Dimensions and Entropies in Chaotic Systems

The theory of nonlinear dynamics and chaotic attractors has been remarkably successful in providing a new paradigm for the understanding of complex and irregular structures in an enormous range of quite different systems [1,2,3,4,5]. The universality of its underlying principles, made it possible that the theory of chaotic dynamics can be applied not only to problems in basically all natural sciences but also in fields like medicine, economics and sociology.

The most dramatic event in the development of the modern theory of the onset of chaos in dynamical systems has been the discovery of universality [1]. Especially well known are the universal numbers α and δ, which in the context of period doubling pertain to the universal scaling properties of the 2^∞ cycle near its critical point, and the rate of accumulation of pitchfork bifurcations in parameter space respectively [1], This type of universality is however local, being limited to behavior in the vicinity of an isolated point either in phase space or in parameter space. In this paper I wish to review some recent progress in elucidating the globally universal properties of dynamical systems at the onset of chaos. This progress has been achieved in collaboration with M.H. JENSEN, A. LIBCHABER, L.P. KADANOFF, T.C. HALSEY, B. SHRAIMAN and J. STAVANS [2-4]. In “global universality” we mean that an orbit in phase space has metric universality as a whole set or that a whole range of parameter space can be shown to have universal properties [5]. Examples that have been worked out recently include the 2^∞ cycle of period doubling, the orbit on a 2-torus with golden-mean winding number at the onset of chaos and the complementary set to the mode-locking tongues in the 2-frequency route to chaos. The approach used is however quite general, as will become apparent below.

An infinite sequence of moments is needed to describe a fractal measure. This fact is widely known today, largely thanks to several speakers at this conference, who either refer to it, or push well beyond. Here, I propose to sketch the extensive early background in my work (before 1968) on the theory of turbulent intermittency. This old story matters, because my general procedure also brings forward a number of topics that have not been duplicated, and calls attention to interesting open issues.

The purpose of this note is to present some recent results and techniques concerning the Hausdorff dimension of various objects. We will report on an estimate for the lower bound of the dimension of a wide class of graphs which includes the Weierstrass-Hardy-Mandelbrot functions, and also on the exact dimension of some objects constructed via random recursions.

Recent studies on chaos have made clear that the concept of chaos is quite different from the probabilistic randomness. Much work has been especially done to understand the internal order in chaos such as topological and fractal ones. The discovery of some routes to chaos has also (contributed to the better understandings of order in germinal chaos, but hereafter the order immersed in the fully developed or grown-stage chaos should be elucidated.

Based on Hausdorff’s original approach to fractional dimensions, we study systems which are not sufficiently characterized by their “fractal” or scaling dimension. We construct informative examples of such sets and relate them to sets observed in the context of dynamical systems.

Fat fractals are fractals with positive measure and integer fractal dimension. Their dimension is indistinguishable from that of nonfractals, and is inadequate to describe their fractal properties. An alternative approach can be couched in terms of the scaling of the coarse grained measure. For the more familiar “thin” fractals, the resulting scaling exponent reduces to the fractal codimension, but for fat fractals it is independent of the fractal dimension. Numerical experiments on several examples, including the chaotic parameter values of quadratic mappings, the ergodic parameter values of circle maps, and the chaotic orbits of area-preserving maps, show a power law scaling, suggesting that this is a generic form. This paper reviews several possible methods for defining coarse-grained measure and associated fat fractal scaling exponents, reviews previous work on the subject, and discusses problems that deserve further study.

A Lorenz cross-section of an attractor in Rⁿ with k > 0 positive Lyapunov exponents is the transverse intersection of the attractor with an n − k dimensional plane. We outline a numerical procedure to compute Lorenz cross-sections of chaotic attractors with k > 1 positive Lyapunov exponents and apply the technique to the attractor produced by the double rotor map, two of whose numerically computed Lyapunov exponents are positive and whose Lyapunov dimension is 3.64. The pointwise dimension of the Lorenz cross-sections is computed approximately as 1.64. This numerical evidence supports a conjecture that the pointwise and Lyapunov dimensions of typical attractors are equal.

