itself in intense mixing of all four colors. Let us now examine carefully the close neighborhood of the right chaotic attractor. The respective region, confined to the rectangle in Figure 4.16(a), is enlarged in Figure 4.16(b). Here, the strongly fractal structure of the basin boundaries is visible even better than before. For the assumed control parameters, the chaotic attractor is very near but still before loss of stability, that is, before the boundary crisis. In Figure 4.16b, the imminent crisis manifests itself in such a way, that the chaotic attractor almost "touches" the boundary of its (blue) basin of attraction. Moreover, one can notice three white points denoted by 'D3T, i = 1,2,3 on the boundary. The symbol D stands for a saddle, while designation by three points means that we deal with a saddle of period 3T. The concept of the boundary crisis of chaotic attractor appeared in the scientific literature in 1983 [2]. Craw of the chaotic attractor defines its sudden change with a small change of the system parameters. Boundary crisis means a sudden destruction of the attractor, the destruction that takes place when the attractor "collides" (in phase space) with the unstable periodic orbit (the saddle) lying on the boundary of its basin of attraction.

Chaos, Bifurcations and Fractals Around Us

76

Xp

.S'n

0.0 D

H

(a)

-0.2

-0.4

-0.6

-1.0

0.0

-0.5

0.5

Xp

(k) Xp 1D 3T -0.1

2D3 T

3D3T -0.4

0.1

0.7

Xp

Xp

crl —*

0.1

(c)

0.7

X^

Fig. 4.16. Boundary crisis of the oscillating chaotic attractor and fractal structure of the basins of attraction; F= 0.075, co =0.7601, h fi=0A.

Vibrating System with Two Minima of Potential Energy

77

Figures 4.16(a) and 4.16(b) show our chaotic attractor just before the "collision" with the 3r-periodic saddle situated on its basin boundary. It follows that even small changes in the driving parameters F, CO will result in the destruction of the attractor together with its basin of attraction. Figure 4.16(c) presents basins of attraction that remain after disappearance of the chaotic attractor (more precisely, after simultaneous vanishing of the chaotic attractors in both potential wells). Thus, green and blue colors disappear, while yellow and red colors, i.e. basins of attraction of the T-periodic nonresonant attractors Sn,Sn, remain. Let us notice that, in the region formerly occupied by the chaotic attractor and its basin, the remaining basins have a strongly fractal structure. It follows that the time-history of the trajectory, which starts in this region, is relevant to the chaotic transient motion. Consequently, it is impossible to predict the final state, i.e. whether the trajectory will be recaptured by the left, or by the right nonresonant attractor. Figure 4.17 illustrates the discussed phenomena by displaying phase portraits of the attractors. First one can see four T-periodic oscillating attractors, Sr,Sr and S'n,S"n, in both potential wells (Figure 4.17(a)). Then, one can notice two nonresonant attractors Sn,Sn, as well as two chaotic oscillating attractors that originate from the resonant attractors (Figure 4.17(b)). Finally, after boundary crisis of the chaotic attractors, only two nonresonant attractors Sn,Sn remain (Figure 4.17(c)). Let us look back for a while on Figure 4.5. We see that the range of frequency co in which both resonant and nonresonant attractors exist, diminishes with increasing amplitude of the driving force F. This frequency range is bounded from the left side by the curve of boundary crisis cru whereas from the right side by the curve of saddle-node bifurcation of the nonresonant attractor, snA. At the same time, the relation cosnA > (Ocn is satisfied. At F = F2 both bifurcation curves (cri and snA) intersect, and for F > F2 we face the opposite relation, namely 0)snA < cocrl. Consequently, in the V-shaped region, there are no oscillating attractors that exist within the single potential well. If so, we intuitively guess that another form of steady-state oscillations of the ball should exist, namely the motion when the ball continuously "jumps" from one potential well to the other.

78

Chaos, Bifurcations and Fractals Around Us

i..

X

(a)

i

i

• s- n

s»

-i.5

-1.5

-1

9

,-0.5

.8.5

.1

,1,5

JC

(b)

x\ 1.5

t

chaotic S;.

chaotic S ;

i

-1.5

-1

I

--«.5

»

0.5

1

J-5

X

I

f?U x\ ' .'•

(c) ' I

s

i .0.3

o

o

-i : 1.5

Fig. 4.17. Phase portraits of the single-well attractors: (a) and (b) prior to and (c) after crisis of the chaotic oscillating attractors.

