I Overture
If have seen less far than other men it is because I have stood behind giants. Edoardo Specchio
1
1.1 Why ChaosBook? 1.2 Chaos ahead 1.3 The future as in a mirror 1.4 A game of pinball 1.5 Chaos for cyclists 1.6 Evolution 1 2 3 7 11 16 1.7 From chaos to statistical mechanics 18 1.8 What is not in ChaosBook 19 1.9 A guide to exercises Summary Further reading Exercises 19 20 22 23
Rereading classic theoretical physics textbooks leaves a sense that there are holes large enough to steam a Eurostar train through them. Here we learn about harmonic oscillators and Keplerian ellipses - but where is the chapter on chaotic oscillators, the tumbling Hyperion? We have just quantized hydrogen, where is the chapter on the classical 3-body problem and its implications for quantization of helium? We have learned that an instanton is a solution of eld-theoretic equations of motion, but shouldnt a strongly nonlinear eld theory have turbulent solutions? How are we to think about systems where things fall apart; the center cannot hold; every trajectory is unstable? This chapter oers a quick survey of the main topics covered in the book. We start out by making promiseswe will right wrongs, no longer shall you suer the slings and arrows of outrageous Science of Perplexity. We relegate a historical overview of the development of chaotic dynamics to Appendix 10, and head straight to the starting line: A pinball game is used to motivate and illustrate most of the concepts to be developed in ChaosBook. This is a textbook, not a research monograph, and you should be able to follow the thread of the argument without constant excursions to sources. Hence there are no literature references in the text proper, all learned remarks and bibliographical pointers are relegated to the Further reading section at the end of each chapter.
Throughout the book indicates that the section requires a hearty stomach and is probably best skipped on rst reading fast track points you where to skip to tells you where to go for more depth on a particular topic indicates an exercise that might clarify a point in the text indicates that a gure is still missingyou are urged to fetch it In the hyperlinked ChaosBook.pdf these destinations are only a click away.
1.1
Why ChaosBook?
It seems sometimes that through a preoccupation with science, we acquire a rmer hold over the vicissitudes of life and meet them with greater calm, but in reality we have done no more than to nd a way to escape from our sorrows. Hermann Minkowski in a letter to David Hilbert
The problem has been with us since Newtons rst frustrating (and unsuccessful) crack at the 3-body problem, lunar dynamics. Nature is rich in systems governed by simple deterministic laws whose asymptotic dy-
2 Overture
namics are complex beyond belief, systems which are locally unstable (almost) everywhere but globally recurrent. How do we describe their long term dynamics? The answer turns out to be that we have to evaluate a determinant, take a logarithm. It would hardly merit a learned treatise, were it not for the fact that this determinant that we are to compute is fashioned out of innitely many innitely small pieces. The feel is of statistical mechanics, and that is how the problem was solved; in the 1960s the pieces were counted, and in the 1970s they were weighted and assembled in a fashion that in beauty and in depth ranks along with thermodynamics, partition functions and path integrals amongst the crown jewels of theoretical physics. This book is not a book about periodic orbits. The red thread throughout the text is the duality between the local, topological, short-time dynamically invariant compact sets (equilibria, periodic orbits, partially hyperbolic invariant tori) and the global long-time evolution of densities of trajectories. Chaotic dynamics is generated by the interplay of locally unstable motions, and the interweaving of their global stable and unstable manifolds. These features are robust and accessible in systems as noisy as slices of rat brains. Poincar, the rst to understand determinise tic chaos, already said as much (modulo rat brains). Once this topology is understood, a powerful theory yields the observable consequences of chaotic dynamics, such as atomic spectra, transport coecients, gas pressures. That is what we will focus on in ChaosBook. The book is a selfcontained graduate textbook on classical and quantum chaos. Your professor does not know this material, so you are on your own. We will teach you how to evaluate a determinant, take a logarithmstu like that. Ideally, this should take 100 pages or so. Well, we failso far we have not found a way to traverse this material in less than a semester, or 200-300 page subset of this text. Nothing to be done.
1.2
Chaos ahead
Things fall apart; the centre cannot hold. W.B. Yeats: The Second Coming
The study of chaotic dynamics is no recent fashion. It did not start with the widespread use of the personal computer. Chaotic systems have been studied for over 200 years. During this time many have contributed, and the eld followed no single line of development; rather one sees many interwoven strands of progress. In retrospect many triumphs of both classical and quantum physics were a stroke of luck: a few integrable problems, such as the harmonic oscillator and the Kepler problem, though non-generic, have gotten us very far. The success has lulled us into a habit of expecting simple solutions to simple equationsan expectation tempered by our recently
1.3
The future as in a mirror 3
acquired ability to numerically scan the state space of non-integrable dynamical systems. The initial impression might be that all of our analytic tools have failed us, and that the chaotic systems are amenable only to numerical and statistical investigations. Nevertheless, a beautiful theory of deterministic chaos, of predictive quality comparable to that of the traditional perturbation expansions for nearly integrable systems, already exists. In the traditional approach the integrable motions are used as zerothorder approximations to physical systems, and weak nonlinearities are then accounted for perturbatively. For strongly nonlinear, non-integrable systems such expansions fail completely; at asymptotic times the dynamics exhibits amazingly rich structure which is not at all apparent in the integrable approximations. However, hidden in this apparent chaos is a rigid skeleton, a self-similar tree of cycles (periodic orbits) of increasing lengths. The insight of the modern dynamical systems theory is that the zeroth-order approximations to the harshly chaotic dynamics should be very dierent from those for the nearly integrable systems: a good starting approximation here is the stretching and folding of bakers dough, rather than the periodic motion of a harmonic oscillator. So, what is chaos, and what is to be done about it? To get some feeling for how and why unstable cycles come about, we start by playing a game of pinball. The reminder of the chapter is a quick tour through the material covered in ChaosBook. Do not worry if you do not understand every detail at the rst readingthe intention is to give you a feeling for the main themes of the book. Details will be lled out later. If you want to get a particular point claried right now, on the margin points at the appropriate section.
Section 1.4
1.3
The future as in a mirror
All you need to know about chaos is contained in the introduction of [ChaosBook]. However, in order to understand the introduction you will rst have to read the rest of the book. Gary Morriss
That deterministic dynamics leads to chaos is no surprise to anyone who has tried pool, billiards or snookerthe game is about beating chaosso we start our story about what chaos is, and what to do about it, with a game of pinball. This might seem a trie, but the game of pinball is to chaotic dynamics what a pendulum is to integrable systems: thinking clearly about what chaos in a game of pinball is will help us tackle more dicult problems, such as computing the diusion constant of a deterministic gas, the drag coecient of a turbulent boundary layer, or the helium spectrum. We all have an intuitive feeling for what a ball does as it bounces among the pinball machines disks, and only high-school level Euclidean geometry is needed to describe its trajectory. A physicists pinball game is the game of pinball stripped to its bare essentials: three equidistantly
Fig. 1.1 A physicists bare bones game of pinball.
4 Overture
placed reecting disks in a plane, Fig. 1.1. A physicists pinball is free, frictionless, point-like, spin-less, perfectly elastic, and noiseless. Pointlike pinballs are shot at the disks from random starting positions and angles; they spend some time bouncing between the disks and then escape. At the beginning of the 18th century Baron Gottfried Wilhelm Leibniz was condent that given the initial conditions one knew everything a deterministic system would do far into the future. He wrote [1], anticipating by a century and a half the oft-quoted Laplaces Given for one instant an intelligence which could comprehend all the forces by which nature is animated...:
23132321
2
1
3
2313
That everything is brought forth through an established destiny is just as certain as that three times three is nine. [. . . ] If, for example, one sphere meets another sphere in free space and if their sizes and their paths and directions before collision are known, we can then foretell and calculate how they will rebound and what course they will take after the impact. Very simple laws are followed which also apply, no matter how many spheres are taken or whether objects are taken other than spheres. From this one sees then that everything proceeds mathematicallythat is, infalliblyin the whole wide world, so that if someone could have a sucient insight into the inner parts of things, and in addition had remembrance and intelligence enough to consider all the circumstances and to take them into account, he would be a prophet and would see the future in the present as in a mirror.
