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The Many Facets of Chaos April 21, 2015 14:57 WSPC/S0218-1274 1530011 International Journal of Bifurcation and Chaos, Vol. 25, No. 4 (2015) 1530011 (15 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127415300116 The Many Facets of Chaos Evelyn Sander Department of Mathematical Sciences, George Mason University, 4400 University Dr., Fairfax, VA 22030, USA [email protected] James A. Yorke Institute for Physical Sci. & Tech. and Departments of Mathematics and Physics, University of Maryland, College Park, MD 20742, USA [email protected] Received November 30, 2014 There are many ways that a person can encounter chaos, such as through a time series from a lab experiment, a basin of attraction with fractal boundaries, a map with a crossing of stable and unstable manifolds, a fractal attractor, or in a system for which uncertainty doubles after some time period. These encounters appear so diverse, but the chaos is the same in all of the underlying systems; it is just observed in different ways. We describe these different types of chaos. We then give two conjectures about the types of dynamical behavior that is observable if one randomly picks out a dynamical system without searching for a specific property. In particular, we conjecture that from picking a system at random, one observes (1) only three types of basic invariant sets: periodic orbits, quasiperiodic orbits, and chaotic sets; and (2) that all the definitions of chaos are in agreement. Keywords: Chaos; Lyapunov exponents; basins of attraction; horseshoes; forced-damped pendulum. by Dr. Evelyn Sander on 05/06/15. For personal use only. Int. J. Bifurcation Chaos 2015.25. Downloaded from www.worldscientific.com 1. Introduction each would be likely to be found, and some of Chaos is observed in so many ways that it can be the methods used for detecting them. We restrict quite confusing for a practitioner to get a reasonable attention to deterministic dynamical systems, in answer to the simple question which the initial conditions determine all future behavior. This paper is certainly not an attempt to “What is the definition of chaos?” be complete — there are many situations which we In fact, chaos cannot now be satisfactorily defined have left out altogether. We also freely admit that mathematically using a single definition, not the particular focus is shaped by personal experi- because chaos is not a single concept, but because ence. These examples are meant to illustrate a point chaos has many manifestations in many different which we do believe is universal: For the defini- situations. There are situations in which many of tion to be useful, the determination of chaos should these manifestations simultaneously apply, but each depend on the viewpoint of the investigator. It must is independent of the others; while these definitions be phrased in terms of the information that is avail- seem to agree in almost all cases, they tend to dif- able to the scientist in question. fer at the margins. In this paper, we discuss a vari- The following is the list of aspects of chaos that ety of manifestations of chaos, the context where we discuss, forming a summary of the rest of the 1530011-1 April 21, 2015 14:57 WSPC/S0218-1274 1530011 E. Sander & J. A. Yorke paper: assumed that there were no homoclinic crossings. When Phragm´en pointed out his error, his attempts • In Sec. 2, we discuss homoclinic orbits and horse- at a correction resulted in his creating the mod- shoes. These aspects of chaos are the earliest ern field of dynamical systems [McGehee, personal types of observed chaotic behavior, first described communication]. In particular, he clearly realized in the writings of Poincar´e. the complex nature of the situation in the following • Section 3 addresses strange chaotic attractors and description of the situation depicted in the bottom their fractal topology. two panels of Fig. 1 [Poincar´e, 1993, pp. I59]: “If • Section 4 describes sensitive dependence on ini- one seeks to visualize the pattern formed by these tial conditions, including scrambled sets of Li– two curves and their infinite number of intersec- Yorke chaos, broad band power spectrum, and tions, each corresponding to a doubly asymptotic Lyapunov chaos. These latter two are commonly solution, these intersections form a kind of lattice- used in the study of time series data. work, a weave, a chain-link network of infinitely fine • Section 5 describes three nonequivalent types mesh; each of the two curves can never cross itself, of entropy: topological entropy, periodic orbit but it must fold back on itself in a very complicated entropy, and metric entropy. way so as to recross all the chain-links an infinite • Section 6 covers important types of robust sets number of times. One will be struck by the complex- which are not attractors but may exhibit chaos: ity of this figure, which I am not even attempting saddles and basin boundaries. to draw.” • Section 7 contains two conjectures on