Vibrating System with Two Minima of Potential Energy

79

4.7. Persistent cross-well chaos Let us observe regions of existence of different attractors in a wide range of forcing parameter F, satisfying the condition F > F2 (Figure 4.18). The V-shaped region mentioned in the previous section, where no single-well oscillating attractors exist, is denoted CH. There is a new attractor, denoted SL that appears in this region. First we observe the steady-state oscillations inside the V-shaped region, prior to the appearance of the attractor S^. We begin with the bifurcation diagram for F = 0.11 with decreasing driving frequency (o Figure 4.19. The initial value of frequency satisfies the relation 0) > cocr[, thus the diagram starts from the r-periodic resonant attractor in either left (Sr) or right (S"r) potential well (Sr in Figure 4.19). With the decrease of frequency, the resonant attractor undergoes a cascade of period doubling bifurcations, and is transformed into a chaotic oscillating attractor, the attractor that occupies a relatively small range of displacement xp within the potential well. The chaotic attractor exists, however, in a very narrow zone of frequency (0 , and it suddenly disappears being replaced by a new form of motion that spreads over both potential wells. In the bifurcation diagram, this motion is illustrated by a wide dark band that covers the displacement range - 1 . 5 < x p <1.5 (Figure 4.19). We conclude that the motion of the ball consists in permanent crossing of the potential barrier (Vmax), and that the motion is not a periodic one, either. When we approach the frequency value COsnA that corresponds to the saddle-node bifurcation, the T-periodic oscillating attractor Sn emerges from the dark region (in Figure 4.19, the related unstable branch of it is also drawn as a dotted line). A sample of the time-history of this irregular motion that embraces both potential wells is shown in Figure 4.20. What can really be noticed is that the motion consists in jumping alternately to and fro over the potential barrier, usually combined with some oscillations. All components of the motion demonstrate irregular, random-like character.

80

Chaos, Bifurcations and Fractals Around Us

F

0.15 F3

0.10 M^

F2

0.06 F1

snB 0.6

0.8

1.0

CO

Fig. 4.18. Regions of existence of various attractors in a wide range of control parameters F - cv, at h= 0.1.

Vibrating System with Two Minima of Potential Energy

0) I

'

'

\

W

0.90 -

'

'

'

'

81

"~

\ 1 . ^ <-

-1.5

-1.0

(£pj

-0.5

0.0

0.5

1.0

1.5

P

Fig. 4.19. Bifurcation diagram at F= 0.11, h= 0.1.

x i

'

•

'

1.5 -

o.o.j|

i" I— ' " " I —

"in

1 II' "[""I"

~" " max

Hill yii IIIINI H I -1.5 1000

1500

2000

2500

t

Fig. 4.20. Sample of the time-history of the chaotic cross-well oscillations.

Fig. 4.20. Sample of the time-history of the chaotic cross-well oscillations.

82

Chaos, Bifurcations and Fractals Around Us

Poincare map of the considered motion complements and confirms the observation (Figure 4.21). The attractor consists of a great number of points, and this number increases with the increasing time of computation. To give evidence by means of numerical experiment that the motion is really chaotic one may: a) prove that the attractor is a "strange attractor", i.e. that it has fractal structure; b) examine time histories x = x(t) of the trajectories that start with very close initial conditions. However, we have mentioned in Section 3.7 that, in order to prove mathematically the exponential sensitivity of the motion to initial conditions, one should calculate Lyapunov exponents.

Fig. 4.21. Poincare map of the cross-well chaotic oscillations - chaotic attractor.

4.8. Lyapunov exponents Exponential dependence on initial conditions means that if we take two initial points that are separated from each other by a small distance d0 at t = 0, then for increasing time t the trajectories that start at these

Vibrating System with Two Minima of Potential Energy

83

points diverge exponentially. It is schematically illustrated in Figure 4.22. The dependence of the distance d between two trajectories upon time t, as well as their initial separation d0 at t = t0, is assumed to be governed by the exponential function

d(t)=do(to)e*. Exponent denoted A is an indicator of the sensitivity to small perturbations of initial conditions, and is referred to as Lyapunov exponent. It may be expressed in the form A = lim-ln—. (4.5) '-»- t d0 If A > 0, the trajectories move apart from each other (as in Figure 4.22); otherwise, if X < 0, the trajectories converge.

/

doXZZZZ^^

d

\ to

,

t

Fig. 4.22. Schematic illustration of the exponential dependence on initial conditions.

Mathematics proves that our system possesses three Lyapunov exponents (the phase space is three-dimensional). At the same time, since we deal with a damped (dissipative) system, the following condition is satisfied

84

Chaos, Bifurcations and Fractals Around Us

5>,.
We define the motion as chaotic if at least one of the Lyapunov exponents is positive. If A, > 0, two other exponents satisfy the condition Aj +A3 <0 (see references [9, 11, 13, 32]). Thus, the trajectories of chaotic motion do not escape to infinity but preserve a recurrence property. They move apart from each other, and then converge again, being confined all the time to the bounded region of the phase space, the region of the chaotic attractor. It is worth noting that the above-cited computer program Dynamics [12] comprises the procedure for calculating the greatest Lyapunov exponent A,. Therefore, it is easy to reveal that, in the irregular crosswell motion visualized in Figures 4.20 and 4.21, the greatest Lyapunov exponent is positive. Consequently, the motion may be referred to as the persistent chaos, and its Poincare map in Figure 4.21 represents the cross-well chaotic attractor. 4.9. Intermittent transition to chaos Let us now return to the Figures 4.18 and 4.19 and consider the route to chaos from the nonresonant attractor Sn to the cross-well chaotic motion. Both figures show that, with the increase of the frequency ft), the nonresonant attractor Sn loses its stability at the saddle-node bifurcation snA, and then it disappears. After that, the system begins to execute the cross-well chaotic oscillations. To explain the sudden change of the character of motion, let us look carefully on the samples of timehistories of the chaotic motion, immediately after disappearance of the Sn attractor. First, on the schematic Figure 4.23, we select a frequency (Ox just prior to the critical 0)snA value, and we also select two other frequencies at higher values inside the chaotic region (frequencies (O2 and ft)3, respectively). No doubt, at ft), the nonresonant jT-periodic Sn attractor does exist (Figure 4.24(a)). Then, at ft)2 and co3, the frequencies that belong to region of the cross-well chaos, somewhat strange phenomenon

