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Physics + Uspekhi 50 (9) 939 + 964 (2007) METHODOLOGICAL NOTES

# 2007 Uspekhi Fizicheskikh Nauk, Russian Academy of Sciences PACS numbers: 05.45. ë a, 05.45.Ac

Dynamical chaos: systems of classical mechanics
A Loskutov
DOI: 10.1070/PU2007v050n09ABEH006341

Contents
1. Introduction 2. Background 3. Hamiltonian mechanics 4. Nonlinear resonance
3.1 Integrable systems; 3.2 Perturbed motion 4.1 Small denominators; 4.2 Universal Hamiltonian; 4.3 Width of the separatrix; 4.4 Internal resonances; 4.5 Overlapping of resonances; 4.6 Higher-order resonances 5.1 The Kolmogorov theorem; 5.2 The Arnol'd diffusion 6.1 Twist map; 6.2 Fixed-point theorem; 6.3 Elliptic and hyperbolic points; 6.4 Splitting of separatrices. Homoclinic tangles 7.1 The Mel'nikov function; 7.2 The Duffing oscillator and nonlinear pendulum 8.1 Ergodicity and mixing; 8.2 Unpredictability and irreversibility; 8.3 Decay of correlations 9.1 The Lorentz gas; 9.2 Scattering billiards with oscillating boundaries. The Fermi acceleration; 9.3 Focusing billiards with oscillating boundaries. Particle deceleration

939 941 942 943 947 949 952 954 956 960 961

5. Elements of the Kolmogorov + Arnol'd + Moser theory 6. The nature of chaos

7. The Mel'nikov method

8. Principal properties of chaotic systems 9. Billiards

10. Conclusion References

Abstract. This article is a methodological manual for those who are interested in chaotic dynamics. An exposition is given on the foundations of the theory of deterministic chaos that originates in classical mechanics systems. Fundamental results obtained in this area are presented, such as elements of the theory of nonlinear resonance and the Kolmogorov + Arnol'd + Moser theory, Â the Poincare + Birkhoff fixed-point theorem, and the Mel'nikov method. Particular attention is given to the analysis of the phenomena underlying the self-similarity and nature of chaos: splitting of separatrices and homoclinic and heteroclinic tangles. Important properties of chaotic systems Ð unpredictability, irreversibility, and decay of temporal correlations Ð are described. Models of classical statistical mechanics with chaotic properties, which have become popular in recent years Ð billiards with oscillating boundaries Ð are considered. It is shown that if a billiard has the property of well-developed chaos, then perturbations of its boundaries result in Fermi acceleration. But in nearly-integrable billiard systems, excitaA Loskutov Physics Department, Lomonosov Moscow State University, Vorob'evy gory, 119992 Moscow, Russian Federation Tel. (7-495) 939 51 56. Fax (7-495) 939 29 88 E-mail: Loskutov@chaos.phys.msu.ru Received 31 January 2007, revised 25 April 2007 Uspekhi Fizicheskikh Nauk 177 (9) 989 + 1015 (2007) Translated by A V Getling; edited by A M Semikhatov

tions of the boundaries lead to a new phenomenon in the ensemble of particles, separation of particles in accordance their velocities. If the initial velocity of the particles exceeds a certain critical value characteristic of the given billiard geometry, the particles accelerate; otherwise, they decelerate.

1. Introduction
For a long time, the concept of chaos was associated with the assumption that, at least, the excitation of an extremely high number of degrees of freedom is necessary in the system. The formation of this idea seems to have been influenced by the concepts of statistical mechanics, in which the motion of an individual gas particle can be predicted in principle but the behavior of a system consisting of a huge number of particles is extremely complex, and therefore a detailed dynamic appr oach is meaningless. This dictated the need for a statistical analysis. But extensive studies have demonstrated that the validity o f t he statis tical l aws a nd s tatistical description is not restricted to highly complex systems with a large number of the degrees of freedom. Random behavior can also be exhibited by entirely deterministic systems with a moderate number of the degrees of freedom. Here, the point is not the complexity of the system or the presence of external noise but the emergence of an exponential instability of motion at certain values of the parameters. The dynamics of systems subject to such an instability is called dynamic stochasticity, or deterministic (dynamic) chaos. Investiga-


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tions in this area are of fundamental importance because they disclose the nature of randomness by supplementing the hypothesi s of molecular chaos with the hypothesis of dynamic stochasticity.  H Poincare [1] was the first to note the relation between statistics and instabilities. However, a statistical approach to the description of systems with numerous degrees of freedom was previously suggested by L Boltzmann [2], who conjectured that the motion of particles in a rarefied gas should be regarded as random and that the entire energetically allowed phase-space domain is accessible to each particle. Such a view of the properties of multiparticle systems, known as the ergodic hypothesis [2 + 4], became the basis of classical statistical mechanics. A rigorous substantiation of this hypothesis could not be found for a long time, however. Some progress in this direction was achieved due to studies by P Ehrenfest [5, 6] (see also Refs [7, 8]). In particular, they allowed establishing applicability limits for the laws of statistical mechanics. However, a well-known study by E Fermi, J Pasta, and S Ulam [9] (see Refs [10, 11] for a more detailed exposition), who made the first attempt to verify the ergodicity hypothesis, put the problem of substantiation of statistical physics in the forefront again. This problem can be partially resolved based on studies by  Poincare (see Ref. [12]), who concluded that the motion of a system is extremely complex in the neighborhood of unstable fixed points in phase space. This was the earliest indication of the possible chaotic properties of nonlinear dynamical systems. Later, G D Birkhoff [13] showed that if the ratio of the frequencies is rational (i.e., at a resonance), stable and unstable fixed points appear in phase space. Higher-order resonances change the topology of phase trajectories and lead to the formation of a chain of `islands.' It turned out that the regular perturbation theory fails to describe such resonances, because the solutions are strongly perturbed near the resonances, and therefore small denominators emerge in the expansion and the series diverge. N S Krylov [14] was the first to deeply investigate the nature of statistical laws. He showed that the property of mixing and the related local instability of nearly all trajectories of the corresponding dynamical systems underlie this nature. In view of this, M Born [15] (see also [16]) suggested that the behavior of the systems is not predictable in classical mechanics. Later, the dynamics due to such instabilities in the systems came to be known as dynamic stochasticity, or deterministic (dynamic) chaos. The word `chaos,' in this meaning, seems to have been introduced by J A Yorke [17] (see Ref. [18], p. 338). But as noted by Ya G Sinai, the word combination `deterministic chaos' was first used by B Chirikov and G Ford in the 1960s. Physically, due to unavoidable fluctuations (i.e., small perturbations of the initial conditions), the initial state of the system is to be specified by some distribution. The problem is in predicting the evolution of the system based on this initial distribution. If the system is stable, such that small perturbations do not increase exponentially with time, its behavior is predictable. In contrast, if the system is subject to exponential instability (which is expressed by saying that the system has a sensitive dependence on the initial conditions), the process allows only a probabilistic description. In essence, precisely these considerations formed the basis of the modern views of dynamical chaos. The discovery that it is chaos, rather than external noise, that mainly determines the behavior of the system was unexpected (see Ref. [20] for a review).

The schools of A N Kolmogorov and A A Andronov, to which a brilliant group of outstanding contemporary mathematicians belongs, have deeply influenced the development of the theory of dynamical chaos. In particular, Kolmogorov's theorem of the preservation of almost periodic motion in weakly perturbed Hamiltonian systems, proved by V I Arnol'd and J Moser and known as the Kolmogorov + Arnol'd + Moser (KAM) theorem (see Refs [21 + 25]), became a keystone in understanding the origin of chaotic behavior. In their early studies, D V Anosov [26] and Sinai [27, 28] showed that dynamical chaos is a widespread phenomenon. In his pioneering investigations of the bifurcations of a saddle + focus separatrix [29, 30], L P Shil'nikov developed, among other things, a special technique for the analysis of the dynamics of systems near saddle-type trajectories and uncovered the extreme complexity of the structures that develop as homoclinic trajectories appear. It was demonstrated that the behavior of systems must be complex in the full neighborhood of the parametric values at which a homoclinic orbit exists. Later, Shil'nikov, L M Lerman, N K Gavrilov, I M Ovsyannikov, D V Turaev, and others developed new methods that allowed describing a finite number of bifurcations that lead to chaotic dynamics (see Refs [31, 32] and the references therein). A new stage in explaining chaotic behavior and its origin in deterministic systems was initiated by Kolmogorov's and Sinai's studies [33 + 35], where the concept of entropy was introduced for dynamical systems. These studies laid the foundations of a consistent theory of chaotic dynamical systems. Various abstract mathematical constructions have played an important role in the development of the theory of deterministic chaos. In particular, S Smale [36], to disprove the hypothesis of the density of systems that exhibit only a periodic-type behavior, constructed a notable example, currently known as the `Smale horseshoe.' This example implies that there exist systems that have both an infinite number of periodic orbits with different periods and an infinite number of aperiodic trajectories [18, 36, 37]. Subsequent to the Smale horseshoe, Anosov's C-systems were found [26, 38], which are characterized by the most pronounced mixing properties. Such systems were generalized by introducing Smale's `Axiom A' [37] (see also Refs [39 + 41] and the references therein) and hyperbolic sets [18, 37, 40 + 42]; these generalizations specified an important class of dynamical systems that have the property of the exponential instability of trajectories (see Ref. [43] for a review). At nearly the same time, mathematical studies began appearing in which attempts were made to substantiate statistical mechanics based on the analysis of billiard systems [27, 28]. Originally, billiards were introduced as simplified models appropriate for the consideration of certain problems of statistical physics [13] (see also the references in Refs [44, 45]). A billiard on a plane is a dynamical system that describes bodies (balls) moving inertially inside a bounded domain, in accordance with the law of equality of the incidence and reflection angles. In essence, planar mathematical billiards are the usual billiards without friction, although with an arbitrary configuration of the table and without pockets. Krylov's problem of mixing in a system of elastic balls [14] was first solved using billiard systems. Furthermore, it was shown that systems corresponding to billiards with scattering


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boundaries have much in common with geodesic flows in negative-curvature spaces, i.e., with Anosov flows. Somewhat later, the class of billiard systems capable of exhibiting chaotic properties was substantially extended (see Refs [45 + 47] and the references therein). It was shown based on a generalization of such systems (a modified two-dimensional Lorentz gas) that motion in purely deterministic systems can be similar to Brownian motion [44, 45]. This result became the first rigorous confirmation of chaotic properties exhibited by dynamical systems (not involving any random mechanism). Further investigations of nonlinear systems, both theoretical and experimental, have shown how typical the chaotic behavior is in systems with few degrees of freedom. It became obvious that chaotic properties can be manifested by a variety of nonlinear systems; if chaos is not revealed, this can merely result from the fact that the development of chaos is restricted either to very small domains in the parameter space or to physically unrealizable domains. How does chaotic motion originate? What is the nature of chaos? Seemingly, there should be many paths toward the onset of chaos. But it became known that the scenarios of chaotization are far from numerous. Moreover, some of them obey universal laws and are independent of the nature of the system. The same development scenarios for chaos are inherent in a variety of objects. The universal behavior resembles the usual second-order phase transitions, and invoking renormalization-group and scaling techniques known in statistical mechanics opens new avenues in investigating chaotic dynamics. This article presents the foundations of the theory of dynamical chaos. We describe the principal results obtained in the field that belong to classical mechanics, such as elements of the theory of nonlinear resonance and the  KAM theory; the Poincare + Birkhoff fixed-point theorem, which is important for understanding the sources of chaotic behavior; and the Mel'nikov method, which allowed obtaining a criterion for the origin of chaos analytically in some cases. Particular attention is given to the nature of chaos. Specifically, we detail the factors that lead perturbed systems to manifest self-similarity, to the splitting of separatrices, and to homoclinic and heteroclinic tangles. We also show that unpredictability, irreversibility, and decay of temporal correlations occur in systems in which such phenomena are observed. In Section 9, we describe models of nonequilibrium classical statistical mechanics with chaotic properties, highly popular today: billiards with oscillating boundaries. The Lorentz gas and stadium-type billiards are considered in detail. An interesting result is presented: the analytic form of billiard-particle acceleration law, i.e., a proof of the presence of Fermi acceleration in billiards with well-developed chaos. But if a billiard system is near an integrable system, such that the curvature of the billiard boundary is not large, small oscillations of the boundary lead to a new phenomenon. A specific, billiard version of Maxwell's demon originates: a billiard particle either accelerates or decelerates depending on the initial conditions. In other words, perturbation of the boundaries of such billiards results in a velocity stratification of the particle ensemble. The modern mathematical techniques used to analyze the chaotic properties of dynamical systems are fairly complex. But in this article, we pursue the aim of giving a general idea of the origin of the phenomenon of deterministic chaos and expose the fundamental concepts underlying the currently

known approaches to chaotic dynamics. Therefore, our presentation is mainly based on geometric techniques and a qualitative approach. Although most of the results described here have been known for a rather long time, we present them in a form appropriate for a nonexpert to comprehend the origins of chaotic dynamics. Therefore, in particular, this article is methodological.