Fig. 1.2 Sensitivity to initial conditions: two pinballs that start out very close to each other separate exponentially with time.
1 Stochastic is derived from Greek stochos, meaning a target, as in shooting arrows at a target, and not always hitting it; targeted ow, with a small component of uncertainty. Today it stands for deterministic drift + diusion. Random stands for pure diusion, with a Gaussian prole. Probabilistic might have a distribution other than a Gaussian one. Boltzmanns Ergodic refers to the deterministic microscopic dynamics of many colliding molecules. Chaotic is the same thing, but usually for a few degrees of freedom.
Leibniz chose to illustrate his faith in determinism precisely with the type of physical system that we shall use here as a paradigm of chaos. His claim is wrong in a deep and subtle way: a state of a physical system can never be specied to innite precision, there is no way to take all the circumstances into account, and a single trajectory cannot be tracked, only a ball of nearby initial points makes physical sense.1
1.3.1
What is chaos ?
I accept chaos. I am not sure that it accepts me. Bob Dylan, Bringing It All Back Home
A deterministic system is a system whose present state is in principle fully determined by its initial conditions, in contrast to a stochastic system. For a stochastic system the initial conditions determine the future only partially, due to noise, or other external circumstances beyond our control: the present state reects the past initial conditions plus the particular realization of the noise encountered along the way. A deterministic system with suciently complicated dynamics can fool us into regarding it as a stochastic one; disentangling the deterministic from the stochastic is the main challenge in many real-life settings, from stock markets to palpitations of chicken hearts. So, what is chaos ?
1.3
The future as in a mirror 5
In a game of pinball, any two trajectories that start out very close to each other separate exponentially with time, and in a nite (and in practice, a very small) number of bounces their separation x(t) attains the magnitude of L, the characteristic linear extent of the whole system, Fig. 1.2. This property of sensitivity to initial conditions can be quantied as |x(t)| et |x(0)| where , the mean rate of separation of trajectories of the system, is called the Lyapunov exponent. For any nite accuracy x = |x(0)| of the initial data, the dynamics is predictable only up to a nite Lyapunov time 1 (1.1) TLyap ln |x/L| , despite the deterministic and, for Baron Leibniz, infallible simple laws that rule the pinball motion. A positive Lyapunov exponent does not in itself lead to chaos. One could try to play 1- or 2-disk pinball game, but it would not be much of a game; trajectories would only separate, never to meet again. What is also needed is mixing, the coming together again and again of trajectories. While locally the nearby trajectories separate, the interesting dynamics is conned to a globally nite region of the state space and thus the separated trajectories are necessarily folded back and can reapproach each other arbitrarily closely, innitely many times. For the case at hand there are 2n topologically distinct n bounce trajectories that originate from a given disk. More generally, the number of distinct trajectories with n bounces can be quantied as N (n) ehn
(b)
Section ??
(a)
where the topological entropy h (h = ln 2 in the case at hand) is the growth rate of the number of topologically distinct trajectories. The appellation chaos is a confusing misnomer, as in deterministic dynamics there is no chaos in the everyday sense of the word; everything proceeds mathematicallythat is, as Baron Leibniz would have it, infallibly. When a physicist says that a certain system exhibits chaos, he means that the system obeys deterministic laws of evolution, but that the outcome is highly sensitive to small uncertainties in the specication of the initial state. The word chaos has in this context taken on a narrow technical meaning. If a deterministic system is locally unstable (positive Lyapunov exponent) and globally mixing (positive entropy) Fig. 1.3 - it is said to be chaotic. While mathematically correct, the denition of chaos as positive Lyapunov + positive entropy is useless in practice, as a measurement of these quantities is intrinsically asymptotic and beyond reach for systems observed in nature. More powerful is Poincars vision of chaos as the e interplay of local instability (unstable periodic orbits) and global mixing (intertwining of their stable and unstable manifolds). In a chaotic system any open ball of initial conditions, no matter how small, will in nite time overlap with any other nite region and in this sense spread
Fig. 1.3 Dynamics of a chaotic dynamical system is (a) everywhere locally unstable (positive Lyapunov exponent) and (b) globally mixing (positive entropy). (A. Johansen) Section ?? Section ??
6 Overture
over the extent of the entire asymptotically accessible state space. Once this is grasped, the focus of theory shifts from attempting to predict individual trajectories (which is impossible) to a description of the geometry of the space of possible outcomes, and evaluation of averages over this space. How this is accomplished is what ChaosBook is about. A denition of turbulence is even harder to come by. Intuitively, the word refers to irregular behavior of an innite-dimensional dynamical system described by deterministic equations of motionsay, a bucket of sloshing water described by the Navier-Stokes equations. But in practice the word turbulence tends to refer to messy dynamics which we understand poorly. As soon as a phenomenon is understood better, it is reclaimed and renamed: a route to chaos, spatiotemporal chaos, and so on. In ChaosBook we shall develop a theory of chaotic dynamics for low dimensional attractors visualized as a succession of nearly periodic but unstable motions. In the same spirit, we shall think of turbulence in spatially extended systems in terms of recurrent spatiotemporal patterns. Pictorially, dynamics drives a given spatially extended system (clouds, say) through a repertoire of unstable patterns; as we watch a turbulent system evolve, every so often we catch a glimpse of a familiar pattern:
=
other swirls
=
For any nite spatial resolution, the system follows approximately for a nite time a pattern belonging to a nite alphabet of admissible patterns, and the long term dynamics can be thought of as a walk through the space of such patterns. In ChaosBook we recast this image into mathematics.
1.3.2
When does chaos matter?
In dismissing Pollocks fractals because of their limited magnication range, Jones-Smith and Mathur would also dismiss half the published investigations of physical fractals. Richard P. Taylor [4, 5]
When should we be mindful of chaos? The solar system is chaotic, yet we have no trouble keeping track of the annual motions of planets. The rule of thumb is this; if the Lyapunov time (1.1)the time by which a state space region initially comparable in size to the observational accuracy extends across the entire accessible state spaceis signicantly shorter than the observational time, you need to master the theory that will be developed here. That is why the main successes of the theory are in statistical mechanics, quantum mechanics, and questions of long term stability in celestial mechanics. In science popularizations too much has been made of the impact of
1.4
A game of pinball 7
chaos theory, so a number of caveats are already needed at this point. At present the theory is in practice applicable only to systems with a low intrinsic dimension the minimum number of coordinates necessary to capture its essential dynamics. If the system is very turbulent (a description of its long time dynamics requires a space of high intrinsic dimension) we are out of luck. Hence insights that the theory oers in elucidating problems of fully developed turbulence, quantum eld theory of strong interactions and early cosmology have been modest at best. Even that is a caveat with qualications. There are applicationssuch as spatially extended (non-equilibrium) systems, plumbers turbulent pipes, etc.,where the few important degrees of freedom can be isolated and studied protably by methods to be described here. Thus far the theory has had limited practical success when applied to the very noisy systems so important in the life sciences and in economics. Even though we are often interested in phenomena taking place on time scales much longer than the intrinsic time scale (neuronal inter-burst intervals, cardiac pulses, etc.), disentangling chaotic motions from the environmental noise has been very hard. In 1980s something happened that might be without parallel; this is an area of science where the advent of cheap computation had actually subtracted from our collective understanding. The computer pictures and numerical plots of fractal science of the 1980s have overshadowed the deep insights of the 1970s, and these pictures have since migrated into textbooks. By a regrettable oversight, ChaosBook has none, so Untitled 5 of Fig. 1.4 will have to do as the illustration of the power of fractal analysis. Fractal science posits that certain quantities (Lyapunov exponents, generalized dimensions, . . . ) can be estimated on a computer. While some of the numbers so obtained are indeed mathematically sensible characterizations of fractals, they are in no sense observable and measurable on the length-scales and time-scales dominated by chaotic dynamics. Even though the experimental evidence for the fractal geometry of nature is circumstantial [2], in studies of probabilistically assembled fractal aggregates we know of nothing better than contemplating such quantities. In deterministic systems we can do much better.
