Nonlinear Science

shown in Figs. la-d. The most immedi- ately striking feature of this set of figures HAMILTONIAN is the existence of nontrivial structure on all scales. Thus, like dissipative systems, Hamiltonian chaos generates strange frac- tal sets (albeit “fat” fractals, as discussed CHAOS and below). On all scales one observes “is- lands,” analogues in this discrete case of the periodic orbits in the phase plane of the simple pendulum (Fig. 2 in the main STATISTICAL text). In addition, however, and again on all scales, there are swarms of dots com- ing from individual chaotic orbits that un- dergo nonperiodic motion and eventually MECHANICS fill a finite region in phase space. In these chaotic regions the motion is “sensitively dependent on initial conditions.” The specific examples of chaotic sys- Figure 2 shows, in the full phase space, tems discussed in the main text—the lo- a plot of a single chaotic orbit followed gistic map, the damped, driven pendu- is the analogue of the coordinate, and through 100 million iterations (again, for lum, and the Lorenz equations—are all the discrete index n plays the role of k = 1.1). This object differs from the dissipative. It is important to recognize strange sets seen in dissipative systems in that nondissipative Hamiltonian systems that it occupies a finite fraction of the full can also exhibit chaos; indeed, Poincare torus, periodic in both p and q. For any phase space: specifically, the orbit shown made his prescient statement concerning value of k, the map preserves the area takes up 56 per cent of the unit area that sensitive dependence on initial conditions in the (p, q) plane, since the Jacobian represents the full phase space of the map. in the context of the few-body Hamil- Hence the “dimension” of the orbit is the tonian problems he was studying. Here The preservation of phase-space vol- same as that of the full phase space, and we examine briefly the many subtleties ume for Hamiltonian systems has the very calculating the fractal dimension by the

of Hamiltonian chaos and, as an illustra- important consequence that there can be standard method gives d f = 2. How- tion of its importance, discuss how it is no attractors, that is, no subregions of ever, the orbit differs from a conventional closely tied to long-standing problems in lower phase-space dimension to which area in that it contains holes on all scales. the foundations of statistical mechanics. the motion is confined asymptotically. As a consequence, the measured value of We choose to introduce Hamiltonian Any initial point (pO, qo) will lie on some the area occupied by the orbit depends chaos in one of its simplest incarnations, particular orbit, and the image of all on the resolution with which this area is a two-dimensional discrete model called possible initial points—that is, the unit measured—for example, the size of the the standard map. Since this map pre- square itself—is again the unit square. In boxes in the box-counting method—and serves phase-space volume (actually area contrast, dissipative systems have phase- the approach to the finite value at in- because there are only two dimensions) space volumes that shrink. For example, finitely fine resolution has definite scaling it indeed corresponds to a discrete ver- the logistic map (Fig. 5 in the main text) properties. This set is thus appropriately sion of a Hamiltonian system. Like the called a “fat fractal,” For our later dis- discrete logistic map for dissipative sys- terval (O, 1) attracted to just two points. cussion it is important to note that the tems, this map represents an archetype for Clearly, for k = O the standard map holes—representing periodic, nonchaotic Hamiltonian chaos. motion—also occupy a finite fraction of The equations defining the standard the phase space. map are time (n) as it should for free motion. The To develop a more intuitive feel for fat orbits are thus just straight lines wrap- fractals, note that a very simple exam- ping around the torus in the q direction. ple can be constructed by using a slight For k = 1.1 the map produces the orbits modification of the Cantor-set technique

242 Los Alamos Science Special Issue 1987

Nonlinear Science

1.0 (a)

P 0.5

.74

THE STANDARD MAP

Fig. 1. Shown here are the discrete orbits of the standard map (for k = 1.1 in Eq. 1) with different colors used to distinguish one orbit from another. increasingly magnified regions of the phase space are shown, starting with the 62 full phase space (a). The white box in (a) is the 0.31 0.43 0.55 region magnified in (b), and so forth. Nontrivial structure, including “’islands” and swarms of dots that represent regions of chaotic, nonpe- 0.744 riodic motion, are obvious on all scales. (Fig- ure courtesy of James Kadtke and David Um- berger, Los Alamos National Laboratory.)

