Elements of Contemporary Theory of Dynamical Chaos: A Tutorial. Part I. Pseudohyperbolic Attractors

Abstract

The paper deals with topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finitedimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large "chaotic sea" with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. In the present paper (which opens a cycle of three our papers), elements of the theory of pseudo-hyperbolic attractors of multidimensional maps are presented. Such attractors, as well as hyperbolic ones, are genuine strange attractors, but they allow the existence of homoclinic tangencies. We give a mathematical definition of a pseudo-hyperbolic attractor for the case of multidimensional maps, from which we derive the necessary conditions for its existence in the three-dimensional case, formulated using the Lyapunov exponents. We also describe some phenomenological scenarios for the appearance of pseudo-hyperbolic attractors of various types in one-parameter families of three-dimensional diffeomorphisms, we propose new methods for studying such attractors (in particular, a method of saddle charts and a modified method of Lyapunov diagrams). We consider also three-dimensional generalized Hénon maps as examples.In the second part, we plan to review the theory of spiral attractors, which compose an important class of attractors often meeting in applications. The third part will be devoted to mixed dynamics -a new type of chaos, which is characteristic, in particular for reversible systems, i.e. systems invariant with respect to the time reversal. It is well known that such systems are met in many problems of mechanics, electrodynamics and other fields of natural science.

Hyperchaos and multistability in the model of two interacting microbubble contrast agents

“…They contain a saddle fixed point with a one-dimensional unstable manifold forming a homoclinic structure resembling a butterfly and figure-eight, respectively. As it was recently shown in [32] these two attractors belong to a class of pseudohyperbolic [31], [37] ("genuinely" chaotic) attractors. Shortly speaking, each orbit on a pseudohyperbolic attractor has a positive Lyapunov exponent and, what is important from a physical point of view, this property persists after small perturbations (changing in parameters).…”

“…Such stability windows indicate the existence of the specific homoclinic bifurcations (cubic homoclinic tangencies or symmetrical pairs of homoclinic tangencies) in the system [30]. All these stability windows indicate that chaotic dynamics in system (1) are not hyperbolic and even not pseudohyperbolic [31][32][33]. In other words, in the accordance with PQ-hypothesis [33] strange attractors in the system under investigation belong to a class of quasiattractors introduced by Afraimovich and Shilnikov in [34].…”

Matching geometric and expansion characteristics of wild chaotic attractors

“…A numerical characteristic for the existence of a wild chaotic attractor is that the sum of its two largest Lyapunov exponents is positive [14,37]. In general, this is not sufficient, because the linearised dynamics transverse to the attractor must also be sufficiently (dominating) contracting; we refer to [15] for precise details.…”

“…The theory of wild chaos tends to consider only saddle-type hyperbolic sets, rather than attractors with this property. More recently, wild chaotic attractors have been a focus of study, which are called pseudo-hyperbolic attractors in [14,15].…”

Robust Devaney chaos in the two-dimensional border-collision normal form

“…Robust chaos refers to the phenomenon that a family of dynamical systems has a chaotic attractor throughout an open region of parameter space [1]. This does not occur for generic families of smooth one-dimensional maps as these have dense windows of periodicity [2], but is typical for systems with sufficiently many dimensions [3,4], a well-known example being the Lorenz system [5,6]. One can impose further requirements on the robustness, such as that the attractor varies continuously with respect to Hausdorff distance [7] or Lebesgue measure [8].…”