Fig. 1.4 Katherine Jones-Smith, Untitled 5, the drawing used by K. JonesSmith and R.P. Taylor to test the fractal analysis of Pollocks drip paintings [3].
1.4
A game of pinball
Formulas hamper the understanding. S. Smale
We are now going to get down to the brass tacks. Time to fasten your seat belts and turn o all electronic devices. But rst, a disclaimer: If you understand the rest of this chapter on the rst reading, you either do not need this book, or you are delusional. If you do not understand it, it is not because the people who wrote it are smarter than you: the most you can hope for at this stage is to get a avor of what lies ahead. If a
Fig. 1.5 Binary labeling of the 3-disk pinball trajectories; a bounce in which the trajectory returns to the preceding disk is labeled 0, and a bounce which results in continuation to the third disk is labeled 1.
8 Overture
statement in this chapter mysties/intrigues, fast forward to a section on the margin, read only the parts that you feel you indicated by need. Of course, we think that you need to learn ALL of it, or otherwise we would not have included it in ChaosBook in the rst place. Confronted with a potentially chaotic dynamical system, we analyze it through a sequence of three distinct stages; I. diagnose, II. count, III. measure. First we determine the intrinsic dimension of the system the minimum number of coordinates necessary to capture its essential dynamics. If the system is very turbulent we are, at present, out of luck. We know only how to deal with the transitional regime between regular motions and chaotic dynamics in a few dimensions. That is still something; even an innite-dimensional system such as a burning ame front can turn out to have a very few chaotic degrees of freedom. In this regime the chaotic dynamics is restricted to a space of low dimension, the number of relevant parameters is small, and we can proceed to step II; we count and classify all possible topologically distinct trajectories of the system into a hierarchy whose successive layers require increased precision and patience on the part of the observer. This we shall do in Section 1.4.2. If successful, we can proceed with step III: investigate the weights of the dierent pieces of the system. We commence our analysis of the pinball game with steps I, II: diagnose, count. We shall return to step IIImeasurein Section 1.5.
1.4.1
Fig. 1.6 Some examples of 3-disk cycles: (a) 12123 and 13132 are mapped into each other by the ip across 1 axis. Similarly (b) 123 and 132 are related by ips, and (c) 1213, 1232 and 1323 by rotations. (d) The cycles 121212313 and 121212323 are related by rotaion and time reversal. These symmetries are discussed in more detail in Chapter ??. (From Ref. [6]) Chapter ?? Chapter ?? Chapter ??
Symbolic dynamics
1.1, page 23
Section 2.1
Chapter ??
Section ??
With the game of pinball we are in luckit is a low dimensional system, free motion in a plane. The motion of a point particle is such that after a collision with one disk it either continues to another disk or it escapes. If we label the three disks by 1, 2 and 3, we can associate every trajectory with an itinerary, a sequence of labels indicating the order in which the disks are visited; for example, the two trajectories in Fig. 1.2 have itineraries 2313 , 23132321 respectively. The itinerary is nite for a scattering trajectory, coming in from innity and escaping after a nite number of collisions, innite for a trapped trajectory, and innitely repeating for a periodic orbit. Parenthetically, in this subject the words orbit and trajectory refer to one and the same thing. Such labeling goes by the name symbolic dynamics. As the particle cannot collide two times in succession with the same disk, any two consecutive symbols must dier. This is an example of pruning, a rule that forbids certain subsequences of symbols. Deriving pruning rules is in general a dicult problem, but with the game of pinball we are luckyfor well-separated disks there are no further pruning rules. The choice of symbols is in no sense unique. For example, as at each bounce we can either proceed to the next disk or return to the previous disk, the above 3-letter alphabet can be replaced by a binary {0, 1} alphabet, Fig. 1.5. A clever choice of an alphabet will incorporate important features of the dynamics, such as its symmetries. Suppose you wanted to play a good game of pinball, that is, get the
1.4
A game of pinball 9
pinball to bounce as many times as you possibly canwhat would be a winning strategy? The simplest thing would be to try to aim the pinball so it bounces many times between a pair of disksif you managed to shoot it so it starts out in the periodic orbit bouncing along the line connecting two disk centers, it would stay there forever. Your game would be just as good if you managed to get it to keep bouncing between the three disks forever, or place it on any periodic orbit. The only rub is that any such orbit is unstable, so you have to aim very accurately in order to stay close to it for a while. So it is pretty clear that if one is interested in playing well, unstable periodic orbits are importantthey form the skeleton onto which all trajectories trapped for long times cling.
a
1
s1
1
s2
(a)
1.4.2
Partitioning with periodic orbits
p sin 1
(s1,p1)
A trajectory is periodic if it returns to its starting position and momentum. We shall refer to the set of periodic points that belong to a given periodic orbit as a cycle. Short periodic orbits are easily drawn and enumeratedsome examples are drawn in Fig. 1.6but it is rather hard to perceive the systematics of orbits from their shapes. In mechanics a trajectory is fully and uniquely specied by its position and momentum at a given instant, and no two distinct state space trajectories can intersect. Their projections onto arbitrary subspaces, however, can and do intersect, in rather unilluminating ways, as in Fig. 1.6 (d). In the pinball example the problem is that we are looking at the projections of a 4-dimensional state space trajectories onto a 2-dimensional subspace, the conguration space. A clearer picture of the dynamics is obtained by constructing a state space Poincar section. e Suppose that the pinball has just bounced o disk 1. Depending on its position and outgoing angle, it could proceed to either disk 2 or 3. Not much happens in between the bouncesthe ball just travels at constant velocity along a straight lineso we can reduce the fourdimensional ow to a two-dimensional map f that takes the coordinates of the pinball from one disk edge to another disk edge. Let us state this more precisely: the trajectory just after the moment of impact is dened by marking sn , the arc-length position of the nth bounce along the billiard wall, and pn = p sin n the momentum component parallel to the billiard wall at the point of impact, Fig. 1.7. Such a section of a ow is called a Poincar section, and the particular choice of coordinates e (due to Birkho) is particularly smart, as it conserves the phase space volume. In terms of the Poincar section, the dynamics is reduced to e the return map P : (sn , pn ) (sn+1 , pn+1 ) from the boundary of a disk to the boundary of the next disk. The explicit form of this map is easily written down, but it is of no importance right now. Next, we mark in the Poincar section those initial conditions which e do not escape in one bounce. There are two strips of survivors, as the trajectories originating from one disk can hit either of the other two disks, or escape without further ado. We label the two strips M0 ,
s1
p sin 2 (s2,p2)
s2
p sin 3
(s3,p3)
(b)
s3
Fig. 1.7 (a) A pinball trajectory is uniquely specied by noting the disk it bounces o, the collision position along the disk wall, and the outgoing angle. (b) Collision sequence (s1 , p1 ) (s2 , p2 ) (s3 , p3 ) from the boundary of a disk to the boundary of the next disk presented in the Poincar section e coordinates. Section ??
Fig. 1.8 (a) A trajectory starting out from disk 1 can either hit another disk or escape. (b) Hitting two disks in a sequence requires a much sharper aim. The cones of initial conditions that hit more and more consecutive disks are nested within each other, as in Fig. 1.9. Section 6
10 Overture
1
0
1 2.5
111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 12 13 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000 111111111111111 000000000000000
M1 . Embedded within them there are four strips M00 , M10 , M01 , M11 of initial conditions that survive for two bounces, and so forth, see Figs. 1.8 and 1.9. Provided that the disks are suciently separated, after n bounces the survivors are divided into 2n distinct strips: the Mi th strip consists of all points with itinerary i = s1 s2 s3 . . . sn , s = {0, 1}. The unstable cycles as a skeleton of chaos are almost visible here: each such patch contains a periodic point s1 s2 s3 . . . sn with the basic block innitely repeated. Periodic points are skeletal in the sense that as we look further and further, the strips shrink but the periodic points stay put forever. We see now why it pays to utilize a symbolic dynamics; it provides a navigation chart through chaotic state space. There exists a unique trajectory for every admissible innite length itinerary, and a unique itinerary labels every trapped trajectory. For example, the only trajectory labeled by 12 is the 2-cycle bouncing along the line connecting the centers of disks 1 and 2; any other trajectory starting out as 12 . . . either eventually escapes or hits the 3rd disk.
