Variety of strange pseudohyperbolic attractors in three-dimensional generalized H'enon maps

TL;DR

This paper investigates the existence and classification of robust strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps, identifying five types and demonstrating their presence in specific map forms.

Contribution

It classifies five types of pseudohyperbolic attractors in 3D maps and shows their occurrence in generalized Hénon maps with specific parameter conditions.

Findings

Identified five types of pseudohyperbolic attractors in 3D maps.

Demonstrated the presence of four attractor types in generalized Hénon maps.

Provided conditions under which these attractors exist.

Abstract

In the present paper we focus on the problem of the existence of strange pseudohyperbolic attractors for three-dimensional diffeomorphisms. Such attractors are genuine strange attractors in that sense that each orbit in the attractor has a positive maximal Lyapunov exponents and this property is robust, i.e. it holds for all close systems. We restrict attention to the study of pseudohyperbolic attractors that contain only one fixed point. Then we show that three-dimensional maps may have only 5 different types of such attractors, which we call the discrete Lorenz, figure-8, double-figure-8, super-figure-8, and super-Lorenz attractors. We find the first four types of attractors in three-dimensional generalized H\'enon maps of form $\bar x = y, \; \bar y = z, \; \bar z = Bx + Az + Cy + g(y,z)$, where $A,B$ and $C$ are parameters ($B$ is the Jacobian) and $g(0,0) = g^\prime(0,0) =0$.