2. Background
T h e s u b j e c t o f o u r a n a l ys i s i s t h e s y s t e m s d e sc r ib e d b y ordinary differential equations x vxY a Y 1

wher e v fv1 Y v2 Y F F F Y vl g is a v ector functi on (typical ly assumed to be smooth), a symbolizes the set of parameters, and x fx1 Y x2 Y F F F Y xl g i s an l-di m ensi onal vector wi t h components x1 Y x2 Y F F F Y xl that characterizes the state of a dynamical system. If the substitution of a function FtY ci , ci const, in Eqn (1) turns them into identities, this function is called a solution. Specifying initial conditions x0 x0 uniquely determines the solution at any instant t, xt FtY x0 X 2

Any solution xt of system of equations (1) can be geometrically represented by a curve in the l-dimensional space of variables x1 Y F F F Y xl . This l-dimensional space is called the phase space of the system (we let M denote this space). Each state of the dynamical system corresponds to a point in M, and each point in M corresponds to a unique state of the system. Changes in the state of the system can be interpreted as the motion of a point (called the representation point) in the phase space. The trajectory of this representation point, i.e., the set of its consecutive positions in the phase space M, is called the phase trajectory. We assume that system (1), whose phase space is M, was in the state x0 at an instant t0 . Then, generally, it is in another state at an instant t T t0 . We let F t x0 denote this new state. Thus, for any t, we define the evolution operator, or the shift map F t X M 3 M of the phase space M into itself. The map F t transforms the system from its state at t0 into the state at t. In other words, solution (2) of Eqns (1) establishes a correspondence between the point xi t0 , i 1Y 2Y F F F Y l, of the phase space M at the instant t0 and a certain phase-space point xi t at an instant t: F t x0 xt X 3

Therefore, in the general case, any phase-space domain O0 transforms in time t into another domain, Ot F t O0 , under the action of the map F t . The map F t X M 3 M is also called the phase flow, and the function vx is the vector field of the given dynamical system whose phase space is M. If the phase flow F t has a secant S, i.e., a certain codimension-1 hypersurface that is transversally intersected by phase curves, then a map F can be defined on S as follows: to any point p of S we associate the closest (next to p) point, p H , of intersection of the phase curve with the same hypersurface S. Then the analysis of the dynamics of the original system reduces to analyzing the properties of the map F,  which is called the succession function, or the Poincare map  (sometimes, the Poincare return map).


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Systems for which div v 0 are called conservative. Most objects considered in classical mechanics belong to this class. Investigating the evolution of conservative systems is of fundamental importance because it is related to a number of problems such as the substantiation of the Boltzmann ergodic hypothesis, planetary motion and the N-body problem, the dynamics of charged particles and plasma heating, etc. We begin our analysis with the Hamiltonian approach, whose advantage is not in its formalism but in a deeper interpretation of the physical essence of the phenomenon in question. In fact, the Hamiltonian approach is a geometrical method of analysis. It has some advantages and allows obtaining solutions to problems that are not amenable to other techniques of analysis.

where j is the angle of deviation from the vertical and g is the acceleration of gravity. The equations of motion of the pendulum are p Àmgl sin j, j paml 2 , or j o2 sin j 0 Y 0 6

3. Hamiltonian mechanics
The Hamiltonian formalism is based on the well-known Ha milt o n e qu at ion s, wh ic h ar e give n b y th e s yste m o f ordinary differential equations qi qH Y qpi pi À qH qq i 4

p where o0 gal is the oscillation frequency. Hamiltonians of th is type a re ty p ic a l of many p r obl e ms and pl a y a fundamental role in classical mechanics. If the full energy of the pendulum, H E, exceeds the maximum value of the potential energy, E Erot b mgl, the momentum p is always different from zero, which leads to an infinite increase in the angle j, i.e., to the rotation of the pendulum. Energies E Eosc ` mgl correspond to oscillations of the pendulum. If E % Es mgl, the oscillation period tends to infinity and the motion follows the separatrix between the two types of motion, oscillation and rotation (Fig. 1a). The equation of the separatrix can be written as  à ps Æ2o0 ml 2 cos js a2, js 4 arctan exp o0 t À p, where the plus and minus signs respectively correspond to the upper and lower branches.
p E Es 0 j a
1

(where i 1Y 2Y F F F Y n), and for which the initial conditions at t t0 are specified as qi t0 qi0 , pi t0 pi0 . Such systems are rich in a variety of motions, from completely integrable dynamics to quasi-periodicity and chaos. Among the fundamental properties of Hamiltonian systems is the conservation of the volume of an arbitrary phase-space domain, i.e., the validity of the Liouville theorem, dq0 dp0 dq dp Y
D0 D
t

E Erot

a

b

E Eosc Àp

O

1

p

J

O
2

2 2

J

1

a

Figure 1. (a) Phase portrait of a nonlinear pendulum and (b) a visual representation of an integrable Hamiltonian system with two degrees of freedom in angle + action variables.

where D0 Y Dt & M.

3.1 Integrable systems The problem of the integrability of Hamiltonian systems is quite complex. There are a number of rather general (although, naturally, not universal) methods that in some cases allow constructing a solution of Eqns (4) or of an approximation to them. Quite comprehensive reviews and monographs describing these methods in detail are available (see, e.g., Refs [48 + 55] and the references therein). Therefore, we do not consider them here; instead, we only present a geometric analysis of integrable systems. Hamiltonian system (4) is completely integrable (and the Hamiltonian H is integrable) if there exists a canonical transformation to angle + action variables, qY p 3 a Y J. Another definition is based on the Liouville theorem on integrable systems: a Hamiltonian system with n degrees of freedom is integrable if n independent integrals in involution are known for this system. Systems with one degree of freedom n 1 are always integrable because their Hamiltonian HqY p E is an integral of motion. A pendulum model is a highly representative example of such systems. Its Hamiltonian can be written as
H p2 À mgl cos j Y 2ml 2 5

In the neighborhood of the points with the coordinates pY j 0Y 2pk, k 0Y Æ1Y Æ2Y F F F Y the family of phase curves consists of ellipses. Such points are therefore called elliptic. The family of trajectories near the points pY j 0Y p 2pk, k 0Y Æ1Y Æ2Y F F F Y is formed by hyperbolas, and such points are called hyperbolic. For an autonomous system with two degrees of freedom, n 2, the integrable system in the angle + action variables JY a has the topology of a two-dimensional torus (Fig. 1b). In view of the constraints o1 o1 J1 Y J2 and o2 o2 J1 Y J2 (if they are present), the frequencies of gyration in circles O1 and O2 for nonlinear systems can vary from torus to torus. Their ratio can also vary: o1 o1 J 1 Y J2 X o2 o2 J 1 Y J2 7

If quantity (7) is rational, o1 ao2 kam (a resonance), the dynamics of the system are periodic: the phase trajectory closes after k windings over the circle O1 and m windings over the circle O2 . If fraction (7) is irrational, o1 ao2 T kam, the phase trajectory covers the torus everywhere densely, and the motion of the system is called quasi-periodic, of almost periodic. Therefore, because J1 and J2 are arbitrary, the phase space is represented by two-dimensional tori, which can be visualized in ordinary three-dimensional space as a set of tori nested in one another, with the major and minor semiaxes specified by the ratios J1 and J2 (Fig. 2).


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Figure 2. General structure of the phase space of an integrable system with two degrees of freedom in the action + angle variables.

For integrable systems with n degrees of freedom, their phase space is 2n-dimensional; in the angle + action variables, it has the topology of a set of n-dimensional tori. Any possible trajectory lies on one of them. Some trajectories may be closed; others cover the corresponding torus everywhere densely. A torus of dimension n 5 2 with given J1 Y J2 Y F F F Y Jn is called resonant if the relation
n i1

ki oi J1 Y J2 Y F F F Y Jn 0 Y

where ki are nonzero integers, holds for a set of frequencies È É oi J1 Y J2 Y F F F Y Jn i 1Y 2Y F F F Y n.

3.2 Perturbed motion The overwhelming majority of Hamiltonian equations (for systems with n degrees of freedom) are not integrable. But a Hamiltonian H Ha Y J can in some cases be divided into a n i n t e g r a b l e p a r t H0 H0 J a n d a p a r t t h a t i s n o t integrable but can be represented as a small perturbation H1 of H0 :
Ha Y J H0 eH1 Y 8 where H0 H0 J, H1 H1 a Y J, and e is a small parameter. Systems whose Hamiltonians can be written in form (8) are called nearly integrable systems. In the chosen variables a and J, the canonical equations that follow from Eqns (4) and (8) have the form q H1 Y Ji À e qai ai q H0 qH1 e Y qJi qJi i 1Y 2Y F F F Y n Y 9

fore, the system shortly returns to the neighborhood of the resonance. The so-called phase oscillations thus set in. Resonances can arise not only between the system and external influences but also between different degrees of freedom of the system itself, which corresponds to the autonomy of the Hamiltonian H1 in Eqn (8). This is the case of so-called internal resonances. If the resonance is not isolated, the overlapping of resonances results in a very complex motion in the system. Moreover, the resonances prevent finding solutions of the equations using the technique of the canonical perturbation theory. In the perturbation theory, the original system is approximated by a close integrable system subject to a small perturbation, and the solution is sought as an expansion in powers of e. The presence of resonances disrupts the convergence of such series, because this approach implicitly assumes that the original equations are integrable. This is not the case in most situations, however. Even very simple systems may not be integrable, and their dynamics may be very complex at certain initial conditions. For example, signatures of dynamical chaos are manifested in the behavior of a nonlinear pendulum [56, 57]. The perturbation theory cannot describe such a complex behavior, which is formally reflected by the divergence of the series. If the initial conditions of the system correspond to regular trajectories (quasi-periodic motion), such trajectories undergo qualitative rearrangements under the action of perturbations in the neighborhood of the resonances. The theory of nonlinear resonance is remarkable for its ability to obtain an analytic criterion for the onset of irregular motion in a Hamiltonian system. This criterion was originally introduced in [58, 59]. We consider the theory of nonlinear resonance from a more general standpoint, following Refs [60, 61] (see also Refs [54, 62]). A more complete exposition of the theory of resonances is given in [63].

4.1 Small denominators We first consider the internal resonance. We decompose the f u n c t i o n H1 a Y J [ s e e E q n ( 8 ) ] i n t o a F o u r i e r s e r i e s , k H1 a Y J k H1 J exp ika , and substitute this decomposition in Hamilton equations (9). This yields k Jj Àie kj H1 exp ika Y k 10 qH k 1 aj oj J e exp ika Y j 1Y 2Y F F F Y n Y qJj k
k where k is a vector with real integer components, H1 are the Fourier coefficients, and oj J qH0 JaqJj . We seek a solution of perturbed system (10) in the form of a series in powers of the small parameter e: I 0 s J j Jj e s Jj Y s1 11 I 0 s e s aj Y j 1Y 2Y F F F Y n X a j aj s1

where e 5 1. If e 0, system (9) is completely integrable, and its solutions cover n-dimensional tori. We now assume that e T 0. How strong are then the changes in the character of the integrable system?

4. Nonlinear resonance
The answer to the question posed in Section 3.2 essentially depends on the relation between the per turbing force frequency and the proper frequency of the system. If the proper frequency is close to the frequency of the external force, this results in an increase in amplitude, leading to a resonance. But for nonlinear systems with perturbation (8), the frequency depends on the amplitude, and hence the system falls out of resonance some time later. This reduces the amplitude, which, in turn, changes the frequency. There-

Next, using expansion (11) and Eqns (10), we select the terms containing equal powers of e. In the zeroth approximation, we 0 0 then find Jj 0, aj oj J0 , j 1Y 2Y F F F Y n. Therefore, J
0 j

const Y

a

0 j

oj J0 t const X

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Because qoj J0 À Á ÀÁ J oj J0 eJ1 % oj J0 e 0 qJm m
1 m

Y

we can easily obtain the first-approximation equations À Á ~k 1 kj H1 J0 exp ikxJ0 t Y Jj À i
k

km ~ km where H1 are the values of H1 with the inclusion of the c o n st an t t h at ap p e a rs i n t h e z e ro t h o rd e r. T h u s, if t he resonance relation kx À mO % 0 holds, perturbation-theory series (11) diverge; this is the essence of the small-denominator problem. To overcome this difficulty, it was suggested that a canonical transformation be used to pass to special (resonant) variables.

a

1 j



qoj J0
m

qJ

0 m

J

1 m

13 qH k J0 ~ À Á 1 exp ikxJ0 t Y qJ j k

k ~k where H1 are the values of the coefficients H1 with the constant taken into account. Equations (13) can be straightforwardly integrated:

4.2 Universal Hamiltonian Let the perturbation H1 be a periodic function of time with a period T 2pan and let the motion be described by the Hamiltonian
H H0 J eH1 aY JY t Y 16

J a

1 j 1 j

À i

qoj J0 km H k J0 ~ 1
m

kj H k J0 ~ 1 exp ikxt const Y kx k qJm
0 k

14

kx2

exp ikxt

where e 5 1. We decompose the function H1 into a Fourier series: Â Ã km H1 J exp ika À mnt X 17 H1 aY JY t
kY m

Ài

qH k J0 exp ikxt ~ 1 X qJj kx k

It can easily be seen that if the condition kx k1 o1 k2 o2 F F F kn on % 0 15

Then Hamilton equations (9) become  à km kH1 J exp ika À mnt Y J Àie
kY m

dH km J Â Ã 1 a oJ e exp ika À mnt Y dJ kY m
À where H1 k condition Y Àm

18

(called the resonance relation) holds, terms with zero or nearzero denominators appear in Eqns (14). This leads to a s s substantial increase in the corrections aj and Jj , which obviously disrupts the convergence of series (11). If the resonance relation is not satisfied in first-approximation equations (13), it can hold for equations of higher approximations, s b 1. The resonance that manifests itself in the sth order of the perturbation theory is called the order-s resonance. We now let the perturbation H1 depend on time periodically (with a period T 2paO), i.e., H1 a Y JY t H1 a Y JY t T . A treatment similar to that used in the preceding case yields  à km H1 a Y JY t H1 J exp ika À mOt X
kY m