Vibrating System with Two Minima of Potential Energy

85

Fig. 4.23. Schematic diagram of the intermittent transition to chaos.

is observed. In long time intervals the system still executes single-well oscillations, close to those illustrated in the previous figure. However, these nearly regular oscillations are interrupted by "bursts" of limited time duration, when the system behavior is chaotic (Figure 4.24(b)). Moreover, time intervals of the regular and chaotic component of motion are unpredictable. We also notice that, as we move away with the frequency 0) from the threshold frequency 0)snA inside the chaotic region, the "bursts" of chaos appear to be longer and more frequent, while the intervals of periodic motion become shorter (Figure 4.24(c)).

86

Chaos, Bifurcations and Fractals Around Us

-0.5 -1.0 -

-1.5 0

(a)

-

200

400

600

800

t

200

400

600

800

t

-1.5 -

0

Fig. 4.24. Samples of time-histories of the intermittent transition to chaos, at the frequencies depicted in Fig. 4.23: (a)co = col; (b)co = co2;

For illustrating time-histories of motion in a considerably longer time interval, we apply again the method of Poincare map. The time-histories of the Poincare displacement x (t) are presented in Figures 4.25(a) and 4.25(b). In this case, time intervals related to the oscillating T-periodic

Vibrating System with Two Minima of Potential Energy

87

~1.5~ 0

200

400

600

800

t

Fig. 4.24(cont). (c) CO = (O3.

motion are mapped as the horizontal lines situated in the vicinity of eitherx p =+l or xp=-l, while "bursts" of the cross-well chaos are visible as dotted, quite dark vertical bands that embrace both potential wells (JC.

s + 1 . 2 , xp . =-1.5).

The considered phenomenon as being one of the possible "routes to chaos", has been called intermittency. 4.10. Large Orbit and the boundary crisis of the cross-well chaotic attractor In this section we explore the nonlinear effects taking place for such system control parameters F, (0 where the cross-well chaotic attractor coexists with a regular attractor denoted SL (see Figure 4.18). Both attractors are illustrated in Figure 4.26, in the Poincare phase plane. For the periodic attractor SL, mapped as one point, the full phase-portrait is also depicted.

88

Chaos, Bifurcations and Fractals Around Us

20

1 *r

i.o

(a)

ffigff

|

p

»

p

-2.0 -

2

-°l^

(b)

-2.0 —

Fig. 4.25. Poincare map representation of the time-histories at the intermittent transition to chaos: (a) co' = a>2; (b) co = ftJ3.

First we notice that the cross-well chaotic attractor is located "inside" the T-periodic orbit; the new periodic attractor also belongs to the category of cross-well motion, but its maximal displacement and velocity

Vibrating System with Two Minima of Potential Energy

89

are larger than those of the cross-well chaos. It looks as if the system did "pay attention" to the existing potential barrier between the two wells.

V(x).

x^

i^

1

^

0

-1 p

i

1.5 •

!

\^

>

+1

sL

|

i

i

i

! i i 1

,

i

i

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

I

-2.0

I

w

P

Fig. 4.26. Potential energy curve V = V(x) and illustration of the two coexisting cross-well oscillations: the Aperiodic Large Orbit SL and the chaotic attractor CH.

90

Chaos, Bifurcations and Fractals Around Us

Let us now have a look at the basins of attraction of the two coexisting attractors, Figure 4.27(a). The T-periodic cross-well attractor SL, also called the "Large Orbit", is represented here by a single point, and its basin of attraction is filled with green color. The chaotic crosswell attractor CH is marked with white color, while its basin of attraction is marked with violet. It is essential for our further considerations that on the boundary of the two basins there exists the point that represents an unstable T-periodic solution (T-periodic saddle). « 1.5

1.0

JL

0.5 0.0

"

-0.5

•«

n,

-1.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Xp

Fig. 4.27(a). Basins of attraction of the chaotic cross-well attractor (violet) and the Large Orbit Si (green); F= 0.144, co = 0.73 h= 0.1 (the chaotic attractor CH is drawn in white).

