Elements of contemporary mathematical theory of dynamical chaos. Part 1. Pseudohyperbolic attractors

TL;DR

This paper introduces the concept of pseudo-hyperbolic attractors in multidimensional maps, providing definitions, conditions, and methods for their study, expanding the understanding of strange attractors in dynamical chaos.

Contribution

It presents a formal definition of pseudo-hyperbolic attractors, necessary conditions for their existence, and new analytical methods, advancing the mathematical theory of chaotic attractors.

Findings

Defined pseudo-hyperbolic attractors for multidimensional maps.

Derived necessary conditions using Lyapunov exponents.

Proposed new methods like saddle charts and Lyapunov diagrams.

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 Henon maps as examples.