Bifurcations and strange attractors

TL;DR

This paper reviews the theory of strange attractors and bifurcations, classifying them into hyperbolic, pseudo-hyperbolic, and quasi-attractors, and discusses their properties, examples, and the complexity of their analysis.

Contribution

It introduces a new type of spiral attractor requiring countably many topological invariants for its description.

Findings

Classifies strange attractors into three groups: hyperbolic, pseudo-hyperbolic, and quasi-attractors.

Provides conditions for the existence of pseudo-hyperbolic attractors like the Lorenz attractor.

Introduces a new spiral attractor with a complex topological structure.

Abstract

We review the theory of strange attractors and their bifurcations. All known strange attractors may be subdivided into the following three groups: hyperbolic, pseudo-hyperbolic ones and quasi-attractors. For the first ones the description of bifurcations that lead to the appearance of Smale-Williams solenoids and Anosov-type attractors is given. The definition and the description of the attractors of the second class are introduced in the general case. It is pointed out that the main feature of the attractors of this class is that they contain no stable orbits. An etanol example of such pseudo-hyperbolic attractors is the Lorenz one. We give the conditions of their existence. In addition we present a new type of the spiral attractor that requires countably many topological invariants for the complete description of its structure. The common property of quasi-attractors and pseudo-hyperbolic ones is that both admit homoclinic tangencies of the trajectories. The difference between them is due to quasi-attractors may also contain a countable subset of stable periodic orbits. The quasi-attractors are the most frequently observed limit sets in the problems of nonlinear dynamics. However, one has to be aware that the complete qualitative analysis of dynamical models with homoclinic tangencies cannot be accomplished.