Let us return for a while to Figure 4.18. One can notice that the region of existence of the cross-well chaotic attractor suddenly ends along the curve denoted cr2. In this section we show that the scenario of destruction of the chaotic attractor appears to be quite different than that considered in the previous section (i.e. along the curve snA). Here, the scenario of boundary crisis is observed again. In Section 4.6, we made

Vibrating System with Two Minima of Potential Energy

91

use of the original definition of crisis as the collision of the chaotic attractor with the unstable orbit sitting on its basin boundary (in the literature, this unstable orbit is referred to as a destroyer saddle). Recently it was found that the "collision" takes place at the homoclinic bifurcation of the unstable orbit. In our case, the relevant unstable orbit (destroyer saddle) is the saddle DL.

v I

10

^

•

-1 n

-2.0

,

J>

T*«S^'^

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Xp

Fig. 4.27(b). Basins of attraction, and the stable and unstable manifolds of the saddle DL prior to its homoclinic bifurcation, F= 0.144, cu = 0.73 /7= 0 . 1 .

We remember that the global homoclinic bifurcation of a saddle takes place when its stable and unstable manifolds become tangent to each other, and then intersect many times (infinitely many if t —> °°) (see reference [19]). Numerical calculations of the manifolds of the saddle DL allow us to identify the critical parameters F, 0) at which the manifolds become tangent. The calculations indicate that, indeed, the homoclinic bifurcation of the saddle DL takes place for the parameters F, at that

92

Chaos, Bifurcations and Fractals Around Us

correspond to the curve denoted cr2, the curve defined by means of computer simulation as the boundary of existence of the cross-well chaotic attractor.

xpf

',

-2.6" ~".5

"t'cP"

• •

-o.£p~ b.o

6.5

>v

1.6~J"" "15

\

Jfcp

Fig.4.27 (c) Basin of attraction of the Large Orbit SL (green), and the stable and unstable manifolds of the saddle DL, after its homoclinic bifurcation, F= 0.144, u = 0.689.

The phenomenon of the boundary crisis of the chaotic attractor, in connection with the homoclinic bifurcation of the destroyer saddle DL, is illustrated in Figures 4.27(a)-4.27(c). Parts (a) and (b) have been performed for the same parameters F, 0) prior to the crisis of the chaotic attractor (F = 0.144, (O = 0.730). Both pictures show basins of attraction of the coexisting attractors, i.e. of Large Orbit SL (green region) and the chaotic attractor CH (violet region), as well as the structure of the manifolds of the saddle DL. One can see that the manifolds neither intersect, nor become tangent. Besides, the stable manifolds WJ,W* coincide with the boundary of both basins, the boundary that is a smooth, regular curve. Instead, the unstable manifold WJ approaches the periodic

Vibrating System with Two Minima of Potential Energy

93

attractor SL, while the other one, WH2, approaching the chaotic attractor, finally takes the shape of the attractor. Figure 4.27(c) is plotted for the parameter values F, CO which correspond to the state after crossing the threshold of homoclinic bifurcation of the saddle DL (F = 0.144, co = 0.689). It is evident that the chaotic attraetor no longer exists, as it has been destroyed together with its (violet) basin of attraction. At the same time, the stable manifolds and the unstable manifold WH2 of the saddle DL intersect. However, the unstable manifold WM2 that formerly approached the chaotic attractor still looks like this attractor (compare with the shape of the attractor marked by white color in Figure 4.27(a)). The problem of geometric similarity of both structures, namely of the chaotic attractor and the unstable manifold that approaches it, as considered by mathematicians, is out of scope of this book. In our computer simulations, this similarity brings us to conclusion that, after the homoclinic bifurcation, resulting motion might appear as a chaotic transient motion. It is illustrated in Figure 4.28.

x

persistent chaos

0

AGO = — 0.01 r - •» i transient chaos

500

1000

1500

2000

2500

t

Fig. 4.28. Sample of the time-history after the boundary crisis of the cross-well chaotic attractor, as a result of a small change Act) of driving frequency.

94

Chaos, Bifurcations and Fractals Around Us

The Figure presents the time-history of the system motion x = x(t) with fixed value of F in the vicinity of the point where a loss of stability of the chaotic attractor takes place. The system exhibits a persistent chaotic motion on the chaotic attractor (0 < t < 500), then a sudden small change of driving frequency (Act) = -0.01) is introduced, and consequently the system crosses the threshold of crisis of the chaotic attractor. However, for a certain, quite a long time interval, the system still exhibits the same type of chaotic motion as before. This is undoubtly the transient chaos, as the chaotic attractor has already been destroyed. Finally, at t = 2000 the transient chaos suddenly disappears, and the system motion restabilizes on the unique existing attractor, i.e. the Tperiodic Large Orbit SL. The considered vibrating system, called the two-well system, is characterized by a huge variety of bifurcational and chaotic phenomena. We have highlighted some of them, still confined to a narrow range of the frequency co , shown in Figure 4.18. It is worth looking at some other attractors, the attractors that appear in a wider range of the driving frequency. 4.11. Various types of attractors of the two-well potential system A collection of all attractors of the two-well system that exist within the range of frequency between co = 1.0 and CO = 0.30 is sketched in Figures 4.29(a) and 4.29(b). For the sake of clarity, periodic attractors are drawn in the form of phase portraits, while the chaotic ones - in the form of their Poincare maps. The attractors designated with numbers 1, 2, ..., 11 exist for control parameter values F, co denoted by the same numbers in Figure 4.18. Point 1 corresponds to a low forcing value F, as it illustrates the coexistence of T-periodic resonant and nonresonant attractors (F = 0.06, CO = 0.74); point 11 pertains to the region where only the cross-well chaotic attractor exists (F = 0.1, co = 0.74). The remaining attractors appear at a higher value of the forcing parameter, F = 0.17. Thus, point 2 lies in the region of existence of the resonant attractor Sr, then at points 3 and 4 one observes two following perioddoubling bifurcations of the attractor. Point 5 is related to the motion in

So1

T—

~

,-

ifj

\ ^

1

0j —

I

j _

\ ——~~~"^

ID

k-

™ ui

CD

L—

3

LL

Fig. 4.29(a). Various types of attractors in the two-well system (points 1-11 marked in Fig. 4.18).