H

à kY m 1

. It can be easily verified that if the 19

koJ À mn % 0

is satisfied in Eqns (18), then a resonance arises. To analyze the dynamics in the neighborhood of the resonance, we isolate the resonant term in decomposition (17) and consider the behavior of the system determined solely by this term. We fix the triplet of numbers k0 , m0 , J0 such that the resonance condition is satisfied exactly, k0 oJ0 m0 n Y 20

In this case, the Hamilton equations become  à km kj H1 exp ika À mOt Y Jj Àie
kY m

and retain only the resonant harmonic in Eqns (18). Then, Â Ã k J Àiek0 H1 0 m0 J exp ik0 a À m0 nt iek0 H 2ek À H
0 Ã k0 m0 1

aj oj J e

qH km  à 1 exp ika À mOt Y j 1Y 2Y F F F Y n X qJj kY m

 à J exp Àik0 a À m0 nt
k0 m 1
0

1 n H 2i
0

 à J exp ic k0 a À m0 nt

As before, the zeroth-approximation solution is given by (12). The first-approximation equations can also be easily obtained and integrated, yielding J a
1 j 1 j

k0 m 1

 Ão J exp Àic k0 a À m0 nt
k0 m0 1

À i

kj H km J0 ~ Â Ã 1 exp ikx À mOt const Y kx À mO kY m
l

2ek 0 H a oJ e e dà H dJ

J sin k0 a À m0 nt c Y
k0 m0 1

qoj J0 kl H ~ q
0 Jl kY m

km 1

J0
2

kx À mO Â Ã qH km J0 exp ikx À mOt ~ 1 Y Ài kx À mO qJ i kY m

 à exp ikx À mOt

d H dJ
k0 m0 1

 à J exp ik0 a À m0 nt

 à J exp Àik0 a À m0 nt d k H dJ 1
0

oJ 2e

m0

J cos k0 a À m0 nt c X


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0 We introduce the notation j k0 a À m0 nt c and H1 k 2jH1 0 m0 j. Then Eqns (18) can be rewritten as 0 J ek0 H1 J sin j Y

E Eosc J a Es

a
rot

b

21
0

i O C

j k0 oJ À m0 n ek

dH J cos j X dJ

0 1

Retaining only the resonant term in expansion (17), we express Hamiltonian (16) as H H0 J eH cos j X
0 1

O C

r

22

We now assume that the quantity J is sufficiently close to J0 , such that the deviation DJ J À J0 is small. In this case, H0 J and oJ can be expanded into series in DJ: H0 J H0 J0 q H0 1 q2 H 0 DJ DJ2 F F F Y qJ 2 qJ 2 23 24

Figure 3. (a) First-order nonlinear resonance in the variables a, J and (b) in the variables specified by Eqn (28). Thin curves, phase oscillations; dot+ dashed curve, unperturbed trajectory J J0 ; thick curve, separatrix [60].

qo 1 q2 o oJ oJ0 DJ2 F F F X DJ qJ 2 qJ 2

We neglect the terms of orders higher than two in Eqn (23) 0 and higher than one in Eqn (24). In addition, we take H1 J at the point J0 , use equality (20), and neglect the term $ e in the second of Eqns (21). Then system (21) and Hamiltonian (22) become d 0 DJ ek0 H1 sin j Y dt doJ0 DJ Y j k0 dJ H H0 J0 oJ0 DJ

25

1 qoJ0 0 DJ2 eH1 cos j X 26 2 qJ

0 0 For convenience, we here use the notation H1 H1 J0 . The variables DJ and j are canonically conjugate for system (25), and system (25) itself is generated by the Hamiltonian

doJ0 0 "1 H k0 DJ2 ek0 H1 cos j Y 2 dJ

27

wh ic h is calle d t he un ive rsal H amilt on ian o f n on lin ear resonance [59]. The canonical transformation that allows passing from the system with Hamiltonian (26) to the universal Hamiltonian of nonlinear resonance has the form aY J 3 k0 a À m0 ntY DJ Y n oJ0 Y " H k0 H À m0 nDJ À k0 H0 J0 X 28

4.3 Width of the separatrix We analyze expression (27) and Eqns (25). A similarity " between H and the Hamiltonian of the nonlinear pendulum in (5) with l 1 can immediately be noted. Indeed, the variable DJ plays the role of the momentum p and the quantity m k0 doJ0 adJÀ1 has the meaning of the 2 e f f e c t i e m a s s . M o r e o v e r , t h e s u b s t i t u t i o n O0 v 2 0 ek0 H1 doJ0 adJ reduces system (25) to
2 j À O0 sin j 0 X

Let k0 1, m0 1 (a first-order resonance). The phase portrait in the original variables a, J is shown in Fig. 3a. The double drop-shaped curve, for a small energy E Eosc , corresponds to local oscillations of the system. In terms of the pendulum analogy, this corresponds to its oscillations. The outer curve corresponds to the energy E Erot , which implies rotation in the case of the pendulum. The separatrix E Es separates these two qualitatively different types of motion. We now pass to the coordinate system specified by transformation (28). The resulting phase portrait is shown in Fig. 3b. We clarify this figure using the pendulum analogy. If the pendulum rotates oppositely to the rotation of the coordinate system, curves located around the point O inside the minor separatrix loop correspond to such motion. If the pendulum corotates with the coordinate system, curves located outside the major separatrix loop correspond to the motion of the pendulum. Horseshoe-shaped closed curves between the two separatrix loops correspond to oscillations of the pendulum. The oscillations occur around the elliptic point denoted as i in Fig. 3b. Both the major and minor separatrix loops pass through the hyperbolic point r. The circle C represents the lower equilibrium of the pendulum. For the original system, the unperturbed trajectory for J J0 corresponds to the circle C. The directions of motion are shown by arrows in Fig. 3b. The maximum distance between the two (major and minor) separatrix loops is known as the width of the nonlinear resonance (or the separatrix width). This quantity can easily be estimated. According to Eqn (27), we have the resonance width with respect to action s À1 0 doJ0 30 max DJ 2eH1 dJ and the resonance width with respect to frequency s doJ0 max DJ 2eH 0 doJ0 X 31 max Do 1 dJ dJ We now find how much the above assumptions are justified and determine the orders of magnitude of quantities 0 (30) and (31). It follows from expression (22) that H0 $ H1 . But because H0 $ J0 oJ0 ,
0 H0 $ H1 $ J0 oJ0 X

29

Equation (29) coincides with Eqn (6) up to a phase lag by p.

32


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n 5 2, however, resonances between dom of the system itself, i.e., inte possible. For clarity, we consider a system freedom n 2. The Hamiltonian of written as
e O

the degrees of freernal resonances, are with two degrees of such a system can be

H H01 J1 H02 J2 eH1 J1 Y J2 Y a1 Y a2 Y and the equations of motion are given by

r

q H1 Y Jj À e qaj

aj

q q H1 H01 H02 e Y qJj qJj

j 1Y 2 X

Figure 4. Nonlinear resonance at k0 4, m0 1.

Therefore, with (30) and (31), we obtain r max DJ e max Do p $ eg Y $ Y J0 g oJ0

According to relation (15), internal resonance occurs if the condition ko1 J01 À mo2 J02 0 is satisfied for certain integers k, m and certain values of the action variables J01 , J02 . We retain only the resonant harmonic in the decomposition  à km H1 J1 Y J2 exp ika1 À ma2 Y H1 33
kY m

where g doJ0 adJJ0 aoJ0 is the so-called nonlinearity parameter. In deriving system (25), we neglected the term $ e in Eqn (21) for j. This can be done [see 0 Eqns (21) and (25)] if dH1 J0 adJ0 e 5 doJ0 adJDJ, or, in view of our estimate of the orders of magnitude, if 0 eH1 aJ0 5 doJ0 adJDJ. Therefore, in accordance with (32) and (33), we find g4eX 34

expand the functions H0j and oj dH0j adJj , j 1Y 2, in the neighborhood of the resonance J01 Y J02 , and use the above approximations to obtain the equations of motion
0 J1 ekH1 sin j Y 0 J2 ÀemH1 sin j Y

do1 J01 do2 J02 DJ1 À m DJ jk dJ1 dJ2 and the universal Hamiltonian

36

2

0 0 In addition, we replaced H1 J with H1 J0 . This substitution is equivalent to the condition J À J0 5 1, or DJ 5 J0 . As can be easily seen from relations (33), the last inequality leads to inequality (34). Retaining only the resonant term and neglecting the nonresonant terms is allowed if other frequencies in expansion (24) do not alter the dynamics qualitatively, i.e., if oJ0 4 doJ0 adJDJ. Next, we use estimate (31) to obtain the condition ge 5 1, which is satisfied for small g and e. With (34), we therefore have

" 1 do1 J01 DJ1 2 1 do2 J02 DJ2 2 eH 0 cos j Y H 1 2 dJ1 2 dJ2 w h e r e w e u s e t h e n o t a t i on DJj Jj À J0j , j 1Y 2, 0 km km H1 exp ic 2H1 J01 Y J02 2H1 J01 Y J02 exp ic, and j ka1 À ma2 c. It can be easily seen that multiplying the first equation of system (36) by m and the second by k, and adding them, we obtain an additional integral of motion mJ1 kJ2 const. Therefore, Eqns (36) are integrable. But we can also proceed differently. Differentiation of the equation for j with respect to time yields expression (29) with 2 0 2 do1 J01 2 do2 J02 O0 eH1 k X m dJ1 dJ2 I f t h e q u a n t i t i e s do1 J01 adJ1 a n d do2 J02 adJ2 h a v e different signs, the expression inside the modulus vanishes, i.e., an additional degeneracy is possible in the system. For systems with more than two degrees of freedom n 5 3, the number of additional integrals of motion is larger, but the dynamics are still governed by an equation similar to Eqn (29). Thus, the ratio of frequencies plays an important role in the analysis of nearly integrable systems: the incommensurability of the frequencies typically determines a quasi-periodic trajectory that densely covers the torus. But a rational frequency ratio results in the emergence of resonances and modifies the structure of invariant surfaces.

e5g5

1 X e

35

Inequality (35) is called the condition of moderate nonlinearity [59, 60]. Thus, the above-presented approximate theory of nonlinear resonance is applicable if inequality (35) is satisfied. Therefore, there is no limit transition to the linear problem g 3 0. Resonance condition (20) can generally be satisfied at any k0 . In this case, all assumptions and formulas remain unchanged. But because the phase is determined by the expression j k0 a À nt c, this leads to a garland of separatrix loops. The number of such loops is k0 . Therefore, the number of elliptic+hyperbolic pairs of points is also equal to k0 (Fig. 4).

4.4 Internal resonances The above description of nonlinear resonance can easily be generalized to systems with many degrees of freedom. For


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4.5 Overlapping of resonances O u r an aly sis o f no n li ne a r re so n an c e w as b ase d on t h e assumption that the quantity J [see Eqn (20)] remains fixed. In other words, we assumed that only one isolated, primary nonlinear resonance exists. But it follows from (19) that the r e s on a nc e r el a t i o n c a n a l s o b e s a ti s f i e d f or o th e r Jm , m 1Y 2Y F F F . In other words, there can be many primary resonances in the system, and Eqn (20) at k0 1 must be rewritten as oJm mn. Therefore, we can introduce the distances between resonances: in action, dJm Jm1 À Jm , and in frequency, dom oJm1 À oJm . If all Jm are sufficiently far from one another, i.e.,
dJm 4 DJm Y 37

where DJm is determined by Eqn (30), then the separatrices of the resonances do not intersect and no interactions between the resonances occur. If the initial conditions of the system are in the domain of one of the resonances, the dynamics can be described in the approximation where the effect of only this resonance is taken into account. If the initial point is in the domain between resonances, the system can be analyzed in the framework of the nonresonance approximation [54, 64]. Similarly, we can formulate the condition of the absence of frequency overlapping (interaction) of resonances: dom 4 Dom X 38

We now assume that condition (37) or (38) is not satisfied and the resonances are close to one another, such that the corresponding separatrices can overlap. What happens in this case, known as the case of strong interaction between resonances? We introduce the parameter K describing the degree of resonance overlapping: K Do DJ $ X do dJ

called secondary resonances, which modify or even completely destroy the integrals of the primary resonances, are possible. A secondary resonance is the resonance between the fundamental frequency of the unperturbed oscillation and the frequencies of harmonics of the primary-resonance phase oscillation. A specific feature of such a resonance is that garlands of separatrix loops appear near the elliptic point inside the primary resonance. The small denominators of the secondary resonances can be removed as this was done for the primary resonance (see Ref. [54] for details). But such an approach has some specific features when applied to secondary resonances. In particular, the width of a secondary resonance depends on e much more strongly than that of the primary resonance [$ e 1a2 , see Eqns (30), (31)]. Therefore, secondary resonances are not important at small e (in the case of small oscillations in the primary resonance). But if the perturbations are sufficiently strong, secondary resonances can affect the dynamics of the system as strongly as the primary resonances do. In addition to secondary resonances, higher-order resonances are possible in systems; they can also substantially affect the motion. Thus, on the whole, the behavior of the system is highly involved and represents a hierarchy of very complex structures. We describe such a structure when we consider the dynamics near resonance separatrices. Our analysis clarifies some issues related to the origin and nature of dynamical chaos. What is the structure of resonances near elliptic and hyperbolic points? This question is directly related to the nature of chaos.