2.5
sin
(a)
1
0 S
111111111111111 000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 123 131 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1 0 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 121 0 132 1 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 111111111111111 000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000 1111111111111111 0000000000000000
1.4.3
Escape rate
0
1
(b)
2.5
0
2.5
s
Fig. 1.9 The 3-disk game of pinball Poincar section, trajectories emanate ing from the disk 1 with x0 = (arclength, parallel momentum) = (s0 , p0 ) , disk radius : center separation ratio a:R = 1:2.5. (a) Strips of initial points M12 , M13 which reach disks 2, 3 in one bounce, respectively. (b) Strips of initial points M121 , M131 M132 and M123 which reach disks 1, 2, 3 in two bounces, respectively. The Poincar e sections for trajectories originating on the other two disks are obtained by the appropriate relabeling of the strips. (Y. Lan) Example ??
What is a good physical quantity to compute for the game of pinball? Such system, for which almost any trajectory eventually leaves a nite region (the pinball table) never to return, is said to be open, or a repeller. The repeller escape rate is an eminently measurable quantity. An example of such a measurement would be an unstable molecular or nuclear state which can be well approximated by a classical potential with the possibility of escape in certain directions. In an experiment many projectiles are injected into a macroscopic black box enclosing a microscopic non-conning short-range potential, and their mean escape rate is measured, as in Fig. 1.1. The numerical experiment might consist of injecting the pinball between the disks in some random direction and asking how many times the pinball bounces on the average before it escapes the region between the disks. For a theorist a good game of pinball consists in predicting accurately the asymptotic lifetime (or the escape rate) of the pinball. We now show how periodic orbit theory accomplishes this for us. Each step will be so simple that you can follow even at the cursory pace of this overview, and still the result is surprisingly elegant. Consider Fig. 1.9 again. In each bounce the initial conditions get thinned out, yielding twice as many thin strips as at the previous bounce. The total area that remains at a given time is the sum of the areas of the strips, so that the fraction of survivors after n bounces, or the survival probability is given by 1 n = = |M0 | |M1 | + , |M| |M| 1 |M|
(n)
sin
1.2, page 23
|M00 | |M10 | |M01 | |M11 | 2 = + + + , |M| |M| |M| |M| (1.2)
|Mi | ,
i
1.5
Chaos for cyclists 11
where i is a label of the ith strip, |M| is the initial area, and |Mi | is the area of the ith strip of survivors. i = 01, 10, 11, . . . is a label, not a binary number. Since at each bounce one routinely loses about the same fraction of trajectories, one expects the sum (1.2) to fall o exponentially with n and tend to the limit n+1 /n = en e . The quantity is called the escape rate from the repeller. (1.3)
Chapter ??
1.5
Chaos for cyclists
Etant donnes des quations ... et une solution particulire e e e quelconque de ces quations, on peut toujours trouver une soe lution priodique (dont la priode peut, il est vrai, tre trs e e e e longue), telle que la dirence entre les deux solutions soit e aussi petite quon le veut, pendant un temps aussi long quon le veut. Dailleurs, ce qui nous rend ces solutions priodiques e si prcieuses, cest quelles sont, pour ansi dire, la seule brche e e par o` nous puissions esseyer de pntrer dans une place u e e jusquici rpute inabordable. e e H. Poincar, Les mthodes nouvelles de la mchanique cleste e e e e
We shall now show that the escape rate can be extracted from a highly convergent exact expansion by reformulating the sum (1.2) in terms of unstable periodic orbits. If, when asked what the 3-disk escape rate is for a disk of radius 1, center-center separation 6, velocity 1, you answer that the continuous time escape rate is roughly = 0.4103384077693464893384613078192 . . ., you do not need this book. If you have no clue, hang on.
1.5.1
How big is my neighborhood?
Not only do the periodic points keep track of topological ordering of the strips, but, as we shall now show, they also determine their size. As a trajectory evolves, it carries along and distorts its innitesimal neighborhood. Let x(t) = f t (x0 ) denote the trajectory of an initial point x0 = x(0). Expanding f t (x0 + x0 ) to linear order, the evolution of the distance to a neighboring trajectory xi (t) + xi (t) is given by the fundamental matrix:
d
xi (t) =
j=1
J t (x0 )ij x0 j ,
J t (x0 )ij =
xi (t) . x0 j
A trajectory of a pinball moving on a at surface is specied by two position coordinates and the direction of motion, so in this case d = 3. Evaluation of a cycle fundamental matrix is a long exercise - here we
Section 6.2
12 Overture
x + x
1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 11111 00000 1111111 0000000 11111 00000 1111111 0000000 11111 00000 1111111 0000000 11111 00000 1111111 0000000 11111 00000 1111111 0000000 11111 00000 1111111 0000000 11111 00000 1111111 0000000 11111 00000 1111111 0000000 11111 00000 1111111 0000000 11111 00000 1111111 0000000 1111111 0000000 1111111 0000000 0 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 0000000
f ( x0)
t
x(t)+J x
just state the result. The fundamental matrix describes the deformation of an innitesimal neighborhood of x(t) along the ow; its eigenvectors and eigenvalues give the directions and the corresponding rates of expansion or contraction, Fig. 4.1. The trajectories that start out in an innitesimal neighborhood separate along the unstable directions (those whose eigenvalues are greater than unity in magnitude), approach each other along the stable directions (those whose eigenvalues are less than unity in magnitude), and maintain their distance along the marginal directions (those whose eigenvalues equal unity in magnitude). In our game of pinball the beam of neighboring trajectories is defocused along the unstable eigendirection of the fundamental matrix M . As the heights of the strips in Fig. 1.9 are eectively constant, we can concentrate on their thickness. If the height is L, then the area of the ith strip is Mi Lli for a strip of width li . Each strip i in Fig. 1.9 contains a periodic point xi . The ner the intervals, the smaller the variation in ow across them, so the contribution from the strip of width li is well-approximated by the contraction around the periodic point xi within the interval, li = ai /|i | , (1.4)
Fig. 1.10 The fundamental matrix J t maps an innitesimal spherical neighborhood of x0 into an ellipsoidal neighborhood nite time t later.
Section ??
Section ??
where i is the unstable eigenvalue of the fundamental matrix J t (xi ) evaluated at the ith periodic point for t = Tp , the full period (due to the low dimensionality, the Jacobian can have at most one unstable eigenvalue). Only the magnitude of this eigenvalue matters, we can disregard its sign. The prefactors ai reect the overall size of the system and the particular distribution of starting values of x. As the asymptotic trajectories are strongly mixed by bouncing chaotically around the repeller, we expect their distribution to be insensitive to smooth variations in the distribution of initial points. To proceed with the derivation we need the hyperbolicity assumption: for large n the prefactors ai O(1) are overwhelmed by the exponential growth of i , so we neglect them. If the hyperbolicity assumption is justied, we can replace |Mi | Lli in (1.2) by 1/|i | and consider the sum
(n)
n =
i
1/|i | ,
where the sum goes over all periodic points of period n. We now dene a generating function for sums over all periodic orbits of all lengths:
(z) =
n=1
n z n .
(1.5)
Recall that for large n the nth level sum (1.2) tends to the limit n en , so the escape rate is determined by the smallest z = e for which (1.5) diverges: ze n (ze ) = . (1.6) (z) 1 ze n=1
1.5
Chaos for cyclists 13
This is the property of (z) that motivated its denition. Next, we devise a formula for (1.5) expressing the escape rate in terms of periodic orbits:
(n)
(z) =
n=1
zn
i
|i |
1
z z2 z2 z2 z2 z + + + + + = |0 | |1 | |00 | |01 | |10 | |11 | z3 z3 z3 z3 + + + + ... (1.7) + |000 | |001 | |010 | |100 | For suciently small z this sum is convergent. The escape rate is now given by the leading pole of (1.6), rather than by a numerical extrapolation of a sequence of n extracted from (1.3). As any nite truncation n < ntrunc of (1.7) is a polynomial in z, convergent for any z, nding this pole requires that we know something about n for any n, and that might be a tall order. We could now proceed to estimate the location of the leading singularity of (z) from nite truncations of (1.7) by methods such as Pad e approximants. However, as we shall now show, it pays to rst perform a simple resummation that converts this divergence into a zero of a related function.