Vibrating System with Two Minima of Potential Energy 95

96

Chaos, Bifurcations and Fractals Around Us

a "periodic window" inside the cross-well chaotic region; it turns out to also be a "cross-well" motion; however, the sequence of singlewell oscillations and the jumps over the potential barrier is so well synchronized that the resulting attractor is periodic with the period 5T. Next, in point 6, we observe two coexisting attractors, namely the cross-well chaotic attractor and the T-periodic Large Orbit. With a further decrease of the frequency, we again face a "periodic window" inside the region of cross-well chaos. Point 7 shows the coexistence of 3r-periodic "window" motion with the Large Orbit, whereas point 8 corresponds again to the coexistence of the Large Orbit and the cross-well chaotic attractor. Point 9 illustrates the case where the Large Orbit is a unique attractor. Point 10 lies in the region beyond the saddle-node bifurcation curve snA, thus the Large Orbit coexists there with the T-periodic nonresonant attractor Sn. The consecutive points 12-20 lie out of the region of the principal resonance (they cover the zone of the so-called subsuperharmonic resonance). We observe here the Large Orbit that coexists with Sn attractor after its first (point 12) and second (point 13) period doubling bifurcation. Points 14-15 correspond to the region where the Large Orbit is the unique attractor again. With a further decrease of frequency, the attractor becomes asymmetric, and then undergoes period doubling bifurcation (point 16, CO = 0.405). As a result of the following cascade of period doublings, the Large Orbit is transformed into a cross-well chaotic attractor (point 17, CO = 0.400). The latter one is quickly destroyed, being replaced by a pair of multiperiodic cross-well attractors (points 18 and 19). After a sequence of consecutive bifurcations, the system will again posses two attractors, a single-well and a cross-well one, but both of multiple periods (point 20, co = 0.30).

18

19

20

L I v

Fig. 4.29(b). Various types of attractors in the two-well system at lower values of the frequency.

17

I—(M

Ns^ N/\

£ S-

|

1 J

16

v^TO v/M WKJ v^KJ

1

N^

15

i

14

r\\/\

13

^1

I

Two Minima of

Chapter 5

Closing Remarks

The two simple mechanical models, i.e. pendulum and two-well potential system, introduced the Readers to fundamental theoretical concepts and essential chaotic phenomena that arise in nonlinear, dissipative oscillators, driven by periodic force. In both systems, the principal resonance curves were bowed to the left, towards the lower frequency, that is, their nonlinearity had a softening property. One may ask: what about oscillators with a hardening type of restoring force characteristics? Indeed, the oscillators with hardening elastic nonlinearity may also exhibit chaotic motion. Recall that "Ueda's strange attractors" were first found in the Duffing system with this type of nonlinearity. However, the systems with hardening type elastic nonlinearity are not as useful in the task of brief introduction to the chaotic dynamics as those with softening nonlinearity. The problem is that, in the classical Duffing system, the strange attractors discovered by Ueda appear in very small regions of system parameters. It would be hard to reveal such fundamental phenomena as routes to chaos or crisis of chaotic attractors, even in a very precise computer experiment. Since the discovery of the chaotic phenomena in the dissipative forced nonlinear oscillators was made with the use of an analog computer, the researchers faced an essential problem: did the irregular solutions occur only in the computational simulations, or did they really exist in the physical world around us? No wonder that, at the next stage of investigations, attention was focused on physical experiments. In the field of mechanics, it was the experiment performed by F. Moon that proved, without a doubt, that chaotic motion may occur in a real, simple mechanical system (see references [8, 9, 13]). 98

Closing Remarks

99

In Moon's experiment, the experimental set-up consisted of a slender cantilever beam placed on a vibration shaker. The free end of the beam was within a nonuniform field of permanent magnets. An approximate equation of motion of the beam was derived in the form of the "twowell" potential equation, the same that was studied in Chapter 4. Comparison of the time-histories of the mechanical device with those obtained by means of an analog computer, published in 1979, was a milestone in the further investigations of the chaotic motion in nonlinear oscillators [9, 10, 13]. Physical experiments played a significant role in mechanics of fluids. The most famous ones are those related to Rayleigh-Benard convection and Taylor-Couette flow between cylinders (see ref. [9, 13]). Chaotic phenomena were found in many technical devices, for instance, in: • wheel-rail systems, • buckled elastic structures, • gyroscopic systems, • aeroelastic systems, and then: • nonlinear acoustical systems, • nonlinear optical systems (lasers), • feedback control systems, • electric circuits, • chemical reactions, • biological systems and many others. Researchers have also begun to discuss the problem of controlling the chaos, and, quite recently, the use of chaotic systems in creating "safe communications" is becoming a point of common interest. Another field of applications of the chaos theory is related to biology and medical problems. In this case, we are not able to derive "equation of motion"; consequently, we cannot apply a computer simulation technique. In contrast, the investigation begins with measurements of the time series of the biological process involved. After collecting a huge amount of the data on, for instance, human heart beats or the tremor of human hands due to Parkinson's disease, teams of experts from various