5. Elements of the Kolmogorov + Arnol'd + Moser theory
A nonlinear resonance indicates that even small perturbations can substantially affect the dynamics of an integrable Hamiltonian system. Resonances modify the topology of phase trajectories and result in the formation of a chain of is lan d s in p h ase sp ac e . T h e p e rt u rb ati on t he o ry c an n o t de s cr i be su ch r es ona n ces , b eca us e r eg ul a r s ol uti o ns a r e strongly disturbed near them, which entails the emergence of sm all denominat ors and t he divergence of series. This  problem was already noted by Poincare, who called it a fundamental problem of classical mechanics. However, it was solved only in the early 1960s, with the advent of the famous Kolmogorov + Arnol'd + Moser (KAM) theory [21 + 25]. To emphasize the physical significance of the KAM theory, we consider the problem of planetary motion around the Sun in accordance with the law of gravity. If the mutual influence of planets is neglected, we have a completely integrable system: planets move in ellipses according to Kepler's laws, and hence the dynamics of the system are in general quasi-periodic. If the interaction of the planets is taken into account, their orbits are deformed ellipses, which precess slowly. The precession is maximum for Mercury. The shift of its perihelion due to the influence of other planets 1 is estimated to be $ 532HH per 100 years [68]. In principle, the interaction of planets should be taken into account using perturbative methods. But if resonances occur, such techniques yield diverging series and therefore do
1

The overlapping parameter K was introduced by Chirikov to characterize the dynamics of Hamiltonian systems [58] (see also Refs [59, 65, 66]). If the interaction is weak, i.e., K 5 1, the motion in the system should be regular and the phase trajectories should generally cover n À m-dimensional tori everywhere densely. But in the case of a strong overlapping, K b 1, the dynamics of the system turn out to be very complex and different from the dynamics in the periodic and quasiperiodic regimes. The validity of the criterion of the onset of irregular motion based on the resonance-overlapping degree was many times confirmed numerically (see Refs [54, 59, 62]). In particular, Ref. [67] reported the origin of chaos with only two resonances overlapping. A specific property of this criterion is that it can relatively simply be used to study particular systems. Indeed, it suffices to apply the above technique in the neighborhood of only one resonance in the approximation of the absence of all others. This makes the resonance-overlapping criterion very convenient in practice. At the same time, it is not universally applicable and in some cases requires more accurate formulations [54, 59, 66].

4.6 Higher-order resonances Up to this point, we have considered only primary resonances. However, if the perturbation e is sufficiently large, so-

We emphasize that we consider only the Newtonian interaction.


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not provide information on the dynamics of the system over long time intervals. This planetary problem, reducible in the general case to the well-known N-body problem, is among the most important problems that have been raised in the course of the development of both mathematics and physics. In 1885, King Oscar II of Sweden and Norway, at the suggestion of G Mittag-Leffler, established a prize for solving this problem: ``Given a system of arbitrarily many mass points that attract each other according to Newton's laws, try to find, under the assumption that no two points ever collide, a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.'' (See Ref. [69] for a more detailed exposition of this and some other interesting points concerning the development of celestial mechanics and nonlinear dynamics.) Thus, the principal problem was not only to find a formal representation of solutions of the N-body problem in the form of series but also to prove their convergence. It is the latter issue that presented the main difficulty. Without dwelling on the history of the question, we only  note that the prize was awarded to Poincare for his inestimable contribution to the solution of the N-body problem and related fundamental problems of dynamics. Very much time was spent to realize that the problem posed is not simple and solving it requires employing novel powerful methods. Such methods were developed by the authors of the KAM theory, A N Kolmogorov, V I Arnol'd, and J Moser. The KAM theory not only permits constructing a converging expansion procedure but also, most importantly, gives a key to understanding the nature of the onset of chaos. In addition, this theory and its consequences proved to be very important in many areas of modern science, such as pure and applied mathematics, mechanics, physics, and even numerical analysis (!). In recent years, substantial progress has been made in the KAM theory, and it remains a field of very active research (see Refs [53, 70, 71] and the references therein). This theory is presented in a large number of reviews and monographs (e.g., Refs [70, 72 + 76]). In Sections 5.1 and 5.2, we therefore describe only some qualitative statements of this theory and a number of its important consequences.

The perturbation must be smooth, i.e., the Hamiltonian H1 must have a sufficient number of derivatives. The system must be outside the neighborhood of a resonance, i.e., kj oj b cjkjÀr Y k k1 Y k2 Y F F F Y kn Y 39
j

where r depends on the number of the degrees of freedom n and the constant c is determined by the magnitude of the perturbation eH1 and the nonlinearity parameter g. We note that the condition of moderate nonlinearity (35) can be obtained from inequality (39). If the above conditions are satisfied, the meaning of the KAM theory is as follows. For most initial conditions, quasiperiodic dynamics are preserved in a nearly integrable system. However, there are initial conditions at which (mainly resonant) tori existing for e 0 break down and the motion becomes irregular. Precisely these tori break down under the perturbation and bring the system to chaos. This is why the theory of nonlinear resonance plays such an important role. The trajectories that originate in the region of destroyed tori can freely move in the energy space, which is manifested on the section surface as numerous randomly scattered points. Such trajectories turn out to be exponentially unstable with respect to small perturbations. At small e, interestingly, the tori remote from the resonance are preserved in the presence of arbitrary smooth perturbations. But the situation changes dramatically with increasing e: the tori start breaking down and the domain of chaos starts expanding. Ultimately, this results in an overlapping of primary resonances and the onset of the phenomenon known in Hamiltonian mechanics as global chaos [54]. As global chaos occurs, no tori that were close to the tori in the unperturbed problem remain in the system, although some other tori can appear. The phase trajectory can move transversally to the chaotic layers in such a system. However, at e 5 1, resonances do not overlap and the solutions lie on weakly deformed invariant tori. There is a qualitative difference in dynamics between systems with two degrees of freedom and those with more n b 2 degrees of freedom.

5.1 The Kolmogorov theorem The famous Kolmogorov theorem (sometimes also called the KAM theorem) is at the heart of the KAM theory. It states that if a completely integrable system is perturbed sufficiently weakly, most nonresonant tori are preserved and only slightly deformed. In this case, the word `most' means that all the resonant tori (corresponding to periodic motion) and p a rt o f n o n r e s o n a n t t o r i c o l la p s e , b u t t h i s se t is s m al l compared to the set of nonresonant tori preserved under the perturbation. The applicability conditions of the KAM theorem are as follows. The unperturbed Hamiltonian must satisfy the nondegeneracy condition 2 qoi det q H0 T 0 Y i 1Y 2Y F F F Y n Y det qJ qJ qJ k i k
which means that the frequencies of the unperturbed system are functionally independent.

5.2 The Arnol'd diffusion In the case of two degrees of freedom, the phase space is fourdimensional, the energy hypersurface (or energy-level space) H E is three-dimensional, and the invariant tori are twod i m e n s i o n al . T h e r e f o r e , su c h t o r i c a n b e re p r e s e n t e d a s e ne rgy le vels H E i mme rse d in th e t hre e -d ime n sion al space of the energy level H E (see Fig. 2). Therefore, the tori divide this space into disconnected domains. Thus, the destroyed tori turn out to be sandwiched between the tori preserved under the perturbation. The phase trajectory that originates at the location of such a destroyed torus (i.e., in the gap between two invariant tori) remains locked there forever. T h is me an s t h at t h e c o rre s po n d in g ac ti on var iab le s are almost unchanged and remain near their initial values in the course of motion. Therefore, for systems with two degrees of freedom satisfying the conditions of Kolmogorov's theorem, for any initial conditions, no evolution occurs and global stability is preserved [72]. If n b 2, invariant tori no longer divide the 2n À 1dimensional energy hypersurface into nonintersecting parts. In such systems, the dynamics are quasi-periodic for most


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initial conditions. But there exist initial conditions at which the action variables recede slowly from their initial values. This can be easily understood from the fact that, at n b 2, the domains of destroyed tori merge, forming a cobweb. The phase point on the hypersurface of a given energy, moving along the threads of this cobweb, can approach any point of this hypersurface arbitrarily close. As investigations show, such an evolution of the action variables is random. This random walk over resonances around invariant tori is called the Arnol'd diffusion [54, 60, 70, 72]. Thus, small perturbations in systems with more than two degrees of freedom not only qualitatively modify the dynamics of the system but also can change the topology of phase trajectories, transforming them into a connected web. A characteristic feature of the Arnol'd diffusion is its universality in the sense that no critical value of the perturbation e necessary for the onset of such diffusion exists. In other words, diffusion always occurs, even at arbitrarily small e. Obviously, the diffusion rate vanishes at progressively reduced perturbations. Therefore, for Hamiltonian systems with n b 2 degrees of freedom, the action variables can evolve slowly, which corresponds to lacking global stability. In the general case, this evolution is fairly slow and can be different in different parts of the phase space. Another peculiarity of systems in which the Arnol'd diffusion is inherent consists in that their dynamics cannot exhibit a sudden transition to global chaos, which originates due to the overlapping of resonances. This is because regions with chaotic behavior in such systems are already unified into a web. In addition, the Arnol'd diffusion is very slow compared to the motion of trajectories in the region of global chaos. The existence of diffusion trajectories was rigorously proved for the first time for a nonlinear Hamiltonian system of a special form [77]. There is no proof of the merging of chaotic trajectories into a web in the general case; however, fairly numerous examples are known in which this phenomenon is observed (see Refs [54, 59, 78 + 81]). The exact upper bound of the Arnol'd diffusion rate was obtained in Ref. [82] (see also Ref. [53] and the references therein). Various cases of diffusion, the accompanying physical phenomena, and estimates of the diffusion rate for various systems are discussed in monographs [53, 54]. We now consider the breakdown of the resonant tori and the origin of chaos in greater detail.

given value of H E, the flow is always three-dimensional. Â This allows considering the corresponding Poincare map instead of the continuous evolution of the system. Such a map can easily be constructed analytically. For an integrable system in the angle + action variables, the phase space is a set of nested tori. We write the flow on one of them as a1 t o1 t a1 0 Y a2 t o2 t a2 0 Y

w h e r e o1 o1 J1 Y J2 qHaqJ1 a n d o2 o2 J1 Y J2 qHaqJ2 . Obviously, a complete revolution along the a2 coordinate takes the time t2 2pao2 . By this time, the a1 variable becomes a1 t t2 a1 t o1 t2 a1 t 2po1 a1 t 2prJ1 X o2

The quantity r o1 ao2 introduced here is called the rotation number. Because motion occurs in the energy-level space, J2 J2 J1 Y E. Therefore, at a given E, the rotation number r is a function of J1 only. We now assume that the surface of a section is the plane a1 Y J1 , i.e., a2 const. Then, for the intersection points xk between theÀphase trajectory and this plane (Fig. 5a), we can Á write xk a1 t kt2 Y J1 . Thus, solutions of the original system on the given torus can be represented as the map P0 that takes an intersection point xk into the point xk1 next in the invariant circle of `radius' J1 . With the notation Jk J1 t kt2 , it is possible to write such a displacement of points as & ak1 ak 2prJk Y P0 X 40 Jk1 Jk X As can be seen from (40), the rotation number r depends on the radius of the circle in general. If r is irrational, then the points xk fill the entire circle as k 3 I. If r lam is rational, the points xk map into one another successively, each m steps (Fig. 5b). Therefore, if this representation is used, we can speak of nonresonant and resonant circles. As we pass from one circle to another, the rotation number changes. To be specific, we assume that rJ increases with increasing J. Then, as the origin of coordinates recedes, the angle through which the circle is rotated increases on average. This results in a twist of the radial line of points under the action of P0 . For this reason, transformation (40) is
a J x x b
k

6. The nature of chaos
The issue of the nature of chaotic behavior traces back to t he f am o us pr o bl e m o f in te rs e c ti on o f s e pa rat ric e s i n dynamical systems. This phenomenon was discovered long  ago by Poincare when he was studying the three-body problem [83]. But to understand the origin of chaos in  systems of classical mechanics, the well-known Poincare + Birkhoff fixed-point theorem [13, 84] should also be invoked ( the mat hemat ics of t he i ssue can be s ur veyed using monograph [85]).

k

a1 J1 J
2

x

k2

x

k1

6.1 Twist map We consider a system with two degrees of freedom, n 2, and a Hamiltonian Hq1 Y p1 Y q2 Y p2 , although the results presented below can be generalized to the multidimensional case. In such a system, H is the total energy. Therefore, at a

a

2

a

 Figure 5. A system with two degrees of freedom (a) and its Poincare map (b).


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called the rotation map (or sometimes the twist map). In the general form, it can be written as P0 g g X An important property of the map P0 is that it is conservative. We now consider perturbed system (9) with Hamiltonian  (8). In the Poincare section, taking the perturbation into account corresponds to adding new terms to the twist map: @ ak1 ak 2prJk ef Jk Y ak Y Pe X 41 Jk1 Jk egJk Y ak Y where f and g are functions periodic in a. As before, the transformation Pe must preserve the area; otherwise, invariant circles do not exist, as a rule. According to the KAM theory, under a small perturbation, e 5 1, most circles with irrational r are preserved, being only slightly deformed. We consider the behavior of the circles in the domain of rational, or resonant, values r lam, for which relation (39) is not valid. It is in this domain, as we already noted, chaotic motion originates.
C
À e

C

e

a

Ce Ge CeÀ Pem Ge p

b

Ge

Pem Ge

Figure 7. (a) Transformation of the curve Ge into the curve Pem Ge under the action of the map Pe and (b) elliptic and hyperbolic points arising in this case.