Section ??
1.5.2
Dynamical zeta function
If a trajectory retraces a prime cycle r times, its expanding eigenvalue is r . A prime cycle p is a single traversal of the orbit; its label is a p non-repeating symbol string of np symbols. There is only one prime cycle for each cyclic permutation class. For example, p = 0011 = 1001 = 1100 = 0110 is prime, but 0101 = 01 is not. By the chain rule for derivatives the stability of a cycle is the same everywhere along the orbit, so each prime cycle of length np contributes np terms to the sum (1.7). Hence (1.7) can be rewritten as
??, page ??
Section 4.5
(z) =
p
np
r=1
z np |p |
r
=
p
n p tp , 1 tp
tp =
z np |p |
(1.8)
where the index p runs through all distinct prime cycles. Note that we have resummed the contribution of the cycle p to all times, so truncating the summation up to given p is not a nite time n np approximation, but an asymptotic, innite time estimate based by approximating stabilities of all cycles by a nite number of the shortest cycles and their repeats. The np z np factors in (1.8) suggest rewriting the sum as a derivative d (z) = z ln(1 tp ) . dz p Hence (z) is a logarithmic derivative of the innite product z np (1 tp ) , tp = 1/(z) = . |p | p (1.9)
14 Overture
Section ?? Section ??
This function is called the dynamical zeta function, in analogy to the Riemann zeta function, which motivates the zeta in its denition as 1/(z). This is the prototype formula of periodic orbit theory. The zero of 1/(z) is a pole of (z), and the problem of estimating the asymptotic escape rates from nite n sums such as (1.2) is now reduced to a study of the zeros of the dynamical zeta function (1.9). The escape rate is related by (1.6) to a divergence of (z), and (z) diverges whenever 1/(z) has a zero. Easy, you say: Zeros of (1.9) can be read o the formula, a zero zp = |p |1/np for each term in the product. Whats the problem? Dead wrong!
1.5.3
Cycle expansions
Chapter ??
Section ??
How are formulas such as (1.9) used? We start by computing the lengths and eigenvalues of the shortest cycles. This usually requires some numerical work, such as the Newtons method searches for periodic solutions; we shall assume that the numerics are under control, and that all short cycles up to given length have been found. In our pinball example this can be done by elementary geometrical optics. It is very important not to miss any short cycles, as the calculation is as accurate as the shortest cycle droppedincluding cycles longer than the shortest omitted does not improve the accuracy (unless exponentially many more cycles are included). The result of such numerics is a table of the shortest cycles, their periods and their stabilities. Now expand the innite product (1.9), grouping together the terms of the same total symbol string length 1/ = = (1 t0 )(1 t1 )(1 t10 )(1 t100 ) 1 t0 t1 [t10 t1 t0 ] [(t100 t10 t0 ) + (t101 t10 t1 )] [(t1000 t0 t100 ) + (t1110 t1 t110 ) +(t1001 t1 t001 t101 t0 + t10 t0 t1 )] . . . (1.10)
Chapter ??
Section ??
The virtue of the expansion is that the sum of all terms of the same total length n (grouped in brackets above) is a number that is exponentially smaller than a typical term in the sum, for geometrical reasons we explain in the next section. The calculation is now straightforward. We substitute a nite set of the eigenvalues and lengths of the shortest prime cycles into the cycle expansion (1.10), and obtain a polynomial approximation to 1/. We then vary z in (1.9) and determine the escape rate by nding the smallest z = e for which (1.10) vanishes.
1.5.4
Shadowing
When you actually start computing this escape rate, you will nd out that the convergence is very impressive: only three input numbers (the
1.5
Chaos for cyclists 15
two xed points 0, 1 and the 2-cycle 10) already yield the pinball escape rate to 3-4 signicant digits! We have omitted an innity of unstable cycles; so why does approximating the dynamics by a nite number of the shortest cycle eigenvalues work so well? The convergence of cycle expansions of dynamical zeta functions is a consequence of the smoothness and analyticity of the underlying ow. Intuitively, one can understand the convergence in terms of the geometrical picture sketched in Fig. 1.11; the key observation is that the long orbits are shadowed by sequences of shorter orbits. A typical term in (1.10) is a dierence of a long cycle {ab} minus its shadowing approximation by shorter cycles {a} and {b}
Section ??
tab ta tb = tab (1 ta tb /tab ) = tab 1
ab a b
,
(1.11)
where a and b are symbol sequences of the two shorter cycles. If all orbits are weighted equally (tp = z np ), such combinations cancel exactly; if orbits of similar symbolic dynamics have similar weights, the weights in such combinations almost cancel. This can be understood in the context of the pinball game as follows. Consider orbits 0, 1 and 01. The rst corresponds to bouncing between any two disks while the second corresponds to bouncing successively around all three, tracing out an equilateral triangle. The cycle 01 starts at one disk, say disk 2. It then bounces from disk 3 back to disk 2 then bounces from disk 1 back to disk 2 and so on, so its itinerary is 2321. In terms of the bounce types shown in Fig. 1.5, the trajectory is alternating between 0 and 1. The incoming and outgoing angles when it executes these bounces are very close to the corresponding angles for 0 and 1 cycles. Also the distances traversed between bounces are similar so that the 2-cycle expanding eigenvalue 01 is close in magnitude to the product of the 1-cycle eigenvalues 0 1 . To understand this on a more general level, try to visualize the partition of a chaotic dynamical systems state space in terms of cycle neighborhoods as a tessellation (a tiling) of the dynamical system, with smooth ow approximated by its periodic orbit skeleton, each tile centered on a periodic point, and the scale of the tile determined by the linearization of the ow around the periodic point, Fig. 1.11. The orbits that follow the same symbolic dynamics, such as {ab} and a pseudo orbit {a}{b}, lie close to each other in state space; long shadowing pairs have to start out exponentially close to beat the exponential growth in separation with time. If the weights associated with the orbits are multiplicative along the ow (for example, by the chain rule for products of derivatives) and the ow is smooth, the term in parenthesis in (1.11) falls o exponentially with the cycle length, and therefore the curvature expansions are expected to be highly convergent.
Fig. 1.11 Approximation to (a) a smooth dynamics by (b) the skeleton of periodic points, together with their linearized neighborhoods. Indicated are segments of two 1-cycles and a 2-cycle that alternates between the neighborhoods of the two 1-cycles, shadowing rst one of the two 1-cycles, and then the other.
Chapter ??
16 Overture
1.6
Evolution
The above derivation of the dynamical zeta function formula for the escape rate has one shortcoming; it estimates the fraction of survivors as a function of the number of pinball bounces, but the physically interesting quantity is the escape rate measured in units of continuous time. For continuous time ows, the escape rate (1.2) is generalized as follows. Dene a nite state space region M such that a trajectory that exits M never reenters. For example, any pinball that falls of the edge of a pinball table in Fig. 1.1 is gone forever. Start with a uniform distribution of initial points. The fraction of initial x whose trajectories remain within M at time t is expected to decay exponentially (t) =
M
dxdy (y f t (x)) et . dx M
The integral over x starts a trajectory at every x M. The integral over y tests whether this trajectory is still in M at time t. The kernel of this integral Lt (y, x) = y f t (x) (1.12)
is the Dirac delta function, as for a deterministic ow the initial point x maps into a unique point y at time t. For discrete time, f n (x) is the nth iterate of the map f . For continuous ows, f t (x) is the trajectory of the initial point x, and it is appropriate to express the nite time kernel Lt in terms of a generator of innitesimal time translations Lt = etA ,
Section ?? Chapter ??
very much in the way the quantum evolution is generated by the Hamiltonian H, the generator of innitesimal time quantum transformations. As the kernel L is the key to everything that follows, we shall give it a name, and refer to it and its generalizations as the evolution operator for a d-dimensional map or a d-dimensional ow. The number of periodic points increases exponentially with the cycle length (in the case at hand, as 2n ). As we have already seen, this exponential proliferation of cycles is not as dangerous as it might seem; as a matter of fact, all our computations will be carried out in the n limit. Though a quick look at long-time density of trajectories might reveal it to be complex beyond belief, this distribution is still generated by a simple deterministic law, and with some luck and insight, our labeling of possible motions will reect this simplicity. If the rule that gets us from one level of the classication hierarchy to the next does not depend strongly on the level, the resulting hierarchy is approximately self-similar. We now turn such approximate self-similarity to our advantage, by turning it into an operation, the action of the evolution operator, whose iteration encodes the self-similarity.