P 0.719 0.4501 0.4532 0.7320

0.694 0.410 0.435 0,460

q 0.7289

243 Nonlinear Science

described in the main text. Instead of and chaotic regions do not fill all of phase deleting the middle one-third of each in- d2x space: a finite fraction is occupied by “in- terval at every scale, one deletes the mid- variant KAM tori.” At a conceptual level, then, the KAM sulting set is topologically the same as theorem explains the nonchaotic behav- the original Cantor set, a calculation of ior and recurrences that so puzzled Fermi,

its dimension yields df = 1; it has the Equation 2 describes the famous Henon- Pasta, and Ulam (see “The Fermi, Pasta, same dimension as the full unit interval. Heiles system, which is non-integrable and Ulam Problem: Excerpts from ‘Stud- Further, this fat Cantor set occupies a fi- and has become a classic example of a ies of Nonlinear Problems’ “). Although nite fraction-amusingly but accidentally simple (astro-) physically relevant Hamil- the FPU chain had many (64) nonlinearly also about 56 per cent-of the unit inter- tonian system exhibiting chaos. On the coupled degrees of freedom, it was close val, with the remainder occupied by the other hand, Eq. 3 can be separated into enough (for the parameter ranges studied) “holes” in the set. two independent N = 1 systems (by a to an integrable system that the invariant To what extent does chaos exist in the KAM tori and resulting pseudo-integrable more conventional Hamiltonian systems properties dominated the behavior over described by differential equations? A tegrable. the times of measurement. full answer to this question would require Although there exist explicit calcula- There is yet another level of subtlety a highly technical summary of more than tional methods for testing for integrabil- to chaos in Hamiltonian systems: namely, eight decades of investigations by math- ity, these are highly technical and gener- the structure of the phase space. For non- ematical physicists. Thus we will have ally difficult to apply for large N. For- integrable systems, within every regular to be content with a superficial overview tunately, two theorems provide general KAM region there are chaotic regions. that captures, at best, the flavor of these guidance. First, Siegel’s Theorem con- Within these chaotic regions there are, in investigations. siders the space of Hamiltonians analytic turn, regular regions, and so forth. For To begin, we note that completely in- in their variables: non-integrable Hamil- all non-integrable systems with N > 3, tegrable systems can never exhibit chaos, tonians are dense in this space, whereas an orbit can move (albeit on very long independent of the number of degrees of integrable Hamiltonians are not. Sec- time scales) among the various chaotic freedom N. In these systems all bounded ond, Nekhoroshev’s Theorem leads to the regions via a process known as “Arnold motions are quasiperiodic and occur on fact that all non-integrable systems have a diffusion.” Thus, in general, phase space hypertori, with the N frequencies (pos- phase space that contains chaotic regions. is permeated by an Arnold web that links sibly all distinct) determined by the val- Out observations concerning the stan- together the chaotic regions on all scales. ues of the conservation laws. Thus there dard map immediately suggest an essen- Intuitively, these observations concern- cannot be any aperiodic motion. Fur- tial question: What is the extent of the ing Hamiltonian chaos hint strongly at a ther, since all Hamiltonian systems with chaotic regions and can they, under some connection to statistical mechanics. As N = 1 are completely integrable, chaos circumstances, cover the whole phase Fig. 1 illustrates, the chaotic orbits in cannot occur for one-degree-of-freedom space? The best way to answer this ques- Hamiltonian systems form very compli- problems. tion is to search for nonchaotic regions. cated “Cantor dusts,” which are nonperi- For N =2, non-integrable systems can Consider, for example, a completely inte- odic, never-repeating motions that wan- exhibit chaos; however, it is not trivial grable N-degree-of-freedom Hamiltonian der through volumes of the phase space, to determine in which systems chaos can system disturbed by a generic non-inte- apparently constrained only by conser- occur; that: is, it is in general not obvi- grable perturbation. The famous KAM vation of total energy. In addition, in ous whether a given system is integrable (for Kolmogorov, Arnold, and Moser) these regions the sensitive dependence or not. Consider, for example, two very theorem shows that, for this case, there implies a rapid loss of information about similar N = 2 nonlinear Hamiltonian sys- are regions of finite measure in phase the initial conditions and hence an effec- tems with equation of motion given by: space that retain the smoothness associ- tive irreversibility of the motion. Clearly, ated with motion on the hypertori of the such wandering motion and effective ir- integrable system. These regions are the reversibility suggest a possible approach analogues of the “holes” in the standard to the following fundamental question of map. Hence, for a typical Hamiltonian statistical mechanics: How can one de- system with N degrees of freedom, the rive the irreversible, ergodic, thermal-