6.2 Fixed-point theorem We temporarily return to the integrable case. At rJ lam, any point of the circle returns to its initial position after m steps, i.e., it is periodic with the period m. We let g denote this res onant circl e and consi der two nonres onant inva ria nt circles g and g À on both sides of g (Fig. 6a). Because rJ increases with J (see Section 6.1.), the irrational rotation numbers of such circles satisfy the respective inequalities r b lam and r ` lam. After applying the map P0 m times m (with the iterated map denoted by P0 ), the points of the circle g are rotated through an angle larger than 2p and the points of the circle g À through an angle smaller than 2p. It therefore appears that relative to g, the map winds g counterclockwise and g À clockwise (Fig. 6a). We now consider perturbed map (41). According to the KAM theory, the circles g and g À are preserved and only slightly deformed under the perturbation. Such closed circles, respectively denoted as Ce and geÀ , are invariant under the transformation Pe :
Pe ge ge Y Pe geÀ geÀ X the the the the

We assume that the parameter e is sufficiently small for relative rotation of ge and geÀ to be preserved under action of Pem . Then, at any radius a const between curves ge and geÀ , a point Jp aY e can be found such that
C C C
À

map Pem rotates its angular coordinates exactly through 2p. Therefore, the angular coordinates of this point are preserved under Pem . At each radius issuing from the center, only one such point is located. Because the perturbation Pe is smooth, these points Jp aY e form a certain closed curve Ge (Fig. 6b) that shrinks to g as e 3 0. The curve Ge is not invariant under Pe . The action of Pem amounts to radially shifting each point of Ge . Thus, a new curve Pem Ge is formed instead of Ge (Fig. 7a). Next, we recall that the map Pe is conservative. Therefore, both curves Ge and Pem Ge must confine equal areas. Hence, the curve Pem Ge can be neither inside nor outside the curve Ge . Therefore, these curves must in general intersect 2 at an even number of points (Fig. 7a). In turn, each of these intersection points is fixed under the perturbed transformation Pem . Â This is the principal meaning of the Poincare + Birkhoff fixed-point theorem [13, 84], according to which the perturbed twist map (41) with the rotation number r lam has 2im, i 1Y 2Y F F F Y fixed points. Thus, as the resonant circle is perturbed (we recall that any point of this circle is fixed m under the transformation P0 ), only an even number 2im of fixed points is preserved. We consider one of the points, p, of intersection between the curves Ge and Pem Ge (Fig. 7a). It is fixed under the map Pem . The transformation Pe acting on p generates a sequence of points pY Pe pY Pe2 pY F F F Y PemÀ1 p; after m iterations, the point p returns to its initial position. On the other hand, each of them is a fixed point for Pem . Thus, there are m fixed points associated with the original point p. Because the curves Ge and Pem Ge intersect at an even number of points, we have a total of 2im fixed points.

a Jp

C C

e

b G

À e

e

Figure 6. (a) Invariant circles g, g , and g À of the unperturbed twist map P0 for r lam, r b lam, and r ` lam, respectively. (b) The result of the action of the perturbation Pe .

6.3 Elliptic and hyperbolic points A detailed consideration of the map Pe in the neighborhood of various points p reveals the following qualitative difference. Points neighboring some points p remain near them and appear to rotate around them, while points located near other p tend to leave their neighborhood. Points of these two sorts alternate (Fig. 7b). Similar motion is observed in the phase space of a nonlinear pendulum (see Section 3.1). For this reason, such poi nt s are respect ively cal led elliptic and hyperbolic. Elliptic points are surrounded by a family of closed trajectories that are invariant under Pem and form `islands'; hyperbolic points are connected by separatrices. This pattern is typical of weakly perturbed nonlinear systems, and it always emerges near a resonance. Each island satisfies the KAM theory. Therefore, most nonresonant circles are preserved. But resonances are also
2

We do not consider the exceptional case of tangency.


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a

r2

b

r r

1

r3

Figure 8. Destruction of tori with rational frequency ratios and the origin of a self-similar structure.

Figure 10. A homoclinic (a) and a heteroclinic (b) trajectory formed by separatrix branches. a q q
HH H

b W w w
s u u

b q q
HH H

Pe q q Pe q
H

HH

q
H

Pe q

HH

a

w
s

u

Pe q w
u

W w
s

s

r w
u

w

s

w

w r w
u

u

Pe q w
s

w r

s

r

Pe2 q

Figure 9. Stable w s and unstable w u directions of a hyperbolic point r  in the Poincare map (a) and the corresponding manifolds W s and W u in the phase space (b).

Figure 11. Map of neighboring points q H and q HH and the formation of loops from separatrices.

 present here. According to the Poincare + Birkhoff fixedpoint theorem, a sequence of alternating elliptic and hyperbolic points appears in the neighborhood of each resonance, although on a smaller scale (Fig. 8). In turn, an island near each of these elliptic points reproduces the entire pattern in miniature. Closed curves in the neighborhood of elliptic points correspond to minor tori. According to the KAM theory, some of these minor tori are preserved; others collapse into smaller ones due to higher-order resonances, and so on, to infinity. Thus, the resonance results in a very complex pattern, which repeats itself on progressively smaller scales, thus being, in a certain sense, self-similar. Following Refs [72, 86], we now consider the phenomena in the neighborhood of hyperbolic points. Each of these points is characterized by four invariant directions (or separatrix branches): two stable ones w s , which enter the hyperbolic point r, and two unstable ones w u , which issue  from r (Fig. 9). Because we are considering the Poincare map, these curves correspond to surfaces, or invariant manifolds, either stable W s or unstable W u , in the original phase space. The oscillation period is infinite at the separatrix. Therefore, if some point q belongs to the stable branch w s , it approaches r exponentially slowly under the action of the map P, i.e.,
k3I

 another, forming smooth structures. In a Poincare map, such a structure appears as a smooth transition from an unstable to a stable separatrix branch. Separatrix branches can either close to the same hyperbolic point (Fig. 10a) or connect several such points, forming garlands (Fig. 10b). In the first case, a stable w s and an unstable w u separatrix branch forms a loop. This loop, known as a homoclinic loop, is a doubly asymptotic trajectory with the property that any point q on such a loop always approaches r, i.e.,
k 3 ÆI

lim P k q 3 r X

lim P k q 3 r X

But if q P w u , it leaves the neighborhood of r exponentially slowly, i.e.,
k3I

lim P

Àk

q 3 rX

6.4 Splitting of separatrices. Homoclinic tangles In the case of an integrable system, stable and unstable manifolds of hyperbolic points can be connected to one

An elliptic point exists inside the homoclinic curve. In the second case, the curves consisting of stable and unstable separatrix branches are called heteroclinic trajectories. The motion along such trajectories implies exponentially slow recession from one hyperbolic point and approach to another. If a perturbation is present, the separatrix branches no longer form smooth homoclinic and heteroclinic junctions but can intersect. The point of the intersection of a stable and an unstable separatrix branches of the same resonance is called a homoclinic point. If a stable and an unstable branch of different hyperbolic points (resonances) intersect, a heteroclinic point appears. We now consider how such points evolve under the action of the map Pe . Let q be a homoclinic point and q H P w s and q HH P w u be its neighboring points (Fig. 11a). Under the action of Pe , the points q H and q HH must be mapped into the points Pe q H and Pe q HH . Where should we find the point Pe q into which q is mapped by Pe ? As can be seen from Fig. 11a, taking the directions of motion in the stable and unstable branches into account, q is located before the points q H and q HH . Therefore, because Pe is continuous, this point must be taken into a point that is also located before the points Pe q H and Pe q HH . This means that a new intersection, i.e., a new homoclinic point Pe q, must arise (Fig. 11b).


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a

b

c

a

Figure 12. Homoclinic tangles in the neighborhood of a hyperbolic point.

Figure 14. Structure of the phase space of perturbed Hamiltonian systems  with two degrees of freedom: (a) in the Poincare map and (b) in the phase space [87].

w w
s

s

w w w
s u

u

r1 w
u

w

s

w r2 w
u

u

w

s

Figure 13. Formation of heteroclinic tangles.

In other words, because the branches w s and w u are invariant, we should remain at both separatrix branches under the action of the map. Therefore, a new intersection point Pe q emerges, and a loop forms between the points q and Pe q. By similar considerations, we can easily find that the point Pe q is mapped into the point Pe2 q, forming another loop (Fig. 11b). Because the new point Pe2 q is closer to the hyperbolic point r, the distance between Pe2 q and Pe q is shorter than the distance between Pe q and q. We recall that the conservation of phase volume is a property of the map Pe . This means that the areas confined by the loops between q and Pe q and between Pe q and Pe2 q must be equal. Therefore, the second loop is more stretched and curved than the first one. Continuing our considerations leads us to the conclusion that an infinite number of intersections of separatrix branches eventually arise, which become progressively closer, and the loops themselves become longer and thinner (Fig. 12a). The stable separatrix branch w s behaves similarly (Fig. 12b). This can easily be understood if we move along w s in the opposite direction. Thus, the pattern in the neighborhood of hyperbolic points is on the whole extremely complex (Fig. 12c). For heteroclinic trajectories, the intersection of separatrices results in the formation of somewhat different structures. The unstable branch w u issuing from the hyperbolic point r1 oscillates in approaching a hyperbolic point r2 located on the right. In contrast, the stable point w s oscillates in receding from r1 (Fig. 13). The stable and unstable branches located in the bottom part of the figure (not shown in full) behave similarly. Thus, due to a perturbation introduced into the integrable system, the separatrices are no longer smooth, as in Fig. 10, but split in an intricate manner. This phenomenon is termed the splitting of separatrices. The structures formed by separatrix branches in the domain of hyperbolic points are called homoclinic and heteroclinic tangles. It is such complex behavior of separatrices that gives rise to chaos in determi-

nistic systems. Invariant tori cannot exist in the domain of homoclinic tangles. There, the systems are not integrable, exhibit the property of the exponential instability of trajectories under small perturbations, and therefore behave chaotically. However, at some different initial conditions from the neighborhood of the surviving tori, the dynamics of the system are regular. We can now imagine the entire pattern arising in phase  space and in the Poincare section of the set of invariant tori. The destruction of resonant circles under perturbations is accompanied by the occurrence of 2im hyperbolic and elliptic points. In the neighborhood of each elliptic point, a family of closed invariant curves is present, and some of them are also destroyed by the perturbation. This leads to a smaller chain of elliptic and hyperbolic points. In the neighborhood of each hyperbolic point, the separatrices split, their tangles appear, etc. (Fig. 14a).  However, all this occurs not in the Poincare section but in the phase space formed by a set of tori. Thus, the general pattern of motion of the phase trajectories turns out to be highly involved (Fig. 14b). This structure repeats itself on progressively smaller scales and is typical of nearly integrable systems. As the perturbation increases, some nonresonant circles also break down. But if the perturbation H1 is small, chaotic trajectories exist only in the phase-space domain bounded by invariant curves. An elementary exposition of the foundations of the theory, the questions concerning the physical aspects of separatrix intersection, and a historical background are given in monograph [69]. A rigorous presentation of related topics is made in Refs [53, 72, 86, 88]. The fixed-point theorem, its qualitative description, and some applications can be found in book [13]. Mathematical approaches to the problem of separatrix intersection are presented in Refs [32, 63, 71, 88 + 90].

7. The Mel'nikov method
The distance between separatrices can be estimated analytically and the conditions for the onset of homoclinic and heteroclinic chaos can be obtained using the Mel'nikov's theory [91] (see also Refs [54, 63, 88, 89]). To simplify the presentation, we consider a two-dimensional autonomous system with a periodic external perturbation x f0 x ef1 xY t Y 42

[ w h e r e x x1 Y x2 , f0 f01 Y f02 , f1 f11 Y f12 , a n d f1 xY t f1 xY t T ] t h a t h a s a u n i q u e h y p e r b o l i c


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point r0 . We assume that the unperturbed system e 0 has a homoclinic loop w0 t (Fig. 10a),
t 3 ÆI

w u tY t0 dtY t0 NtY t0 r
p

lim w0 t r0 X

r

0

The perturbation results in splitting this loop, such that the s u incoming w0 and outgoing w0 branches no longer form a single curve. To find the condition of the presence of chaos, it is necessary to perturbatively calculate the distance DtY t0 between the branches of the separatrix at some instant t0 . s Then, if DtY t0 does not change its sign, the branches w0 and u w0 do not intersect. But if the sign of DtY t0 changes at some t0 , homoclinic tangles appear (Fig. 12c) and chaotic motion sets in. This is considered formally in Section 7.1.

w0 w s t Y t 0 Figure 15. Determining the distance between the separatrix branches, DtY t0 N d [54].

form as DtY t0 f0 d. We rewrite this expression as D Ds À Du Y where w À ith r M w0 follows 45

7.1 The Mel'nikov function The method proposed in Ref. [91] is based on a comparison of first-order terms in the expansion of separatrix solutions into series in the perturbation parameter e. In other words, to calculate DtY t0 , it is sufficient to find a stable branch w s and an unstable branch w u in the first approximation. s Let w sY u tY t0 w0 t À t0 ew1 Y u tY t0 . Then, using Eqn (42), we obtain, in the first order in e,
s Á À dw1 Y u s Mw0 w1 Y u ef1 w0 t À t0 Y t Y dt

s D sY u tY t0 f0 w1 Y u . N o w , d i f f e r e n t i a t i n g D s s s e s p e c t t o t i m e y i e l d s D s f0 w1 f0 w1 Á s s w0 w1 f0 w1 . Therefore, because w0 f0 , it from (43) that À Á s s s D s f0 w1 f0 w1 Mw0 f0 w1 À Á s f0 Mw0 w1 f0 f1

or
s D s Sp Mw0 f0 w1 f0 f1 Sp Mw0 D s f0 f1 X 46

43

where Mw0 is the H qf01 f qx1 Mw0 f d qf 02 qx1

Jacobi matrix I qf01 qx2 g g qf02 e qx2

Asymptotically, we have the distance s D s tY t0 t 3 I f0 r0 w1 0 X If the unperturbed system is weakly dissipative, Sp M % 0. Therefore, integrating Eqn (46) over the semi-interval t0 Y I, we obtain I D s t0 Y t0 À f0 f1 dt X
t
0

taken at the unperturbed homoclinic trajectory w0 t À t0 . If a perturbation is introduced into an integrable system, the original hyperbolic point r0 displaces to a point rp . Then, obviously, the stable and unstable branches satisfy the relation
t3I

lim w s lim w u rp X
t 3 ÀI

Similarly, for D we find that t0 f0 f1 dt X D u t0 Y t 0
ÀI

u

We introduce the vector dtY t0 w s tY t0 À w u tY t0 X In the first approxim ation, dtY t0 w tY t0 À w tY t0 . According to Ref. [91], the distance DtY t0 between the stable and unstable branches of the separatrix is defined as the projection of d onto the normal N to the unperturbed loop w0 at the instant t (Fig. 15): DtY t0 Nd X 44
s 1 u 1

Because the sought function D is determined by Eqn (45), we finally obtain I DÀ f0 f1 dt X 47
ÀI

be

At e 0, using Eqn (42), we can determine the vector N to NtY t0 Àf02 w0 f01 w0 X

The obtained function (sometimes called the Mel'nikov function) describes the splitting of separatrices of a hyperbolic point due to the introduction of a small perturbation into an integrable system. If function (47) is sign-alternating, the stable and the unstable branches intersect forming homoclinic tangles, which results in the onset of chaotic dynamics. Despite its relative simplicity, the Mel'nikov method has proven itself in applications (in particular, to weakly dissipative systems). Moreover, for certain problems, it is possible to calculate integral (47) analytically and compute the threshold of the onset of chaos.

