Strange attractors for the family of orientation preserving Lozi maps

TL;DR

This paper proves the existence of strange attractors with chaotic properties for a family of orientation-preserving Lozi maps, extending previous results and analyzing their stability and structure in parameter space.

Contribution

It extends the existence of strange attractors to orientation-preserving Lozi maps and characterizes their properties and parameter dependence.

Findings

Existence of strange attractors in an open parameter subset.

Attractors are maximal and are closures of unstable manifolds.

Attractors vary continuously with parameters in the Hausdorff metric.

Abstract

We extend the result of Michal Misiurewicz assuring the existence of strange attractors for the parametrized family $\{f_{(a,b)}\}$ of orientation reversing Lozi maps to the orientation preserving case. That is, we rigorously determine an open subset of the parameter space for which an attractor $\mathcal{A}_{(a,b)}$ of $f_{(a,b)}$ always exists and exhibits chaotic properties. Moreover, we prove that the attractor is maximal in some open parameter region, and arises as the closure of the unstable manifold of a fixed point, on which $f_{(a,b)}|_{\mathcal{A}_{(a,b)}}$ is mixing. We also show that $\mathcal{A}_{(a,b)}$ vary continuously with parameter $(a,b)$ in the Hausdorff metric.