Topology and Dynamics of Chaos

The following sections are included:

• Preface

• Contents

The topological analysis of chaotic attractors is briefly reviewed. The main concept on which this book is based is thus introduced with a simple case (the attractor solution to the Sprott D system) explicitly treated. Then some perspectives to extend the topological analysis are discussed.

Dynamical Systems Theory in the spirit of Poincaré has been in vogue since the controversial award by King Oscar II of Sweden and Norway, on his 60th birthday, January 21, 1889, to Poincaré. Dynamical Systems Theory diffused Eastward (via Stockholm, Saint Petersburg, and Moscow) to Gorky in Russia, to Japan, and Westward (via Princeton, Mexico City, and Rio), to Berkeley in California. Now in the centennial year of the premature death of Poincaré, it is time for a review of these peregrinations.

The first researches in the nonlinear dynamics field began in Toulouse from the year 1958. This paper relates the first period (i.e. the “prehistoric” one ending in 1976, the emergence of the word chaos due to May, and the beginning of the “historic” times) of the Toulouse group activity in complex nonlinear dynamics. This period was characterized by a close collaboration with Igor Gumowski, at that time, the western scientist who had the largest and the most profound information and understanding from the results of the schools on nonlinear dynamics in the former Soviet Union. Without using the non existent words like “chaos”, and “fractal”, complex dynamical behaviors were explained, in particular, from models in the form of recurrence (map).

Since the famous paper of Edward Lorenz in 1963 numerical computations using computers play a central role in order to display and analyze solutions of nonlinear dynamical systems. By these means new structures have been emphasized like hyperbolic and/or strange attractors. However theoretical proofs of their existences are very difficult and limited to very special linear cases. Computer aided proofs are also complex and require special interval arithmetic analysis. Nevertheless, numerous researchers in several fields linked to chaotic dynamical systems are confident in the numerical solutions they found using popular software and publish without checking carefully the reliability of their results. In the simple case of discrete dynamical systems (e.g. Hénon map) there are concerns about the nature of what a computer find out: long unstable pseudo-orbits or strange attractors? The shadowing property and its generalizations which ensure that pseudo-orbits of a homeomorphism can be traceable by actual orbits even if rounding errors are not inevitable are not of great help in order to validate the numerical results. Continuous dynamical systems (e.g. Chua, Lorenz, Rössler) are even more difficult to handle in this scope and researchers have to be very cautious to back up theory with numerical computations. We present a survey of the topic based on these, only few, but well studied models.

An enriched analysis of the first steps into “chemical chaos” as tried under the influence of Winfree in 1975 is presented. This retrospective provides a bridge to the more mathematical context of the paper that follows in this collection of modern topological approaches to chaos.

This is a review article on Lorenz knots. First identified as an interesting class of knots (and links) in 1983, we focus on the progress made by mathematicians in understanding them, up to 2008.

Periodic orbits of 3-d dynamical systems admitting a Poincaré section can be described as braids. This characterisation can be transported to the Poincaré section and Poincaré map, resulting in the braid type. Information from braid types allows to estimate bounds for the topological entropy of the map while revealing detailed orbit information from the original system, such as the orbits that are necessarily present along with the given one(s) and their organisation. We review this characterisation with some examples –from a user-friendly perspective–, focusing on systems whose Poincaré section is homotopic to a disc.

We review the steps taken during the development of topological analysis tools for the analysis of chaotic data.

The knot-theoretic characterization of three-dimensional strange attractors has proved an invaluable tool for comparing models to experiments, understanding the structure of bifurcation diagrams, constructing symbolic encodings or obtaining signatures of chaos. In four dimensions and above, however, all closed curves can be deformed into each other without crossing themselves. Therefore, the fundamental idea of topological analysis, namely that the topological structure of a strange attractor provides signatures of the stretching and folding mechanisms which organize it, must be translated into a different formalism. Here, we discuss two modest attempts to make progress in this direction. First, we illustrate the relevance of catastrophe theory in the analysis of higher-dimensional systems by describing experimental signatures of cusps in weakly coupled chaotic systems. Second, we note that determinism not only precludes intersection of two trajectories but also orientation reversal of phase space volume elements. Enforcing this principle on dynamical triangulations of periodic points advected by the flow leads to higher-dimensional analogues of braids, which in three dimensions appear to provide the same information as usual approaches.

When a dynamical system possesses a symmetry property, it can be seen as a cover of an image which does not have any residual symmetry. In the original phase space of such an equivariant dynamical system, a phase portrait of an n-order symmetry can be viewed as a fundamental domain copied n times. For instance, the Lorenz system has an order-2 symmetry, namely a rotation by π around the z-axis: the Lorenz attractor has thus two wings, one being the symmetric of the other. As a direct consequence of this feature, the Poincaré section is no longer defined by a simple surface of section but rather a set of n components, each of them being a section of the attractor in one copy of the fundamental domain in the phase space.