If we introduce the operator by the relation x y x1 y2 À x2 y1 , Eqn (44) can be represented in a simpler

7.2 The Duffing oscillator and nonlinear pendulum As the first example of using the above analytic approach and estimating its accuracy, we consider two models that are used


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to te s t v a r i ou s a pp l ic a ti on s of th e m et ho ds o f a n al y ti c treatment of nonlinear systems: the Duffing oscillator [92] and a nonlinear pendulum with dissipation and periodic excitation. The Duffing oscillator is a system with two stable equilibrium states divided by a single separatrix and with excitation. The equation describing such a system is called the Duffing equation: x dx À x x 3 g cos ot X 48

If o O, the values of d at which D can change sign can be explicitly written as d ` dc 3 jB À Aj X 2k 2 51

For o T O, we must additionally consider the cases of commensurate and incommensurate frequencies o and O. In the former case, d ` dc 3A B X 2k 2

Equation (48) has been comprehensively studied, on the one hand, numerically and analytically and, on the other hand, experimentally (see, e.g., Refs [88, 93 + 95] and the references therein). At d 0, Eqn (48) always has chaotic solutions near the separatrix, which are bounded by invariant curves. In the dissipative case, no such curves exist, and hence, with the passage of time, a chaotic trajectory finds its way to the attraction region of a stable focus or limit cycle. In this case, transient chaos is observed in the system. The Mel'nikov function for Eqn (48) can be estimated without difficulty (see Ref. [96]). We rewrite Eqn (46) in a different form, x y, y x À x 3 À dy g cos ot. The unperturbed Hamiltonian is then given by H0 y 2 a2 À x 2 a2 x 4 a4. The equations of motion at the separatrix can be written as [96] p p sinh t 2 X Y y0 t À 2 x0 t cosh t cosh2 t Using these relations, we next find f0 f1 y0 g cos ot À dy0 . The integral of this function can be analytically calculated as p sin ot0 4 dX Dt0 2 pgo cosh poa2 3 Hence, the condition of intersection of separatrices can be written as p 3 2 pgo X 49 d ` dc 4 cosh poa2 A generalized Duffing equation is also often considered; it describes an oscillator with a parametrically perturbed cubic term, x dx À x b1 Z cos Otx g cos ot Y where Z 5 1 is the amplitude and O is parametric perturbation. This equation it has two excitation frequencies, o and The Mel'nikov function for Eqn (50) Ref. [97])
3

For incommensurate frequencies (see Ref. [98]), 3 3 jA À Bj ` dc ` 2 A B X 2k 2 2k For a nonlinear pendulum that obeys the equation x ax sin x g cos ot, the Mel'nikov function can also be calculated analytically [56, 57], 1 2pg cos ot0 Dt0 À4aB Y Y1 Æ cosh poa2 2 where BrY s is the Euler b function. Based on this, the threshold of the onset of chaotic motion can also be determined. As found in Refs [56, 96, 98], the obtained analytic results agree very well with numerical estimates based on calculations of the maximal Lyapunov exponent. For example, at b 8, g 0X114, Z 0X03, and o O 1X1 [see Eqn (50)], we obtain dc 0X3798 from (51). A numerical analysis yields dc 0X378 as the threshold of the onset of chaos. Thus, in some cases, the Mel'nikov method allows analytically determining the threshold of the onset of chaos, in particular, for multidimensional systems [99]. We emphasize, however, that this is a local criterion that can be used in the neighborhood of unperturbed separatrix loops.

8. Principal properties of chaotic systems
We now describe important properties inherent in chaotic systems: unpredictability, irreversibility, and decay of correlations. However, we start by introducing the concept of mixing.

50

the frequency of the is interesting because O. can be written as (see

Dt0 ÀA sin ot0 B sin Ot0 C Y where A pkgo sech B po Y 2

8.1 Ergodicity and mixing We assume that the trajectories of a dynamical system are localized in a certain bounded domain D of the phase space. The motion of the dynamical system is said to be ergodic (or the dynamical system is called ergodic) if, for an arbitrary integrable function hx, at almost all initial conditions x0 , the temporal and phase averages of hx coincide: 1T t lim hF x0 dt hx dm Y T3I T 0 D 52 mD dm 1 X
D

1 pO pbZk 4 O 2 4 O 2 cosech Y 24 2 s 2 2 k C k 2d Y X 3 b

Thus, if the motion of a conservative system is ergodic, a phase trajectory, with the passage of time, uniformly and densely covers the hypersurface specified by all integrals of motion in the n-dimensional phase space. A characteristic property of the ergodic motion of a Hamiltonian system is the invariance of the shape of a small domain. As time passes,


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D At
0

D At At
1 2

D

a system, its behavior can naturally be considered chaotic. Moreover, the property of mixing has three very important consequences observed in natural systems Ð unpredictability, irreversibility, and decay of temporal correlations.

B t t0 t t1

B t t2

B

Figure 16. Evolution of a small domain during mixing.

such a domain simply moves over the torus intersecting each of its segments infinitely many times. It is obvious, however, that `ergodic' does not necessarily mean `chaotic.' In addition, the motion of a chaotic system must also be locally unstable, which corresponds to the presence of homoclinic and heteroclinic tangles. Because of such an instability, an initial elementary volume, moving over the entire energetically accessible hypersurface (as for ergodic flows), undergoes strong deformations with time. Such systems are called systems with mixing. The concept of mixing is very simple in essence. For illustration, we consider Gibbs's classic example. We assume that a vessel contains 30% ink and 70% water and these fluids are not mixed at the initial instant. We then thoroughly shake the contents of the vessel. It is natural to expect that some time later (after the mixing time has elapsed), any portion of the obtained mixture will consist of 30% ink and 70% water. Formalization of such a process leads to the concept of mixing. We consider two arbitrary domains A and B with measures mA and mB inside the domain D & M and assume that the domain B remains motionless, while the domain A evolves under the action of the transformation F t : At F t A (Fig. 16). Let At B be the collection of all parts of At that were at inside the motionless domain B at an instant t. Dynamical system (1) is called a system with mixing (and, accordingly, the phase flow F t is said to be mixing) if there exists the limit À Á m At B mA X 53 lim t3I mB We clarify the meaning of this definition. First, we recall that because the considered motion is conservative, the measure of the domain A is preserved in the course of motion, i.e., mAt mA. Moreover, mA mAt amD is the relative volume À Á by the domain A in D. At the occupied same time, the ratio m At B amB is the relative volume of the domain B occupied by the pieces of At that found their way there by time t. It follows from definition (53) that these two ratios coincide as t 3 I, irrespective of the sizes, shapes, and mutual positions of the domains A and B. A mixing flow arises if phase-space points that are close to one another initially move in exponentially diverging trajectories. This fact was noted long ago by E Hopf as he analyzed motion on a negative-curvature manifold [100]. It can be easily understood that mixing is inherent in dynamical systems in the domain of homoclinic and heteroclinic tangles. Mixing implies ergodicity. But the converse is not true: ergodicity does not imply mixing. Therefore, mixing is a much stronger property than ergodicity. If mixing occurs in

8.2 Unpredictability and irreversibility Th e unp r ed i cta bi l i ty an d i rr e ve r s i bi l i ty o f th e tem por a l evolution of systems described by reversible dynamic equations (e.g., Hamiltonian equations) can be described based on the concepts of mixing and openness. As is known, because of unavoidable fluctuations and external perturbations, the state of any physical system can be known only to a certain finite accuracy. This can be interpreted as the distribution of initial conditions x0 in some domain Oe & D with a characteristic size e. Then, if the system has the property of mixing, such an initial distribution spreads over the entire domain D as t 3 I. In other words, if the position of a phase point is initially known to a finite accuracy (i.e., we only knew at t 0 that this points belonged to a certain region Oe ), it is impossible to predict the position of this point after a sufficiently long lapse of time. Thus, unpredictability originates in systems with mixing. The sensitivity of the system to initial conditions (their small perturbations) plays a crucial role here. If an isolated Hamiltonian system is ergodic, then it passes through all energetically possible states in the course of time and its phase trajectory visits all accessible domains of the phase space. The time of residence in a given domain is then proportional to the volume of this domain. As t 3 I, any of these states is observable. If the number of the degrees of freedom of the system is sufficiently large, then the time in which the system reaches a certain particular state (assumed to be the initial one) can be very long. This obviously implies the irreversibility of closed systems with very many degrees of freedom. This explanation of irreversibility is now generally accepted (see also Refs [101, 103] in this context). For low-dimensional open systems, external fluctuations play an important role in the emergence of irreversibility. If mixing occurs in the system, then, inside any accessible part of a domain D in the phase space M, parts of various initial domains A & D find themselves after the lapse of a sufficiently long time (see Fig. 16). Therefore, if we only know that the particle is located at a finite time within a small domain with a size of the order of e, we cannot indicate where it was at the initial time. In other words, the openness and, therefore, the unavoidable inaccuracy in determining the current state of the system play an important role in this case. It can be easily seen that there is no contradiction with the existence of reversible equations of motion that describe motion with mixing (i.e., random motion, in a sense). The point is that irreversibility is related to coarsening or rounding of some quantities. In this case, part of the information contained in the exact solution is lost from the very beginning. The state of such a system can only become progressively less definite with its motion. For this reason, describing chaotic systems immediately necessitates using a coarsened analysis. Thus, the concept of mixing corresponds to our intuitive idea of systems with complex and irregular regimes of motion. But not all chaotic motion must be mixing. Definition (53) imposes a number of strict limitations that are not always satisfied in reality.


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8.3 Decay of correlations A consequence of mixing in a system is a very important property that is frequently used as the criterion of chaotic behavior: the decay of temporal correlations. We consider arbitrary integrable functions f and g in a certain domain D of the phase space M of a dynamical system. We write their phase averages as 1 f x dx1 F F F dxn Y hfi VD D 1 gx dx1 F F F dxn X hgi VD D
Let the initial state of the system be specified by the initial conditions x0 x0 . Then, after the lapse of some time, the system comes to the state xt that depends on x0 in view of the uniqueness of the solution, x xx0 Y t. We define the correlator of the functions f and g as Á À Á À 1 f xx0 Y t gx0 dx0 X f xx0 Y t gx0 VD D This can be rewritten in the form Á À Á À 1 f x g x0 xY t dx X f x g x0 xY t VD D It can be easily shown (see, e.g., Ref. [103]) that if mixing occurs in the system, then we have 54 f x gx0 h f ihgi for arbitrary functions f and g as t 3 I. Property (54) is called the decay of temporal correlations. In other words, a system with mixing `forgets' its initial conditions with the passage of time. The decay of correlations is among the very important consequences of chaotic behavior and consists of the fact that after the lapse of a time interval equal to the correlation-decay time, the values of the phase variables become statistically independent.

Let M be the Euclidian plane. As a billiard table Q, we take a certain domain Q & M with a piecewise-smooth boundary qQ. The billiard dynamical system in Q is generated by the free motion of a mass point (billiard ball) with elastic reflections from the boundary in accordance with the law `the incidence angle is equal to the reflection angle.' If the ball reaches a corner, its further motion is not uniquely defined or is not defined at all. As a rule, the boundary of all billiards consists of m components qQi , i 1Y 2Y F F F Y m. If, for each component qQi , we introduce unit normals nq at each point q P qQi directed inside the domain Q, they determine the curvature kq of the curve qQi at all regular points q. The component qQi is scattering if kq b 0. At kq 0 and kq ` 0, we respectively obtain a neutral and a focusing component of the billiard boundary. We let the union of all scattering, neutral, and focusing boundaries be respectively denoted by qQ , qQ 0 , and qQ À . If the billiard boundary consists only of sufficiently smooth convex curves inside Q, such that qQ qQ , then the billiard system is called a scattering (or Sinai) billiard. It is for scattering billiards that the property of mixing was proved for the first time [108]. A billiard system whose boundary consists of focusing components Q qQÀ or neutral and q focusing components qQ qQ 0 qQ À is called focusing. Under certain conditions, such a billiard can also have the property of mixing [45, 107, 109]. For arbitrary relations between the billiard-boundary parameters, the dynamics of a particle are determined by the relation between the times spent by the particle in the neighborhoods of qQ and qQ À . The exact meaning of this statement is formulated in Refs [45, 106, 109].