1.6
Evolution 17
1.6.1
Trace formula
In physics, when we do not understand something, we give it a name. Matthias Neubert
Recasting dynamics in terms of evolution operators changes everything. So far our formulation has been heuristic, but in the evolution operator formalism the escape rate and any other dynamical average are given by exact formulas, extracted from the spectra of evolution operators. The key tools are trace formulas and spectral determinants. The trace of an operator is given by the sum of its eigenvalues. The explicit expression (1.12) for Lt (x, y) enables us to evaluate the trace. Identify y with x and integrate x over the whole state space. The result is an expression for tr Lt as a sum over neighborhoods of prime cycles p and their repetitions
tr Lt =
p
Tp
r=1
(t rTp ) r det 1 Mp
.
(1.13)
Fig. 1.12 The trace of an evolution operator is concentrated in tubes around prime cycles, of length Tp and thickness 1/|p |r for the rth repetition of the prime cycle p. Section ??
This formula has a simple geometrical interpretation sketched in Fig. 1.12. After the rth return to a Poincar section, the initial tube Mp has been e stretched out along the expanding eigendirections, with the overlap with r the initial volume given by 1/ det 1 Mp 1/|p |, the same weight we obtained heuristically in Section 1.5.1. The spiky sum (1.13) is disquieting in the way reminiscent of the Poisson resummation formulas of Fourier analysis; the left-hand side is es t , while the right-hand side the smooth eigenvalue sum tr eAt = equals zero everywhere except for the set t = rTp . A Laplace transform smooths the sum over Dirac delta functions in cycle periods and yields the trace formula for the eigenspectrum s0 , s1 , of the classical evolution operator:
0+
dt est tr Lt 1 s s =0
= =
tr
1 = sA
Tp
p r=1
er(ApsTp ) . r det 1 Mp
(1.14)
The beauty of trace formulas lies in the fact that everything on the righthand-sideprime cycles p, their periods Tp and the stability eigenvalues of Mp is an invariant property of the ow, independent of any coordinate choice.
Section ??
1.6.2
Spectral determinant
The eigenvalues of a linear operator are given by the zeros of the appropriate determinant. One way to evaluate determinants is to expand them in terms of traces, using the identities
4.1, page 70
18 Overture
d d 1 ln det (s A) = tr ln(s A) = tr , ds ds sA
(1.15)
and integrating over s. In this way the spectral determinant of an evolution operator becomes related to the traces that we have just computed:
Chapter ??
det (s A) = exp
p r=1
trln(1zL)
1 esTp r r r det 1 Mp
.
(1.16)
det(1zL)
z
z
= exp
Fig. 1.13 Spectral determinant is preferable to the trace as it vanishes smoothly at the leading eigenvalue, while the trace formula diverges. Section ??
The 1/r factor is due to the s integration, leading to the replacement Tp Tp /rTp in the periodic orbit expansion (1.14). The motivation for recasting the eigenvalue problem in this form is sketched in Fig. 1.13; exponentiation improves analyticity and trades in a divergence of the trace sum for a zero of the spectral determinant. We have now retraced the heuristic derivation of the divergent sum (1.6) and the dynamical zeta function (1.9), but this time with no approximations: formula (1.16) is exact. The computation of the zeros of det (s A) proceeds very much like the computations of Section 1.5.3.
1.7
From chaos to statistical mechanics
Chapter ??
The replacement of dynamics of individual trajectories by evolution operators which propagate densities feels like a bit of mathematical voodoo. Actually, something very radical has taken place. Consider a chaotic ow, such as the stirring of red and white paint by some deterministic machine. If we were able to track individual trajectories, the uid would forever remain a striated combination of pure white and pure red; there would be no pink. What is more, if we reversed the stirring, we would return to the perfect white/red separation. However, that cannot bein a very few turns of the stirring stick the thickness of the layers goes from centimeters to ngstrms, and the result is irreversibly A o pink. Understanding the distinction between evolution of individual trajectories and the evolution of the densities of trajectories is key to understanding statistical mechanicsthis is the conceptual basis of the second law of thermodynamics, and the origin of irreversibility of the arrow of time for deterministic systems with time-reversible equations of motion: reversibility is attainable for distributions whose measure in the space of density functions goes exponentially to zero with time. By going to a description in terms of the asymptotic time evolution operators we give up tracking individual trajectories for long times, by trading it in for a very eective description of the asymptotic trajectory densities. This will enable us, for example, to give exact formulas for transport coecients such as the diusion constants without any probabilistic assumptions (in contrast to the stosszahlansatz of Boltzmann). A century ago it seemed reasonable to assume that statistical mechanics applies only to systems with very many degrees of freedom. More
1.8
What is not in ChaosBook 19
recent is the realization that much of statistical mechanics follows from chaotic dynamics, and already at the level of a few degrees of freedom the evolution of densities is irreversible. Furthermore, the theory that we shall develop here generalizes notions of measure and averaging to systems far from equilibrium, and transports us into regions hitherto inaccessible with the tools of equilibrium statistical mechanics. The concepts of equilibrium statistical mechanics do help us, however, to understand the ways in which the simple-minded periodic orbit theory falters. A non-hyperbolicity of the dynamics manifests itself in powerlaw correlations and even phase transitions.
Chapter ??
1.8
What is not in ChaosBook
This book oers a breach into a domain hitherto reputed unreachable, a domain traditionally traversed only by mathematical physicists and pure mathematicians. What distinguishes it from pure mathematics is the insistence on computability and numerical convergence of methods oered. A rigorous proof, the end of the story as far as a mathematician is concerned, might state that in a given setting, for times in excess of 1032 years, turbulent dynamics settles onto an attractor of dimension less than 600. Such a theorem is of a little use for a working scientist, especially if a numerical experiment indicates that within the span of the best simulation the dynamics seems to have settled on a (transient?) attractor of dimension less than 3.
1.9
A guide to exercises
God can aord to make mistakes. So can Dada! Dadaist Manifesto
The essence of this subject is incommunicable in print; the only way to develop intuition about chaotic dynamics is by computing, and the reader is urged to try to work through the essential exercises. As not to fragment the text, the exercises are indicated by text margin boxes such as the one on this margin, and collected at the end of each chapter. The problems that you should do have underlined titles. The rest (smaller type) are optional. Dicult problems are marked by any number of *** stars. If you solve one of those, it is probably worth a publication.2 By the end of a (two-semester) course you should have completed at least three small projects: (a) compute everything for a one-dimensional repeller, (b) compute escape rate for a 3-disk game of pinball, (c) compute a part of the quantum 3-disk game of pinball, or the helium spectrum, or if you are interested in statistical rather than the quantum mechanics, compute a transport coecient. The essential steps are: Dynamics (1) count prime cycles, Exercise 1.1
??, page ??
2
To keep you on your toes, some of the problems are nonsensical, and some of the solutions given are plainly wrong
20 Overture
(2) pinball simulator, Exercise 6.1, Exercise ?? (3) pinball stability, Exercise 8.1, Exercise ?? (4) pinball periodic orbits, Exercise ??, Exercise ?? (5) helium integrator, Exercise 2.10, Exercise ?? (6) helium periodic orbits, Exercise ??, Exercise ?? Averaging, numerical (1) pinball escape rate, Exercise ?? (2) Lyapunov exponent, Exercise ?? Averaging, periodic orbits (1) cycle expansions, Exercise ??, Exercise ?? (2) pinball escape rate, Exercise ??, Exercise ?? (3) cycle expansions for averages, Exercise ??, Exercise ?? (4) cycle expansions for diusion, Exercise ?? (5) desymmetrization Exercise ?? (6) semiclassical quantization Exercise ?? (7) ortho-, para-helium, lowest eigen-energies Exercise ?? Solutions for some of the problems are given in Appendix ??. A clean solution, a pretty gure, or a nice exercise that you contribute to ChaosBook will be gratefully acknowledged. Often going through a solution is more instructive than reading the chapter that problem is supposed to illustrate.