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disciplines of knowledge try to perform a deep analysis of the results. First, they have to answer the question: are the processes chaotic, or do they possess stochastic properties? The further work is laborious, arduous and costly, but it is stimulated by the hope that this research may lead to new methods in diagnostics, prophylaxis and, then, therapy of some human heart and brain diseases. Although our attention is focused on the macroscopic world around us, let us look also for a while into cosmos, on the mysterious irregularity of motion of Hyperion, one of the moons of the planet Saturn. This is how I. Stewart ends his "fairy-tale" on the strange behavior of this newly discovered celestial body in his famous book "Does God play Dice?" (see reference [16]): "One moon, Hyperion, is unusual. It is irregular in shape, a celestial potato. Its orbit is precise and regular; but its attitude in orbit is not. Hyperion is tumbling. Not just end over end, but in a complex and irregular pattern. Nothing in this pattern defies Newton's law: the tumbling of Hyperion obeys the laws of gravitation and dynamics. Both its position in orbit, and its attitude, are determined by the identical physical laws, the same mathematical equations. Its position corresponds to a regular solution of those equations; but its attitude corresponds to an irregular solution. Hyperion's tumbling is due not to random external influences, but to dynamical chaos. Why is Hyperion chaotic? For that matter, why are all the other bodies regular? Is it the potato-like shape? Are all potatoes chaotic? Not at all. The reasons are more subtle, more complicated, and much more interesting. Hyperion's chaotic motion is a cosmic coincidence. At various times in the history of the Solar System, other bodies have evolved into, and back out of, a period of dynamical chaos. But it so happens that Hyperion is undergoing this process at precisely the time when the human race has become interested in it."

Bibliography

[1] Argyris, J., Faust, G. and Haase, M. An Exploration of Chaos, vol. VII of the series: Texts on Computational Mechanics. J. Argyris (Editor), North-Holland, Amsterdam 1994. [2] Grebogi, C , Ott, E. and Yorke, J.A. (1983). Crises, sudden changes in chaotic attractors and transient chaos, Physica D7, pp. 181-200. [3] Guckenheimer, J. and Holmes, P.J. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York 1983. [4] Hayashi, Ch. Nonlinear Oscillations in Physical Systems. Princeton University Press, Princeton, N.J 1985. [5] Holmes, P.J. (1979). A nonlinear oscillator with a strange attractor, Phil. Trans. Roy. Soc. London, A292(1394), pp. 419-448. [6] Mandelbrodt, B. The Fractal Geometry of Nature. W.H. Freeman, San Francisco 1982. [7] McDonald, S.W., Grebogi, C , Ott, E. and Yorke, J.A. (1985). Fractal basin boundaries, Physica D17, pp. 125-153. [8] Moon, F.C. (1980). Experiments on chaotic motion of a forced nonlinear oscillator - strange attractors. ASMEJ. of Applied Mechanics, 47, pp. 638-644. [9] Moon, F.C. Chaotic Vibrations, An Introduction for Applied Scientists and Engineers. John Wiley & Sons, New York 1987. [10] Moon, F.C. and Holmes, P.J. (1979). A magnetoelastic strange attractor. J. Sound and Vibration, 65(2), pp. 275-296. [11] Nayfeh, A.H.. and Balachandran, B. Applied Nonlinear Dynamics. John Willey & Sons, Inc., New York 1995. [12] Nusse, H.E. and Yorke, J.A. Dynamics: Numerical Explorations. 2nd ed., SpringerVerlag, New York 1998. [13] Ott, E. Chaos in Dynamical Systems. Cambridge University Press, Cambridge 1993. [14] Ruelle, D. Elements of Differentiate Dynamics and Bifurcation Theory. Academic Press, San Diego/London 1989. [15] Schuster, H.G. Deterministic Chaos. An Introduction. Physik-Verlag, Weinheim 1984. 101