9. Billiards
The concept of dynamical chaos was rigorously substantiated for the first time in the 1970s based on a fairly simple model of st a ti st i c al p h y si c s, b ill ia rd s (s e e S e c t io n 1 ). Co n s id e r in g billiard models traces back to studies by J Hadamard [104], who investigated motion on a twisted negative-curvature surface. Later, billiards as dynamical systems were studied by D Birkhoff [13]. A more complete analysis of the problems related to the dynamics of mass points in a bounded region was given by N S Krylov [14]. The issues that arise in analyzing billiard problems are closely related to Boltzmann's ergodic hypothesis; for this reason, billiard models still attract appreciable interest. Fairly general conditions of the onset of chaos in billiards have been obtained, and the remarkable properties of billiards have been described (see Refs [45 + 47, 60, 105 + 107] and the references therein). In this section, we consider a simple generalization of billiard systems Ð billiards with oscillating boundaries. As we demonstrate below, this generalization leads to unexpected and very interesting phenomena.

9.1 The Lorentz gas A variation of the scattering billiards is a system defined in an unbounded dom a in D and con si sting of m any cir cula r, infinitely heavy scatterers Bi with boundaries qQi , of radius R, situated at the sites of an infinite periodic triangular lattice with a step a (Fig. 17). The billiard in the domain Q D n ir1 Bi is called the Lorentz gas. For qQ const, such a billiard has been studied fairly comprehensively (see, e.g., Refs [44,45, 47] and the references therein). The properties of this model can be qualitatively different depending on the scatterer radius and the parameter a. If

R a

Figure 17. Two-dimensional model of the Lorentz gas. Scatterers of radius R are located at the sites of a triangular lattice with a period a.


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R 4 aa2, the billiard is said to have an infinite horizon. If p R 5 aa 3, the motion of the particle is restricted to one cell, and hence p system has a bounded horizon. If the aa2 ` R ` aa 3, the free path of the particle is limited, but it can freely move in space. In this case, the Lorentz gas has an open horizon. It was proved for the Lorentz gas with bounded and open horizons that the motion of the particle in a planar, twodimensional case has the property of mixing and, moreover, reduces to Brownian motion. A positive diffusion coefficient exists in such a system [44]; it can be expressed in terms of the correlation function of the particle velocity. Thus, Brownian dynamics can follow from a purely deterministic behavior. For an infinite horizon, the statistical properties of the Lorentz gas change because of the increasing probability of long paths. In particular, the decrease of correlations is slowed down, and the mean free path does not converge [44, 45, 110 + 112]. At the same time, for a Lorentz gas with a bounded or open horizon, an exponential decay of correlations is observed.

9.2 Scattering billiards with oscillating boundaries. The Fermi acceleration T he Lo re nt z gas is a sys te m lyin g w ith in t he sc op e o f nonequilibrium statistical mechanics. Among its variations is the model first suggested in the context of the Fermi acceleration [113, 114], in which the scatterers oscillate with a small amplitude. A generalized treatment of this problem implies a physically more natural analysis of billiards with boundaries disturbed according to a certain law. What do the oscillations of the boundaries result in? The particle in such billiards experiences both head-on and head + tail collisions with the boundary. In the first case, the reflection from the boundary occurs if the particle and the boundary move opposite to each other. In the second case, the collisions occur if the particle and the boundary move in the same direction. The head-on collisions `heat' the particle, i.e., increase its velocity. In contrast, head + tail collisions result in `cooling' the particle, i.e., in decreasing its velocity. The mechanism of particle acceleration due to collisions with moving massive scatterers was first suggested by Fermi [115] to account for the origin of high-energy cosmic particles (see Ref. [116] for a review). Fermi's idea was that charged particles colliding with chaotically moving magnetic clouds in interstellar space should accelerate on average. If we regard the cloud as a massive body, we can easily understand the reasons for the acceleration. If the velocities of the clouds with which the particle collides are distributed at random, it can be said that the number of clouds moving in some direction is equal to the number of oppositely moving clouds. Therefore, the particle predominantly collides with counter-moving clouds and hence more frequently gains energy that loses it. An effective acceleration, called the Fermi acceleration, thus occurs. Later, the Fermi + Ulam and other models were developed (see Refs [54, 117 + 125]) to account for the origin of this phenomenon to a greater or lesser degree. Billiards with perturbed boundaries can be considered a generalization of the Fermi + Ulam model. In particular, an investigation of the properties of elliptic and circular billiards shows [126 + 128] that the increase in velocity is limited in this case, as in the Fermi + Ulam model. Billiards were also considered in the domain formed by a rectangle with corners replaced with circular quadrants of radius R, with a

periodically oscillating side [129]. The collision with the boundary is not absolutely elastic, the particle loses part of its velocity, and the loss is proportional to a certain constant d d 5 1. This model is close to the Fermi + Ulam model but the presence of the rounded corners introduces some randomization into the particle dynamics. The relaxation of the system to an equilibrium state was studied. Similar investigations were previously carried out in Refs [130, 131] using the Fermi + Ulam model. It was shown that the quantity Pt Et À EI, i.e., the deviation of the average energy of the system from its equilibrium value decreases exponentially in this model, Pt $ exp Àtat, which is generally natural for most physical systems. An analysis of this quantity for a billiard model [129] shows that the relaxation of the system to the equilibrium state is slower, and PÁt decreases in À accordance with the law Pt $ exp Àtatb , where b ` 1 and b decreases with an increase in R. The results in Refs [113, 114] have clarified the reasons for the decrease in the relaxation rate of the system. As the radii of the arcs are increased, the motion becomes more chaotic, which results in the acceleration of particles. Therefore, the relaxation of the system to the equilibrium, being related to the dissipation of the particle energy, is slower. Such an approach allows determining the quantity b and, accordingly, the rate of relaxation to the equilibrium for systems with predominantly chaotic dynamics. In the general case, the situation is much more complex: the Fermi acceleration is not inherent in an arbitrary chaotic billiard with disturbed boundaries [114]. The point is that particle deceleration can also be observed in focusing billiards depending on the initial velocity of the particle [132]. If the initial velocity (which generally depends on the geometry of the billiard) exceeds a certain critical value, the particle is accelerated on average. On the contrary, if the initial velocity is less than the critical value, the billiard particles are decelerated. Thus, it becomes possible to separate the ensemble of particles by their velocity and decelerate particles by disturbing the boundaries. This effect of the billiard version of Maxwell's demon was first revealed and described in Refs [114, 132] based on the example of a `stadium' billiard (i.e., a billiard whose domain Q resembles the usual stadium in shape; see Fig. 18). The dynamics of scattering billiards with disturbed boundaries were described in detail in Refs [113, 114, 133]. For this reason, we do not consider them here, dwelling on focusing billiards. We only note that the obtained results suggest a universal presence of particle acceleration in chaotic billiards with oscillating boundaries. Moreover, based on the investigations carried out, the hypothesis that deterministic randomness is a sufficient condition for the Fermi acceleration was put forward [113, 114]. This assumption has been confirmed by several investigators and, in part, was recently substantiated [125, 134, 135].

9.3 Focusing billiards with oscillating boundaries. Particle deceleration Th e m e cha ni s m s of the ons et of cha os a r e d if fer ent i n scattering and focusing billiards [45, 47]. In the first case, a parallel beam encountering a scattering component starts diverging immediately. In the second case, after a reflection from the focusing surface, the beam converges at the focusing point. In such billiards, chaos sets in if the convergence time of the beams is shorter than the divergence time (i.e., the beam of trajectories is defocused).


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a
b Vn l R
1

x

1.0 0.8 0.6

a

Vn F jn a VnÀ1 aà n a

cn

Vn

1

b
Vn cnÀ
1 À1

1.0 x 0.8 0.6 0.4 0.2

b

jn a
à n

1 Ã n1

Vn cn
1

1

0.4 0.2 0

n

a

n

Vn a

0

à xn

xn

x

0.1p 0.2p 0.3p 0.4p 0.5p 0 c

0.1p 0.2p 0.3p 0.4p 0.5p c

Figure 18. The construction of a map for stadium billiards.

Figure 19. Phase portrait of a stadium billiard with a boundary in the form of a circular arc (a) and a parabola (b), at a 0X5, b 0X01, and l 1 [132].

We first consider billiards with unperturbed boundaries [132]. We assume that the curvature of its focusing components is small, l4a4bX 55

Let these components be formed by arcs of radius R, with the angular measure 2F (Fig. 18a), symmetric with respect to the vertical axis of the billiard. Geometric considerations lead to the relations R a 2 4b 2 a8b and F arcsin aa2R. In particular, the condition for chaos is satisfied for such a billiard if the arc completing the focusing component to the entire circle belongs to the domain Q (see Ref. [45]). For b 5 a, this yields la2R % 4blaa 2 b 1. We introduce dynamic variables as shown in Fig. 18a. Let the positive direction be counterclockwise for the angles jn à and an and clockwise for the angle an . If the boundary of the à billiard is motionless, the incidence angle an is equal to the reflection angle an . Let Vn be the particle velocity and tn the instant of the nth collision with the boundary. To construct a map describing the dynamics of the particle in such a billiard, two cases must be considered: (1) after a collision with a focusing component, the particle collides with it again (pair collisions) and (2) the next collision occurs with another focusing component. In the case of pair collisions, a geometric analysis [132] leads to a map of the form (Fig. 18b)
à an1 an Y

a

n1 n1

a

à n1

Y 2R cos an X Vn

56

j t

jn p À 2an mod 2p Y tn

n1

Figure 19a presents a numerically obtained phase portrait of a stadium billiard specified by maps (56) and (57). As the coordinate, we chose the quantity xn 1a2 R sin jn aa equal to the projection of the point of collision between the particle and the focusing component onto the 0x axis, normalized to the width of the billiard. The gray scale in the figure represents the number of points in the given region of phase space. To construct the graph, we used four trajectories. One of them starts in the stochastic region and includes 5 Â 108 iterations. Three others start in the regular region and contain 107 iterations each. The initial conditions are marked with crosses in the graph. It can be seen from Fig. 19 that stable fixed points surrounded by invariant curves exist in the system. The dynamics of particles in the neighborhood of these points are regular and are represented by motion along such invariant curves. The regions corresponding to different resonances are divided by separatrices surrounded by a stochastic layer. The width of this layer is determined by the degree of nonlinearity of the system. The particle starting its motion in such a layer visits all regions accessible to it in a chaotic manner. As the nonlinearity increases, the fixed points lose their stability; as a result, a globally stochastic domain forms, for which the entire phase space is accessible to a particle. For an analytic treatment of map (56), (57), we make some simplifications. We approximate the focusing component of the billiard system by a segment of a parabola, wx 4bxx À aaa 2 ; as the coordinates, we choose the angle c between the velocity vector and the vertical, c P Àpa2Y pa2, and the projection x of the particle + boundary collision point onto the 0x axis, x P 0Y a. With approximation (55), we then find 58 cn À 2bxn1 Y n1 Á À where bx arctan w H x is the inclination angle of the tangent at the coll ision point. In t his case, obviously, b % 4b2x À aaa 2 . We now make the substitution x xaa, x P 0Y 1. In these variables, the map acquires the form c x
n1

If jjn1 j ` F, the particle continues a cascade of collisions with one component. Otherwise, the n 1th collision occurs with another focusing component. For a transition from one focusing component to another, the map can be written as ! xà à an1 arcsin sin cn F À n1 cos cn Y R a
n1 n1

x

n1

xn l tan c

n1

mod a Y

a

à n1

Y
à n1

57 Y

xn

l tan cn mod 1 Y a
1

j t

cn À a tn

Rcos jn cos jn1 À 2 cos F l Y n1 Vn cos cn  à where cn an À jn , xn Ra cos cn sin an sin F À cn , à and xn1 xn l tan cn mod a.

c

n1

8b cn À 2xn a

59

À 1 X

Clearly, one of the families of fixed points É the billiard can of È be written as x 1a2, cs arctan maal , m 1Y 2Y F F F . In the configuration space, these points correspond to collisions


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of the particle with the center of the focusing component. We analyze the stability of such points. For this, we linearize the m a p u s i n g t h e s u b s t i t u t i o n xn Dxn 1a2, cn Dcn arctan maal . Then, an expansion into a series in Dc yields Dx
n1

Dxn

l 2 Dcn ODcn Y a cos2 cs 16b Dx a
n1

Dc

n1

Dcn À

Y

where cs arctan maal . In this case, the transformation matrix is H I l 1 f a cos2 cs g f g Af gX d 16b 16bl e À 1À 2 a a cos2 cs We can easily verify that det A 1. Therefore, the obtained map preserves the measure. Hence, we can find that the condition of stability of fixed points is cos2 cs 5 4blaa 2 , or m 2 4 la4b À l 2 aa 2 . The destruction of all resonances and transition to chaos occurs if 4blaa 2 b 1. It can be understood without difficulty [132] that the phase-space dynamics of the billiard system in the neighborhood of a stable point are described by a twist map with the rotation number r  à arccos 1 À 8blaa cos cs 2 . The time between successive collisions of the particle with the boundary is t % laV cos cs , where V is the particle velocity. Therefore, the rotation period is T
rot