Much progress has been done in the reconstruction of the geometry of strange attractors, from experimental single time-series, exploiting embedding techniques [l], which make possible, for instance, the estimation of fractal dimensions and metric entropies. A particularly relevant aspect of these procedures, which has not yet been pointed out, concerns the role of filtering. In fact, not only any measurement of experimental signals is to some extent filtered, due to the finite instrumental bandwidth, but often an explicit intervention of the observer is present as well, motivated by the need of “cleaning” the system’s output from the presence of noise.

Our purpose is to describe a new class of methods for computing the “capacity dimension” and related quantities for point-sets. The techniques presented here build on existing work which has been described in the literature. The novelty of our methods lies first in the approach taken to the definition of computation of dimension (namely, via Monte Carlo calculation of the volume of an ε-cover of the point-set), and second in the use of data structures which result in extremely efficient codes for vector computers such as the Cyber 205 (the computation is reduced to the sorting and searching of one-dimensional arrays so that a calculation employing one million points requires less than 2 minutes).

A technique for deriving the metric entropy of strange attractors from estimates of the mutual information in scalar time series is presented and applied to experimental and model data. The results are accurate enough to determine if a system is chaotic.

Algorithms for estimating Lyapunov exponents from experimental data monitor the divergence of nearby phase space orbits. These algorithms rely on the assumption that the dynamics on intermediate length scales are “close” to the dynamics on infinitesimal length scales. We have studied two one-dimensional maps in which intermediate length scale dynamics may result in inaccurate exponent estimates. This effect is found to be small enough so that the exponent estimates are still good characterizations of the systems. Similar effects are likely to be present whenever a finite quantity of data is used for Lyapunov exponent estimation.

Two methods for estimating the Lyapunov exponents of attractors reconstructed from a time series are compared. A method due to Wolf et al. for computing the largest Lyapunov exponent λ₁ is found to be robust with reasonable changes in input parameters. In contrast, a least-squares method suggested by Eckmann and Ruelle yields estimates for the Lyapunov exponents that vary considerably depending on the embedding dimension of the attractor. It appears that only the Wolf algorithm is suitable for the analysis of experimental data.

Since the subject matter of the presentation given at the conference is (or will be) well represented elsewhere [1–3], here we just give a brief account.

Three different methods for calculating the dimension of attractors are analyzed. An approach to error-estimation is presented and is used on various data sets. In some cases it is shown that the errors can become very large.

We use box-counting methods to attempt to reliably calculate the generalized dimensions (including box-counting dimension, i.e., capacity dimension, and information dimension) for the Henon attractor (a = 1.4, b = 0.3). In order to investigate possible errors arising in more general situations, we have analyzed the asymptotic behavior of the cover of the attractor as the number of iterates considered approaches infinity. The error in estimating the box-counting dimension depends in part on the geometric shape of the “boxes” used, and we give a heuristic derivation of the rate of approach. We introduce the use of disks rather than squares to minimize errors in estimates of the number of “boxes” required. The resulting dimension estimates have very small fitting errors: the points in a log-log plot are quite well fit by a straight line. However, what would happen for even smaller box sizes cannot be estimated.

In recent years the characterization of fractals, strange attractors, dynamical chaos has received much attention. A particularly attractive measure is the fractal dimensionality of the dynamical system trajectory. This is a (generally noninteger) number that characterizes the trajectory (viewed as a collection of causally connected discrete points that are embedded in a D-dimensional space, such that d ≤ D). Depending on its definition, it may reflect purely geometrical aspects (capacity), but also probabilistic, information-type features of the system. For practical purposes the Balatoni-Rényi generalized dimension [1] d_q, q ≥ 0, subsumes all currently used definitions of d (see however, [2]). A number of algorithms have been proposed to determine d [3–10]; they are reasonably successful, but only for d ≤ 3.