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[16] Stewart, I. Does God Play Dice? The New Mathematics of Chaos. Penguin Books, London 1990. [17] Szemplinska-Stupnicka, W. The Behavior of Nonlinear Vibrating Systems; vol. I Fundamental Concepts and Methods: Applications to Single-Degree-of-Freedom Systems. Kluwer Academic Publishers, Dordrecht 1990. [18] Szemplinska-Stupnicka, W. and Rudowski, J. (1993). Steady-states in the twinwell potential oscillator: Computer simulations and approximate analytical studies. CHAOS, Int. J. Nonlinear Science, 3(3), pp. 375-385. [19] Szemplinska-Stupnicka, W. and Janicki, K.L. (1997). Basin boundary bifurcations and boundary crisis in the twin-well Duffing oscillator: scenarios related to the saddle of the large resonant orbit. Int. J. Bifurcation and Chaos 7(1), pp. 129-146. [20] Szemplinska-Stupnicka, W. and Tyrkiel, E. (1997). Sequences of global bifurcations and the related outcomes after crisis of the resonant attractor in a nonlinear oscillator. Int. J. Bifurcation and Chaos 7(11), pp. 2437-2457. [21] Szemplinska-Stupnicka, W., Zubrzycki, A. and Tyrkiel, E. (1999). Properties of chaotic and regular boundary crisis in dissipative driven nonlinear oscillators, Nonlinear Dynamics, 19, pp. 19-36. [22] Szemplinska-Stupnicka, W., Tyrkiel, E. and Zubrzycki, A. (2000). The global bifurcations that lead to transient tumbling chaos in a parametrically driven pendulum, Int. J. Bifurcation and Chaos 10(9), pp. 2161-2175. [23] Szemplinska-Stupnicka, W., Tyrkiel, E. and Zubrzycki, A. (2001). On the stability "in the large" and unsafe initial disturbances in a nonlinear oscillator, Computer Assisted Mech. Engng. Sci., 8, pp. 155-168. [24] Szemplinska-Stupnicka, W. and Tyrkiel, E. (2002). The oscillation-rotation attractors in a forced pendulum and their peculiar properties, Int. J. Bifurcation and Chaos, 12(1), pp. 159-168. [25] The Science of Fractal Images. Eds. H.O. Peitgen and D. Saupe. Springer-Verlag, New York 1988. [26] Thompson, J.M.T. and Stewart, H.B., Nonlinear Dynamics and Chaos. John Wiley & Sons, Chichester 1986. [27] Thompson, J.M.T., Stewart, H.B. and Ueda, Y. (1994). Safe, explosive and dangerous bifurcations in dissipative dynamical systems, Phys. Rev., E 49 (2), pp. 1019-1027,. [28] Tyrkiel, E., Szemplinska-Stupnicka, W, and Zubrzycki, A. (2000). On the boundary crises of chaotic attractors in nonlinear oscillators, Computer Assisted Mech. Engng. Sci., 7, pp. 743-755. [29] Ueda, Y. (1979). Randomly transitional phenomena in the system governed by Duffing's equation, J. Stat. Phys., 20(2), pp. 181-196. [30] Ueda, Y. (1980). Steady motions exhibited by Duffing's equation: a picture book of regular and chaotic motions, in New Approaches to Nonlinear Problems in Dynamics, ed. P.J. Holmes, SIAM, Philadelphia, pp. 331-322.

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[31] Vibrations. Ed. S. Kaliski. Polish Scientific Publishers PWN, Warsaw - Elsevier, Amsterdam / Oxford / New York / Tokyo, 1992. [32] Wiggins, S., Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer-Verlag, New York, 1990.

Index

homoclinic, 33, 61-63, 67, 72, 91-93 local, 17 period-doubling, 26-29, 49, 55, 74, 79, 94, 96 saddle-node, 17, 24, 28, 39,41, 49, 55, 57, 77, 79, 84, 96 symmetry-breaking, 26, 29, 40

attractor, 5, 8-10, 16, 17, 22, 24-37, 3 9 ^ 4 , 46, 49, 55, 57-59, 61-63, 65-67, 69,71-79, 82, 84, 87, 89-98 asymmetric, 55, 96 chaotic, 27, 41-44,46, 49, 55, 57, 72, 74-79, 82, 84, 87, 89, 98 cross-well, 81, 82, 84, 87, 90, 96 nonresonant, 17, 26, 28, 31, 33, 37, 41, 55, 62, 66, 72, 74, 75, 77, 84, 94, 96 oscillation-rotation, 34, 35, 37 periodic, 24, 27, 34, 41, 55, 61, 74, 77, 79, 84, 87, 90, 94 resonant, 17,24,26,28, 31,55, 57-59, 61, 62, 66, 71-75, 77, 79, 84, 94 single-well, 55, 57, 78, 79, 96 strange, 5, 8, 10, 20, 44,49, 82, 98 symmetric, 26 unsymmetric, 26

Cantor set, 44, 46, 47, 48 chaos, 1, 4, 5, 27, 28, 33, 40, 49, 55, 62, 68,70,71,79,84-86,88,89,94, 96,99 cross-well, 70, 79, 84, 87, 89, 96 persistent, 79, 84 single well, 68 transient, 33, 62, 68, 70, 94 crisis, 27, 28, 39, 49, 55, 57, 72, 74-78, 87, 90-94 boundary, 27,28, 49, 55,72,74, 75, 76, 77, 87, 90, 92, 93

basin, 28, 30-37, 46, 49, 58-67, 69, 74-77, 90-93 boundaries, 31,33, 35,59,60, 62, 63, 66, 75, 77, 90-92 of attraction, 28, 30-37, 46, 49, 58-67, 69, 74-77, 90-93 bifurcation, 17, 24, 26, 27-29, 33, 3 9 ^ 1 , 49, 55, 57, 59, 61-67, 69, 71-74,77,79,81,84,91-94,96 diagram, 24, 26-28,40, 41, 49, 72-74, 79, 81 global, 33, 61, 63-65, 67, 69, 71 heteroclinic, 64, 66, 69, 71, 72

dimension, 44, 46-48 equation, 1-3, 5, 6, 11-16, 20,49-53, 99 differential, 1, 3, 5, 6, 13, 14, 51 linear, 13, 14 logistic, 2 nonlinear, 13, 15, 16 force, 1, 2, 6, 11, 12, 14, 19, 20, 50, 51, 52, 57, 77 damping, 6, 11, 12 105

