Generalized Attracting Horseshoes and Chaotic Strange Attractors

TL;DR

This paper introduces generalized attracting horseshoes as a new framework for understanding chaotic strange attractors in smooth and piecewise smooth maps, providing simplified proofs applicable to well-known chaotic systems.

Contribution

It presents a novel concept of generalized attracting horseshoes for describing chaotic attractors, extending the applicability of existence proofs to a broader class of maps.

Findings

Applicable to Henon and Lozi maps

Simplifies proofs of chaotic attractor existence

Works for arbitrary finite rank attractors

Abstract

A generalized attracting horseshoe is introduced as a new paradigm for describing chaotic strange attractors (of arbitrary finite rank) for smooth and piecewise smooth maps f from Q to Q, where Q is a homeomorph of the unit interval in real m-space for any integer m > 1. The main theorems for generalized attracting horseshoes are shown to apply to Henon and Lozi maps, thereby leading to rather simple new chaotic strange attractor existence proofs that apply to a range of parameter values that includes those of earlier proofs.