the types More than 60 years after Poincar´e’s work, of behavior observable for a dynamical system Smale [1967] rigorously described homoclinic tan- selected at random. We give a formal definition gles by showing that the behavior of homoclinic tan- for what we mean by observable. In broad strokes, gles is at least as complicated as the behavior of a we conjecture that for observable dynamical very simple iterated map that he called a horseshoe systems: the basic invariant sets have periodic, for its geometric shape. In particular, he was then quasiperiodic behavior, or chaotic behavior; and able to fully define all dynamical behavior under multiple concepts of chaotic sets will agree for the horseshoe map. In particular, he showed that observable dynamical systems. In contrast to although the map is deterministic, the behavior expected typical behavior, we list some com- of the orbits is as random as a coin flip, includes monly studied concepts which represent nontyp- infinitely many periodic orbits, and displays expo- ical behavior. nential divergence of trajectories. The set of points with these hallmarks of chaos (known as the chaotic by Dr. Evelyn Sander on 05/06/15. For personal use only. set) also has an interesting fractal structure. The 2. Homoclinic Orbits and chaos in this case can be transient, meaning that the Horseshoes chaotic set is unstable. After creating the seemingly Int. J. Bifurcation Chaos 2015.25. Downloaded from www.worldscientific.com If an orbit of a system containing an initial con- arbitrary map, Smale demonstrated that for every dition x has the same limit both in forward and fixed point (or periodic orbit) with a homoclinic backward time, x is called a homoclinic point.If crossing, there is a horseshoe map embedded in the the stable and unstable manifolds cross each other behavior of the system. Therefore, any system with at a homoclinic point, then there are infinitely- a homoclinic crossing includes the type of random many homoclinic points limiting on each other, and behavior described above. The relationship between chaos occurs near this set of points. This situation is homoclinic points and chaos motivates much of the referred to as a homoclinic tangle.Figure1 shows a study of homoclinic points and their bifurcations. homoclinic bifurcation, meaning the progression as Verifying chaos by checking for homoclinic a parameter is varied starting with no intersections crossings is in some cases feasible when other man- between stable and unstable manifolds, followed by ifestations — such as power spectrum or Lyapunov a tangency between manifolds, and ending with a exponent — cannot be used. This method has homoclinic tangle. This was the type of chaos that even been used in infinite-dimensional contexts, Poincar´e initially missed in his treatise on the sta- such as for parabolic partial differential equations bility of the solar system, which had won him a and delay differential equations [Lani-Wayda & prize given by King Oscar III of Sweden. Poincar´e Walther, 1995, 1996]. (More general results in this 1530011-2 April 21, 2015 14:57 WSPC/S0218-1274 1530011 The Many Facets of Chaos by Dr. Evelyn Sander on 05/06/15. For personal use only. Int. J. Bifurcation Chaos 2015.25. Downloaded from www.worldscientific.com 2 Fig. 1. The formation of a homoclinic tangle in the H´enon map: (x, y) → (ρ − x − 0.3y, x)for−4 <x<4, −3 <y<3, left to right, up to down ρ =2.0, 2.01725, 2.01875, 2.0246. The stable manifold for a fixed point is shown in red. The unstable manifold contains the attractor, in yellow. In the first figure, the stable and unstable manifolds do not cross and the attractor is disconnected. In the second figure, the stable and unstable manifolds are tangent, and the attractor becomes connected. The manifolds cross in the third and fourth figures, becoming a tangled mess limiting on themselves infinitely often. Homoclinic tangles are one of the best known behaviors associated with chaos. spirit show the existence of Shilnikov heteroclinic is, for example, true for H´enon maps [Devaney & orbits for delay equations [Lani-Wayda, 2001].) Nitecki, 1979] and in some rigorous computational Although the horseshoe is most often used to study proofs of chaotic behavior [Arai & Mischaikow, homoclinic tangles, some dynamical systems can be 2006]. There are shortcomings of testing for a shown to have a horseshoe directly without first chaotic set via chaotic orbits. As mentioned above, showing the existence of a homoclinic orbit. This the set may be unstable, referred to as transient 1530011-3 April 21, 2015 14:57 WSPC/S0218-1274 1530011 E. Sander & J. A. Yorke chaos. In addition, the existence of a homoclinic structure, but these structures are nontypical and crossing is not a quantitative measure of either the highly degenerate1 [Grebogi et al., 1984].) To say amount of chaos in a system or its robustness under that an attractor “looks fractal” has been made variation of a parameter.
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