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

Abstract

The paper is devoted to topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finite-dimensional 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 series of three of our papers), elements of the theory of pseudohyperbolic attractors of multidimensional maps and flows are presented. Such attractors, as well as hyperbolic ones, are genuine strange attractors, but they allow the existence of homoclinic tangencies. We describe two principal phenomenological scenarios for the appearance of pseudohyperbolic attractors in one-parameter families of three-dimensional diffeomorphisms, and also consider some basic examples of concrete systems in which these scenarios occur. We propagandize new methods for studying pseudohyperbolic attractors (in particular, the method of saddle charts, the modified method of Lyapunov diagrams and the so-called LMP-method for verification of pseudohyperbolicity of attractors) and test them on the above examples. We show that Lorenz-like attractors in three-dimensional generalized Hénon maps and in a nonholonomic model of Celtic stone as well as figure-eight attractors in the model of Chaplygin top are genuine (pseudohyperbolic) ones. Besides, we show an example of four-dimensional Lorenz model with a wild spiral attractor of Shilnikov–Turaev type that was found recently in [Gonchenko et al., 2018].