Summary
This text is an exposition of the best of all possible theories of deterministic chaos, and the strategy is: 1) count, 2) weigh, 3) add up. In a chaotic system any open ball of initial conditions, no matter how small, will spread over the entire accessible state space. Hence the theory focuses on describing the geometry of the space of possible outcomes, and evaluating averages over this space, rather than attempting the impossible: precise prediction of individual trajectories. The dynamics of densities of trajectories is described in terms of evolution operators. In the evolution operator formalism the dynamical averages are given by exact formulas, extracted from the spectra of evolution operators. The key tools are trace formulas and spectral determinants. The theory of evaluation of the spectra of evolution operators presented here is based on the observation that the motion in dynamical systems of few degrees of freedom is often organized around a few fundamental cycles. These short cycles capture the skeletal topology of the motion on a strange attractor/repeller in the sense that any long orbit can approximately be pieced together from the nearby periodic orbits of nite length. This notion is made precise by approximating orbits
1.9
A guide to exercises 21
by prime cycles, and evaluating the associated curvatures. A curvature measures the deviation of a longer cycle from its approximation by shorter cycles; smoothness and the local instability of the ow implies exponential (or faster) fall-o for (almost) all curvatures. Cycle expansions oer an ecient method for evaluating classical and quantum observables. The critical step in the derivation of the dynamical zeta function was the hyperbolicity assumption, i.e., the assumption of exponential shrinkage of all strips of the pinball repeller. By dropping the ai prefactors in (1.4), we have given up on any possibility of recovering the precise distribution of starting x (which should anyhow be impossible due to the exponential growth of errors), but in exchange we gain an eective description of the asymptotic behavior of the system. The pleasant surprise of cycle expansions (1.9) is that the innite time behavior of an unstable system is as easy to determine as the short time behavior. To keep the exposition simple we have here illustrated the utility of cycles and their curvatures by a pinball game, but topics covered in ChaosBook unstable ows, Poincar sections, Smale horseshoes, syme bolic dynamics, pruning, discrete symmetries, periodic orbits, averaging over chaotic sets, evolution operators, dynamical zeta functions, spectral determinants, cycle expansions, quantum trace formulas, zeta functions, and so on to the semiclassical quantization of helium should give the reader some condence in the broad sway of the theory. The formalism should work for any average over any chaotic set which satises two conditions: 1. the weight associated with the observable under consideration is multiplicative along the trajectory, 2. the set is organized in such a way that the nearby points in the symbolic dynamics have nearby weights. The theory is applicable to evaluation of a broad class of quantities characterizing chaotic systems, such as the escape rates, Lyapunov exponents, transport coecients and quantum eigenvalues. A big surprise is that the semi-classical quantum mechanics of systems classically chaotic is very much like the classical mechanics of chaotic systems; both are described by zeta functions and cycle expansions of the same form, with the same dependence on the topology of the classical ow.
22 Further reading But the power of instruction is seldom of much ecacy, except in those happy dispositions where it is almost superuous. Gibbon
Further reading
Nonlinear dynamics texts: This text aims to bridge the gap between the physics and mathematics dynamical systems literature. The intended audience is Henri Roux, the perfect physics graduate student with a theoretical bent who does not believe anything he is told. As a complementary presentation we recommend Gaspards monograph [8] which covers much of the same ground in a highly readable and scholarly manner. As far as the prerequisites are concernedChaosBook is not an introduction to nonlinear dynamics. Nonlinear science requires a one semester basic course (advanced undergraduate or rst year graduate). A good start is the textbook by Strogatz [9], an introduction to the applied mathematicians visualization of ows, xed points, manifolds, bifurcations. It is the most accessible introduction to nonlinear dynamicsa book on dierential equations in nonlinear disguise, and its broadly chosen examples and many exercises make it a favorite with students. It is not strong on chaos. There the textbook of Alligood, Sauer and Yorke [10] is preferable: an elegant introduction to maps, chaos, period doubling, symbolic dynamics, fractals, dimensionsa good companion to ChaosBook. Introductions more comfortable to physicists are the textbooks by Ott [12], with the bakers map used to illustrate many key techniques in analysis of chaotic systems, and by Tl and M. Gruiz [11], where chaotic dynamics is ine troduced through classical mechanics. They are perhaps harder than the above two as rst books on nonlinear dynamics. Sprott [13] and Jackson [14] textbooks are very useful compendia of the 70s and onward chaos literature which we, in the spirit of promises made in Section 1.1, tend to pass over in silence. An introductory course should give students skills in qualitative and numerical analysis of dynamical systems for short times (trajectories, xed points, bifurcations) and familiarize them with Cantor sets and symbolic dynamics for chaotic systems. A good introduction to numerical experimentation with physically realistic systems is Tullaro, Abbott, and Reilly [15]. Korsch and Jodl [16] and Nusse and Yorke [17] also emphasize handson approach to dynamics. With this, and a graduate level-exposure to statistical mechanics, partial dierential equations and quantum mechanics, the stage is set for any of the one-semester advanced courses based on ChaosBook. The courses taught so far start out with the introductory chapters on qualitative dynamics, symbolic dynamics and ows, and then continue in dierent directions: Deterministic chaos: Chaotic averaging, evolution operators, trace formulas, zeta functions, cycle expansions, Lyapunov exponents, billiards, transport coecients, thermodynamic formalism, period doubling, renormalization operators. A graduate level introduction to statistical mechanics from the dynamical point view is given by Dorfman [32]; the Gaspard monograph [8] covers the same ground in more depth. Driebe monograph [33] oers a nice introduction to the problem of irreversibility in dynamics. The role of chaos in statistical mechanics is critically dissected by Bricmont in his highly readable essay Science of Chaos or Chaos in Science? [34]. Spatiotemporal dynamical systems: Partial dierential equations for dissipative systems, weak amplitude expansions, normal forms, symmetries and bifurcations, pseudospectral methods, spatiotemporal chaos, turbulence. Quantum chaos: Semiclassical propagators, density of states, trace formulas, semiclassical spectral determinants, billiards, semiclassical helium, diraction, creeping, tunneling, higher-order corrections. For more on this topic, hop to the quantum chaos introduction, Chapter ??. Periodic orbit theory: This book puts more emphasis on periodic orbit theory than any other current nonlinear dynamics textbook. The role of unstable periodic orbits was already fully appreciated by Poincar [18, 19], who noted that hidden in the apparent e chaos is a rigid skeleton, a tree of cycles (periodic orbits) of increasing lengths and self-similar structure, and suggested that the cycles should be the key to chaotic dynamics. Periodic orbits have been at core of much of the mathematical work on the theory of the classical
Exercises 23 and quantum dynamical systems ever since. We refer the reader to the reprint selection [20] for an overview of some of that literature. If you seek rigor: If you nd ChaosBook not rigorous enough, you should turn to the mathematics literature. The most extensive reference is the treatise by Katok and Hasselblatt [21], an impressive compendium of modern dynamical systems theory. The fundamental papers in this eld, all still valuable reading, are Smale [22], Bowen [23] and Sinai [25]. Sinais paper is prescient and oers a vision and a program that ties together dynamical systems and statistical mechanics. It is written for readers versed in statistical mechanics. For a dynamical systems exposition, consult Anosov and Sinai [24]. Markov partitions were introduced by Sinai in Ref. [26]. The classical text (though certainly not an easy read) on the subject of dynamical zeta functions is Ruelles Statistical Mechanics, Thermodynamic Formalism [27]. In Ruelles monograph transfer operator technique (or the PerronFrobenius theory) and Smales theory of hyperbolic ows are applied to zeta functions and correlation functions. The status of the theory from Ruelles point of view is compactly summarized in his 1995 Pisa lectures [28]. Further excellent mathematical references on thermodynamic formalism are Parry and Pollicotts monograph [29] with emphasis on the symbolic dynamics aspects of the formalism, and Baladis clear and compact reviews of the theory of dynamical zeta functions [30, 31]. If you seek magic: ChaosBook resolutely skirts number-theoretical magic such as spaces of constant negative curvature, Poincar tilings, modular doe mains, Selberg Zeta functions, Riemann hypothesis, . . . Why? While this beautiful mathematics has been very inspirational, especially in studies of quantum chaos, almost no powerful method in its repertoire survives a transplant to a physical system that you are likely to care about. Sorry, no shmactals: ChaosBook skirts mathematics and empirical practice of fractal analysis, such as Hausdor and fractal dimensions. Addisons introduction to fractal dimensions [36] oers a well-motivated entry into this eld. While in studies of probabilistically assembled fractals such as Diusion Limited Aggregates (DLA) better measures of complexity are lacking, for deterministic systems there are much better, physically motivated and experimentally measurable quantities (escape rates, diusion coecients, spectrum of helium, ...) that we focus on here. Rat brains: If you were wandering while reading this introduction whats up with rat brains?, the answer is yes indeed, there is a line of research in neuronal dynamics that focuses on possible unstable periodic states, described for example in Ref. [3740].