[132]. The particles can either accelerate or decelerate depending on their initial velocity. We consider this phenomenon in more detail. Let the focusing components of the billiard be disturbed such that the velocity of their motion is the same in magnitude at each point and directed normally to them. We also assume that the boundary oscillates according to a periodic law (Fig. 18a): R R0 r0 f ot Z, where o is the oscillation frequency and Z is the phase. Then the velocity of the boundary is Ut R. In addition, we assume that the displacement of the boundary is small, i.e., r0 5 l. Then the map corresponding to such a billiard system is given by [114, 132] q 2 Ã 2 Vn VnÀ1 4VnÀ1 cos an Un 4Un Y 61 V nÀ 1 Ã sin an Y an arcsin Vn W Ã an1 an Y b b b a jn1 jn p À 2an mod 2p Y at jjn1 j 4 F b 2R cos an b b Y tn1 tn 62 Vn and, if jjn p À 2an j b F, cn an À jn Y Ã RÂ xn sin an sin F À cn Y cos cn
à xn1 xn l tan cn mod a Y à an1 arcsin sin cn F 


2p 2pl 1 t X 60 Â 2Ã r V arccos 1 À 8blaa cos cs cos cs

The phase portrait of the billiard with parabolic focusing components specified by approximate map (59) is shown in Fig. 19b. By comparing it with Fig. 19a based on the exact map (56), (57), we can see that in this case, the trajectory is uniformly distributed over the entire stochastic domain. In the first case, a higher density corresponds to the domain c 5 0, while the domain c 4 pa2 is virtually empty. The difference occurs because in deriving map (59), we assumed that the depth of the focusing component b is small. As a result, if this approximation is used instead of the exact map, collisions at large c occur with a higher probability on the right-hand part of the arc (see Fig. 18), and hence the angle c decreases. Thus, the particles are pushed out to the domain of small c and predominantly move in the neighborhood of the regular domains. If the boundary of the billiard is perturbed, the particle can pass from a chaotic domain to a regular domain, and vice versa. If the particle velocity is much larger than the velocity of the boundaries, it can be shown [132] that the trajectory of the representation point in the neighborhood of stable fixed points of the phase space is similar to a spiral. In this case, the rotation frequency remains the same as in the unperturbed system. In addition, it can be easily understood that a resonance between such a rotation and the perturbation of the boundaries sets in at a certain particle velocity. The presence of the resonance in the system under study leads to an unexpected result Ð the selection of the billiard particles according to their velocity

b b b R b b b b à b b jn1 cn À an1 Y b b b Rcos jn cos jn1 À 2 cos F l b b Y tn1 tn Xb Vn cos cn cos c
n

x

à n1

! Y

W b b b b b b b b b b b b b b b a

63

We note that relation (55) was not used in deriving this map. The only approximation assumed was the smallness of the displacements of the billiard boundary (i.e., we took only the leading terms into account). Expressions (62) correspond to a series of successive collisions of the particle with one focusing component and expressions (63) to a transition from one component of the boundary to another. We analyze map (61) + (63) in two cases: for completely chaotic and nearly integrable billiards. In the first case, the billiard is a classical stadium, i.e., F pa2 (see Fig. 18), and its boundary consists of two halfcircles and two parallel straight lines tangent to them. The following parameters were chosen for this billiard: a 0X5, b 0X25, l 1, r0 0X01, o 1, and V0 0X1. The particle velocity was calculated as the average over an ensemble of 5000 trajectories with various initial conditions. The initial conditions were chosen at random at the focusing component such that the velocity vector of the particle was directed inward for the billiard table Q. As follows from a numerical analysis (curve 1 in Fig.p20a), the obtained dependence has the form Vn $ n. It is interesting to note that the same dependence is also observed in the Lorentz gas [113, 114]. In the nearly integrable case, the parameter b is fairly small, the curvature of the focusing component introduces only a weak nonlinearity into the system, and relation (55) is satisfied. In such a configuration, the phase space has domains of regular and chaotic dynamics (see Fig. 19).


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fairly high velocities (curve 5) corresponding to the maximum velocity in the ensemble are present. As a result, the average velocity (curve 4) increases. Therefore, a certain critical velocity exists in this billiard system; particles with velocities below the critical value decelerate and those with velocities above the critical value accelerate. The critical velocity can be expressed as Vc ol  ÃX cos cs arccos 1 À 8blaa cos cs 2 64

4

2 2.0 4.0 3.5 3.0 2.5 2.0 1.5 1.0

3

6.0

8.0

Â105

10.0 n b

V

c

Vc

0.5 0.6 0.7 0.8 0.9 1.0 a

0

0.002 0.004 0.006 0.008 b

Figure 20. (a) Dependence of the average particle velocity on the number n of collisions with the boundary in a stadium billiard with oscillating walls [114, 132]. Two curves 1 correspond to a billiard with well-developed chaotic properties b 0X25. Curves 2 + 5 correspond to a nearly integrable billiard b 0X01, V0 1 (curves 2 and 3) and V0 2 (curves 4 and 5). (b) Dependence of the critical particle velocity Vc on the parameters a and b (see Fig. 18) in a nearly integrable stadium billiard with oscillating boundaries. Dashed curves are numerical calculations; solid curves are approximation (64) [136].

Figure 20b shows the dependence of critical velocity (64) on the characteristic parameters of the billiard, the width a and the depth b of the focusing component (Fig. 18a). The dashed curves represent the dependences found numerically and the solid curves are the approximation based on Eqn (64). It can be seen that the critical velocity approaches zero with an increase in b, i.e., particle deceleration is not observed in the corresponding billiards. The same is the case if the parameter a decreases. This is because such variations in the parameter increase the nonlinearity in the system and, accordingly, the degree of chaotization in the billiard dynamics. Thus, in the billiard system considered, the particles in the ensemble are separated in accordance with their velocity: some of them are accelerated and others are decelerated.

10. Conclusion
T h e o r e t i c a l a n a l ys i s o f n o n l i n e ar s y st e m s w i t h c o m p l e x , particularly chaotic, behavior is among the most important areas of modern physics. This is because chaotic processes are inherent in a very wide class of phenomena. In addition, nonlinear dynamics, in the framework of which the concept of deterministic chaos is developed, combines extremely diverse applications suggested by the natural sciences, from the theory of excitable media to the problems of information processing and storage [137]. The nature and consequences of chaos remain among the most challenging issues in these fields. Substantial progress has been achieved by now in the investigation of classical mechanics systems. In particular, a possibility for renormalization was found for the resonances of perturbed motion near the separatrix [138, 139]. Notable advances were achieved in the KAM theory [53, 70]; its generalizations to infinite-dimensional systems were found [140, 141]; results concerning the presence of diffusion in systems described by partial differential equations were obtained [142, 143]; consequences of the KAM theory for reversible systems were extracted [144]; and quantum versions of the KAM theory were developed. For diffusion trajectories, a structure was discovered that came to be known as the `stochastic web' [60, 62, 145]. Among the areas in which the theory presented here finds direct application, The author is most familiar with the Nbody problem, celestial mechanics, and systems of classical mechanics; we therefore outline the range of issues that belong precisely to these areas. Applications to other fields are described in some publications in the reference list. Â In his time, Poincare called the N-body problem a fundamental unresolved problem of classical mechanics. V M Alekseev substantially contributed to the development and investigation of this problem [146]. The KAM theory cannot be directly used to analyze planetary problems,

In a sufficiently small neighborhood of fixed points, the billiard system has a definite rotation period Trot [see Eqn (60)]. At the same time, the oscillation period of the boundary is T 2pao. Thus, at a certain velocity Vr , a resonance is observed in the system. As follows from the analysis in Ref. [132], the dependence of the particle velocity on the number of collisions is different for the two sides of the resonance. For the initial velocity V0 ` Vr , the particle velocity decreases to the final value Vfin ` Vr and the velocity distribution of particles in the interval 0Y Vfin approaches a stationary distribution. But if V0 b Vr , the particle velocity can reach high values. In this case, the distribution of particles is time-dependent and increases indefinitely. Figure 20a (curves 2 + 5) shows the dependence of the particle velocity on the number of collisions. Based on 5000 realizations for each initial velocity, three curves were plotted for the velocities (mean, minimum, and maximum) achieved by the nth collision with the boundary. The range of velocity variation can be determined thus. As follows from the figure, at V0 ` Vr , the average particle velocity (curve 2) gradually decreases and approaches a constant. The maximum particle velocity (curve 3) decreases to Vfin and fluctuates around this value. In the case where V b Vr , the minimum particle velocity also decreases. This means that particles that have found their way to the smallvelocity domain are also present in the ensemble. According to the results of numerical analysis, the fraction of such particles is about 75%. At the same time, particles with


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because proper degeneration is present there. But generalizations are known [23, 147] that allow applying the results of the KAM theory to certain systems from a relatively narrow class. In the past 25 years, some extremely important results  were obtained in celestial mechanics: the Painleve hypothesis on collisionless singularities was proved [148 + 150], new remarkable solutions (choreographies) of the N-body problem were found (see Ref. [151]), and resonances were proved to play a key role in the dynamics of celestial bodies [152]. In addition, large-scale chaos was revealed in the Solar system, which seems to be among the most important factors responsible for the formation of the asteroid belt and for the diffusion of comets from the outer regions of the Solar system [153 + 155] (astronomical implications of this phenomenon are described in Refs [156, 157]). It was also established that chaos is present in the system of Saturn's satellites [158]. We mention the notable result that the Arnol'd diffusion is present in the three-body problem [159, 160] (see Ref. [161] for a discussion of the diffusion in the N-body problem). Reference [162] presents a proof of the Arnol'd diffusion in the D'Alembert's problem of the rotation of an oblate planet around a motionless star. A recently published collection [163] presents some new results that refer to the development of chaos in celestial mechanics and the N-body problem. The problem of migration of minor bodies in the Solar system is among the incompletely resolved problems of celestial mechanics closely related to instability and chaos (see Refs [54, 154, 164, 165] and the references therein); in particular, these are the formation of Kirkwood gaps [166] (i.e., gaps in the distribution of asteroids near resonances with Jupiter's mean motion) [167 + 169], of the Kuiper (or Edgeworth + Kuiper) belt [170, 171], and of the Oort cometary cloud [172]. Closely related to these issues is the asteroidalhazard problem (highly popular in recent years), i.e., the evaluation of the probability of migration of minor bodies into the terrestrial-planet zone [164, 173, 174]. We also note theoretical issues of the finiteness of the number of relative equilibria in the classical N-body problem and of collisional trajectories. Among the very interesting phenomena that call for explanation and are related to the theory of chaotic dynamical systems, we note certain complex configurations of the major-planet rings [164, 175]. In recent years, the interest in this problem was revived by the Cassini mission. The stability of the Solar system remains an unresolved fundamental problem of celestial mechanics, which has commanded investigators' attention for a long time [164, 176, 177]. Recent studies have shown that although the Solar system is not stable, it can nonetheless be considered stable over time intervals shorter than its lifetime (about five billion years). Within several billion years, the orbital diffusion of terrestrial planets can result in the recession of Mercury or its collision with Venus (see Ref. [178]). At the same time, the motion of major planets is to be fairly regular. We note, however, that many other effects unaccounted for in the Newton equations (frictional interaction with interstellar dust, relativistic corrections, etc.) must manifest themselves on such a time scale. Therefore, the question of the stability of the Solar system in the formulation in question is only indirectly related to the actual planetary motion. Billiard-type systems with perturbed boundaries described in Section 9 belong to a relatively fresh area of mathematical physics, which opens wider horizons in inves-

tigating certain fundamental problems of classical statistical mechanics (see Ref. [137]). In particular, it presents a new view of the problem of cosmic-particle acceleration to high energies, the Fermi acceleration. Ever since the classic study by Fermi [115], this problem has attracted the attention of researchers in various branches of physics Ð optics [179, 180], plasma physics [181, 182], astrophysics [183 + 185], etc. Similar ideas were recently invoked to account for some new experimental results obtained in atomic physics [186]. Specially configured billiards are also used to study the phenomenon of Fermi acceleration [113, 114, 134]. Such systems have the distinctive feature that they could be used, after a suitable modification, in experimental investigations [187 + 189]. The authors of Refs [113, 114], based on their results, put forward the hypothesis that if a billiard with a fixed boundary is chaotic, switching on a disturbance of its boundary leads to the Fermi acceleration. This hypothesis was tested and partially confirmed (see Refs [125, 134, 135]). In focusing nearly integrable billiards for which the curvature of the boundary components introduces only a weak nonlinearity into the system, periodic disturbances of the boundaries lead to an interesting new phenomenon, particle separation by velocity. Depending on the initial velocity, particles in the billiard system either accelerate or decelerate. In the case of acceleration, the particle distribution is time-dependent and increases unlimitedly. For decelerating particles, an equilibrium velocity distribution is established. This property of a billiard system can be regarded as a billiard version of Maxwell's demon: the periodic perturbations of the boundaries result in velocity losses by slow (`cold') particles, to a certain nonzero level, and the acceleration of fast (`hot') particles. The physical mechanism decelerating the particles is not yet completely understood [136]. Acknowledgments I am obliged to V B Braginskii for his suggestion to write this article, the useful discussions, and the valuable remarks, and also to M L Gorodetskii for critically reading the manuscript and suggesting some improvements in the text.

References
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