244 Los Alamos Science Special Issue 1987 Nonlinear Science

equilibrium motion assumed in statistical 1.0 A “FAT” FRACTAL mechanics from a reversible, Hamiltonian microscopic dynamics? Fig. 2. A singles chaotic orbit of the standard Historically, the fundamental assump- map for k = 1.1. The picture was made by di- viding the energy surface Into a 512 by 512 grid tion that has linked dynamics and statis- and iterating the initial condition 108 times. tical mechanics is the ergodic hypothesis, The squares visited by this orbit are shown which asserts that time averages over ac- P in black. Gaps in the phase space represent tual dynamical motions are equal to en- portions of the energy surface unavailable to semble averages over many different but the chaotic orbit because of various quasiperi- equivalent systems. Loosely speaking, odic orbits confined to tori, as seen In Fig. 1. this hypothesis assumes that all regions (Figure courtesy of J. Doyne Farmer and David of phase space allowed by energy con- Umberger, Los Alamos National Laboratory.) servation are equally accessed by almost 0.0 all dynamical motions. 0.0 1.0 What evidence do we have that the er- godic hypothesis actually holds for re- this topic, see “The Ergodic Hypothesis: alistic Hamiltonian systems? For sys- A Complicated Problem of Mathematics tems with finite degrees of freedom, the and Physics.”) KAM theorem shows that, in addition Among the specific issues that should to chaotic regions of phase space, there be addressed in a variety of physically are nonchaotic regions of finite measure. realistic models are the following. These invariant tori imply that ergodicity does not hold for most finite-dimensional ● How does the measure of phase space Hamiltonian sytems. Importantly, the occupied by KAM tori depend on N ? few Hamiltonian systems for which the Is there a class of models with realistic KAM theorem does not apply, and for interactions for which this measure goes which one can prove ergodicity and the to O? Are there non-integrable models approach to thermal equilibrium, involve for which a finite measure is retained by “hard spheres” and consequently contain the KAM regions? If so, what are the non-analytic interactions that are not re- characteristics that cause this behavior? alistic from a physicist’s perspective. ● How does the rate of Arnold diffusion For many years, most researchers be- depend on N in a broad class of mod- lieved that these subtleties become irrele- els? What is the structure of important vant in the thermodynamic limit, that is, features—such as the Arnold web-in the the limit in which the number of degrees phase space as N approaches infinity? of freedom (N) and the energy (E) go to ● If there is an approach to equilibrium, infinity in such a way that E/N remains how does the time-scale for this approach a nonzero constant. For instance, the depend on N? Is it less than the age of the universe? KAM regions of invariant tori may ap- proach zero measure in this limit. How- ● Is ergodicity necessary (or merely suf- ever, recent evidence suggests that non- ficient) for most of the features we as- trivial counterexamples to this belief may sociate with statistical mechanics? Can a exist. Given the increasing sophistication less stringent requirement, consistent with of our analytic understanding of Hamilto- the behaviour observed in analytic Hamil- nian chaos and the growing ability to sim- tonian systems, be formulated? ulate systems with large N numerically, Clearly, these are some of the most chal- the time seems ripe for quantitative inves- lenging, and profound, questions current- tigations that can establish (or disprove!) ly confronting nonlinear scientists. ■ this belief. (For additional discussion of

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