We discuss different methods of characterizing turbulent channel flow in terms of “the number of independent degrees of freedom”. These methods all suggest that the dimension of this system is greater than 10.

Helical instabilities in an electron-hole plasma in Ge in parallel dc electric and magnetic fields are known to exhibit chaotic behavior. By fabricating probe contacts along the length of a Ge crystal we study the spatial structure of these instabilities, finding two types: (i) spatially coherent and temporally chaotic helical density waves characterized by strange attractors of measured fractal dimension d ~ 3, and (ii) beyond the onset of spatial incoherence, instabilities of indeterminately large fractal dimension d ≥ 8. In the first instance, calculations of the fractal dimension provide an effective means of characterizing the observed chaotic instabilities. However, in the second instance, these calculations do not provide a means of determining whether the observed plasma turbulence is of stochastic or of deterministic (i.e., chaotic) origin.

By solving the recursion relation of a reaction-diffusion equation on a lattice, we find two distinct routes to turbulence, both of which reproduce commonly observed phenomena: the Feigeribaum route, with period-doubling frequencies; and a much more general route with noncommensurate frequencies and frequency entrainment, and locking. Intermittency and large-scale aperiodic spatial patterns, also observed in physical systems, are reproduced in this new route. The fractal dimension has been estimated to be about 2.6 in the oscillatory instability and about 6.0 in the turbulent regime.

In order to characterise quantitatively the behaviour of dissipative dynamical systems we have to determine the values of information dimension, metric entropy and Lyapunov exponents associated with the limit sets in phase space. For numerically integràble dynamical systems such as iterated maps and systems of ordinary differential equations, methods are available which lead to the determination of Lyapunov exponents with an accuracy generally depending only on the power of the utilized computer. We furthermore have the values of information dimension and metric entropy by applying the conjectured formulas relating their values to the Lyapunov exponents.

The instabilities in the electrical conductivity of barium sodium niobate (BSN) crystals are studied in the presence of ac and dc fields. Transitions from quasiperiodicity into chaos via phase locking are observed. Phase portraits, Poincaré sections and return maps constructed from measured voltage signals illustrate the emergence of a strange attractor from a torus. The dimension and entropy of the attractor are determined as a function of the control parameter.

High quality experimental data have been taken on a convection cell containing a dilute ³He−⁴He solution. We discuss some problems with the determination of dimension and entropy for experimental data, and compare the results to detailed Poincaré sections. At the chaotic transition, we show the behavior of dimension and entropy as a function of Rayleigh number.

We present a study of the geometrical and measure properties of attractors generated by Rayleigh-Bénard convection with two oscillators present. One oscillator was induced by the flow while the second was imposed externally. The winding number was tuned to two different irrational numbers. The evolution of the attractors with increasing non-linear coupling between the oscillators was examined. Their dynamical properties were extracted and compared with those of circle maps.

We discuss briefly some aspects of ‘open flow systems’ in the context of deterministic chaos. This note is mostly a statement of the difficulties in characterizing such flows, especially at high Reynolds numbers, by dynamical systems. Brief comments will be made on the fractal geometry of turbulence.

The quantification of complex dynamical phenomena associated with motions on strange attractors has made available a tool of considerable power for the analysis of systems which display aperiodic or apparently random temporal behavior. Until recently, aperiodic phenomena were described primarily in terms of snapshots of time sequences, power spectra, or correlation functions. These made possible some qualitative or pictorial analyses but did not provide simple numerical criteria suitable for more quantitative studies. During the past decade, spectral studies have made possible the identification of a few characteristic routes to apparently chaotic behavior in hydrodynamic[l–3], chemical [4], optical [5–7, 16, 29], and electronic [8, 9] systems. There were very strong indications that the complex motions, characterized by broadband spectra, to which these routes led, were in fact motions on strange attractors.

The study of complex systems may be performed by analysing experimental data recorded as a series of measurements in time of a pertinent and easily accessible variable of the system. In most cases, such variables describe a global or averaged property of the system.