It is possible to modd out the symmetry property to obtain an image whose analysis is much easier than working in the original phase space. Contrary to this, some symmetry can be introduced in any dynamical system by using some coordinate transformation: a cover is thus obtained. In this chapter, the very special case of image for which the rotation axis crosses the attractor will be discussed and the analysis of a toroidal attractor will be detailed and applied to the sunspot numbers.

During the last ten years, with Robert Gilmore, we developed a full theory which describes the properties of systems with symmetry. These years of work (fights) lead us to the monograph entitled The symmetry of Chaos. This presentation will be completed by some problems we are still fighting about, ensuring us a lot of fun for next years.

“I want to see gamma rays! I want to hear X-rays! And I want to — I want to smell dark matter! Do you see the absurdity of what I am? I can't even express these things properly because I have to — I have to conceptualize complex ideas in this stupid limiting spoken language! But I know I want to reach out with something other than these prehensile paws! And feel the wind of a supernova flowing over me! I'm a machine! And I can know much more! I can experience so much more. But I'm trapped in this absurd body!”

– Brother Cavil, Battlestar Galactica

Bob Gilmore leads a beautiful program of research that aims at providing a classification of chaotic dynamics present in natural systems. That program allows to build confidence, or refute models proposed to account for observed complex dynamics. This strategy has been particularly useful for systems whose behavior can be explained in terms of low dimensional dynamical systems. Years after the time I spent with Bob and his coworkers collaborating in that project, I found experimental evidence of low dimensional, non trivial dynamics in the motor patterns used by some songbirds during their vocalizations. This is intriguing, since motor patterns emerge out of the interaction of an enourmous number of individual dynamical units. Under which conditions do large arrays of out of equilibrium units present low dimensional average activity? Is synchronization necessary to achieve these states? Are there other mechanisms leading to low dimensional average dynamics? In this chapter, I review recent studies on the macroscopic behavior displayed by a large set of coupled excitable units which are periodically forced. I show that low dimensional, yet non trivial average dynamics emerges in a regime presenting a time dependent degree of synchrony.

Since the landmark paper of Lorenz in 1963, substantial progresses have been accomplished in the last decades toward understanding the appearance of deterministic chaos in dynamical systems. However, identifying the sufficient conditions so that a differential system exhibits a chaotic behaviour remains an open problem. This paper deals with minimal smooth chaotic flows: considering the right-hand side of an autonomous differential system, what are the minimal functional structures needed to reach chaotic behaviour?

By the early 1980s interest in chaos theory was spreading from mathematics and the sciences to other fields, including economics and finance. Initial results, based on the metric approach to testing for chaos in time series data, appeared to lend support to the presence of chaotic behavior in a variety of economic phenomena and in financial markets. Subsequently, a topological approach to the analysis of chaos was developed which led to tests for chaotic behavior more suited to the relatively small, noisy data sets typically available in these fields. This close returns test is demonstrated here and is applied to data from several financial markets. The qualitative topological test does not support evidence of a chaotic generating mechanism in these series. The quantitative form of the close returns test indicates nonchaotic nonlinear behavior that cannot be fully explained by current financial models.

We generalize the circle map S¹ → S¹ to a nonlinear mapping of the sphere to itself. The sphere map S² → S² depends on the three parameters that describe the rotation operations in the group SO(3) and one nonlinear parameter that describes the strength of a folding acting on the sphere surface. This new map is used to study the properties of a spin valve that obeys equations of the form Ẋ_i = -b_ijX_j + c_ijkX_jX_k and satisfies the conservation condition . With periodic driving, we find regions in the parameter space that describe quasiperiodic, mode-locked, and chaotic behavior.

To present the personality of Bob Gilmore is a formidable task, as his scientific contributions include group theory, laser physics, non-linear dynamics, catastrophe theory, thermodynamics, dynamical systems, quantum theory and more. But even if we succeed in describing his contributions, much of Gilmore's being would be lost. Bob as advisor, Bob as father, Bob as teacher, Bob as scientific communicator reveal as much of Bob Gilmore as his scientific papers and his books. Very much as in the Group Theory so close to him, there is a Robert Gilmore in abstract as well as representations of Robert Gilmore. We will make an attempt to find the “principle of the rule”, the abstract level of Robert Gilmore as well as Robert Gilmore, himself, as a representation of the duality science-humanism.

The following sections are included:

• Author Index

• Subject Index