106

Chaos, Bifurcations and Fractals Around Us

drag, 2, 6 external, 12, 50 gravity, 11, 50 fractal, 12, 31, 33, 35, 36, 44, 46,47, 49, 60, 64-66, 75-77, 82 boundaries, 31, 35, 60, 66, 75 dimension, 44, 46, 47 structure, 12, 33, 35, 36, 64, 66, 75-77, 82 Hyperion, 100 intermittency, 87 intermittent transition, 49, 85, 86, 88 jump phenomenon, 17 Lyapunov exponent, 8, 44, 49, 82-84 manifolds, 23, 24, 30-33, 57-60, 62-64,71,91-93 stable, 23, 24, 30-33, 58-60, 62, 63,71,91-93 unstable, 23, 30-33, 58-60, 62, 71, 91-93 Melnikov criterion, 49, 57, 59-62, 66 motion, 1-3,11-21, 28, 34, 37, 39, 40, 41, 44-^6, 50-53, 55, 57, 60-62, 64-66, 70, 71, 74, 77, 79, 82, 84-86, 93, 94, 96, 98, 99 chaotic, 2, 20, 28, 39, 40, 41, 44-46, 61, 65, 66, 77, 82, 84, 93, 94, 98, 99 cross-well, 84, 86 periodic, 18, 19, 57, 66, 85, 86 single-well, 53, 55, 66 transient, 8, 17, 39, 60-62, 64-66, 70,71,77,93 oscillations, 2, 5-7, 11, 13-15, 17, 20, 23,39,42,53,54,59,70,71,77, 79, 81, 82, 84, 85, 89, 96 chaotic, 42, 81, 82 cross-well, 81, 82, 84, 89

forced, 14, 15, 20 free, 14, 15 nonlinear, 11, 15, 23 nonperiodic, 8 periodic, 6, 54, 59, 70 regular, 85 steady state, 5-8, 15, 17, 39, 71, 77,79 subharmonic, 7 pendulum, 1,11-13, 15, 20, 21, 37, 39, 41, 49, 55, 98 periodic window, 37, 41, 96 phase, 14, 18, 19, 21, 26, 28, 29, 34, 35, 44, 54, 55, 57, 60, 72, 74, 75, 77, 83, 84, 87, 94 plane, 18, 19, 21, 28, 35,44, 57, 60, 66, 87 portrait, 18, 26, 29, 34, 54, 55 space, 72, 74, 75, 77, 83, 84, 87, 94 Poincare\ 17-21, 24, 28, 34, 35, 41-44, 57, 58, 72, 75, 82, 84, 86-88, 94 coordinates, 19 displacement, 24, 73, 86 map, 18-21, 34,41-44, 57, 75, 82, 84,86 plane, 19-21, 28, 35, 58, 87, 88, 94 potential energy, 20,49, 50, 52-54, 57, 75 resonance, 14-17, 24, 55-57, 62, 65, 71,96 curve, 16, 17,24, 55, 57 principal, 15, 16, 24, 55, 56, 62, 65,71,96 routes to chaos, 28, 84, 87, 98 saddle, 17,22-24, 28, 30-34, 39,41, 55, 57-64, 71, 75, 77, 79, 84, 90-93 destroyer, 91, 92 hilltop, 57-64 sampling time, 19 sensitivity, 33, 36, 44, 45, 82, 83 exponential, 82

Index solution, 8,12,14-24, 34, 44-46, 55, 57, 63, 90 chaotic, 46 stable, 16, 21-23, 46, 55 unstable, 12, 16, 17, 20, 21, 23, 24, 46, 57, 63, 90 system, 1-3, 5, 6, 12, 14-17,19, 20, 24, 28, 33, 37, 39, 40,44, 49-53, 55,57,61,62,66,70,71,74,75, 83-85, 87, 89, 94-99 autonomous, 5, 14, 20 chaotic, 33, 61, 99 conservative, 14 damped, 15, 83 deterministic, 2 dissipative, 2, 14, 83 Duffing, 6-8, 98 dynamical, 1-3, 5, 8, 28, 44 linear, 6, 16, 17 nonautonomous, 5, 14, 17 nonlinear, 16, 17 two-well, 49, 50, 94, 95, 97, 98 vibrating, 49, 94 trajectory, 18,44, 46, 55, 59-61, 66, 74,77 unpredictability, 1, 2, 33, 49,71, 73, 74

107