Exercises
(1.1) 3-disk symbolic dynamics. As periodic trajectories will turn out to be our main tool to breach deep into the realm of chaos, it pays to start familiarizing oneself with them now by sketching and counting the few shortest prime cycles (we return to this in Section ??). Show that the 3-disk pinball has 32n itineraries of length n. List periodic orbits of lengths 2, 3, 4, 5, . Verify that the shortest 3-disk prime cycles are 12, 13, 23, 123, 132, 1213, 1232, 1323, 12123, . Try to sketch them. (1.2) Sensitivity to initial conditions. Assume that two pinball trajectories start out parallel, but separated by 1 ngstrm, and the disks are of A o radius a = 1 cm and center-to-center separation R = 6 cm. Try to estimate in how many bounces the separation will grow to the size of system (assuming that the trajectories have been picked so they remain trapped for at least that long). Estimate the Whos Pinball Wizards typical score (number of bounces) in a game without cheating, by hook or crook (by the end of Chapter ?? you should be in position to make very accurate estimates).
References
3 [1] G.W. Leibniz, Von dem Verhngnisse. 3 a We tend to list all source literature [2] D. Avnir, O. Biham, D. Lidar and O. Malcai, Is the Geometry of we found a useful reading for a given chapter. Not all of them are necessarily Nature Fractal?, Science 279, 39 (1998). cited in the Further reading section. [3] R. Kennedy, The Case of Pollocks Fractals Focuses on Physics, New York Times (Dec. 2, 2006). [4] R. P. Taylor, A. P. Micolich and D. Jonas, Fractal analysis of Pollocks drip paintings, Nature 399, 422 (1999). [5] K. Jones-Smith and H. Mathur, Fractal Analysis: Revisiting Pollocks drip paintings, Nature 444, E9 (2006); R. P. Taylor, A. P. Micolich and D. Jonas, Fractal Analysis: Revisiting Pollocks drip paintings (Reply), Nature 444, E10 (2006). [6] P. Cvitanovi, B. Eckhardt, P.E. Rosenqvist, G. Russberg and P. c Scherer, Pinball Scattering, in G. Casati and B. Chirikov, eds., Quantum Chaos (Cambridge U. Press, Cambridge 1993). [7] K.T. Hansen, Symbolic Dynamics in Chaotic Systems, Ph.D. thesis (Univ. of Oslo, 1994); http://ChaosBook.org/projects/KTHansen/thesis [8] P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cambridge U. Press, Cambridge 1998). [9] S.H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wesley 1994). [10] K.T. Alligood, T.D. Sauer and J.A. Yorke, Chaos, an Introduction to Dynamical Systems (Springer, New York 1996) [11] T. Tl and M. Gruiz, Chaotic Dynamics: An Introduction Based e on Classical Mechanics (Cambridge U. Press, Cambridge 2006). [12] E. Ott, Chaos in Dynamical Systems (Cambridge U. Press, Cambridge 1993). [13] J. C. Sprott, Chaos and Time-Series Analysis (Oxford University Press, Oxford, 2003) [14] E. Atlee Jackson, Perspectives of nonlinear dynamics (Cambridge U. Press, Cambridge, 1989). [15] N.B. Tullaro, T.A. Abbott, and J.P. Reilly, Experimental Approach to Nonlinear Dynamics and Chaos (Addison Wesley, Reading MA, 1992). [16] H.J. Korsch and H.-J. Jodl, Chaos. A Program Collection for the PC, (Springer, New York 1994). [17] H.E. Nusse and J.A. Yorke, Dynamics: Numerical Explorations (Springer, New York 1997). [18] H. Poincar, Les mthodes nouvelles de la mchanique cleste (Guthiere e e e
26 References
Villars, Paris 1892-99) [19] For a very readable exposition of Poincars work and the develope ment of the dynamical systems theory see J. Barrow-Green, Poincar e and the Three Body Problem, (Amer. Math. Soc., Providence R.I., 1997), and F. Diacu and P. Holmes, Celestial Encounters, The Origins of Chaos and Stability (Princeton Univ. Press, Princeton NJ 1996). [20] R.S. MacKay and J.D. Miess, Hamiltonian Dynamical Systems (Adam Hilger, Bristol 1987). [21] A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge U. Press, Cambridge 1995). [22] S. Smale, Dierentiable Dynamical Systems, Bull. Am. Math. Soc. 73, 747 (1967). [23] R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Springer Lecture Notes in Math. 470 (1975). [24] D.V. Anosov and Ya.G. Sinai, Some smooth ergodic systems, Russ. Math. Surveys 22, 103 (1967). [25] Ya.G. Sinai, Gibbs measures in ergodic theory, Russ. Math. Surveys 166, 21 (1972). [26] Ya.G. Sinai, Construction of Markov partitions, Funkts. Analiz i Ego Pril. 2, 70 (1968). English translation: Functional Anal. Appl. 2, 245 (1968). [27] D. Ruelle, Statistical Mechanics, Thermodynamic Formalism, (AddisonWesley, Reading MA, 1978). [28] D. Ruelle, Functional determinants related to dynamical systems and the thermodynamic formalism, (Lezioni Fermiane, Pisa), preprint IHES/P/95/30 (March 1995). [29] W. Parry and M. Pollicott, Zeta Functions and the Periodic Structure of Hyperbolic Dynamics, Astrisque 187188 (Socit Mathmatique e ee e de France, Paris 1990). [30] V. Baladi, Dynamical zeta functions, in B. Branner and P. Hjorth, eds., Real and Complex Dynamical Systems (Kluwer, Dordrecht, 1995). [31] V. Baladi, Positive Transfer Operators and Decay of Correlations (World Scientic, Singapore 2000). [32] R. Dorfman, From Molecular Chaos to Dynamical Chaos (Cambridge U. Press, Cambridge 1998). [33] D.J. Driebe, Fully Chaotic Map and Broken Time Symmetry (Kluwer, Dordrecht, 1999). [34] J. Bricmont, Science of Chaos or Chaos in Science?, available on www.ma.utexas.edu/mp arc/c/96/96-116.ps.gz [35] V.I. Arnold, Mathematical Methods in Classical Mechanics (SpringerVerlag, Berlin, 1978). [36] P. S. Addison Fractals and chaos: an illustrated course, (Inst. of Physics Publishing, Bristol 1997). [37] S.J. Schi, et al. Controlling chaos in the brain, Nature 370, 615 (1994). [38] F. Moss, Chaos under control, Nature 370, 615 (1994).
27
[39] J. Glanz, Science 265, 1174 (1994). [40] J. Glanz, Mastering the Nonlinear Brain, Science 227, 1758 (1997). [41] Poul Martin Mller, En dansk Students Eventyr [The Adventures of a Danish Student] (Copenhagen 1824).