Robust chaos in orientation-reversing and non-invertible two-dimensional piecewise-linear maps

TL;DR

This paper extends the understanding of robust chaos in two-dimensional piecewise-linear maps by including non-invertible and orientation-reversing cases, revealing chaos is not solely dependent on topological properties.

Contribution

It generalizes previous constructions to encompass non-invertible and orientation-reversing maps, providing a unified view of robust chaos in these systems.

Findings

Identified parameter regions with robust chaos

Demonstrated chaos can occur independently of map invertibility

Mapped bifurcation points destroying chaotic attractors

Abstract

This paper concerns the two-dimensional border-collision normal form -- a four-parameter family of piecewise-linear maps generalising the Lozi family and relevant to diverse applications. The normal form was recently shown to exhibit a chaotic attractor throughout an open region of parameter space. This was achieved by constructing a trapping region in phase space and an invariant expanding cone in tangent space, but only allowed parameter combinations for which the normal form is invertible and orientation-preserving. This paper generalises the construction to include the non-invertible and orientation-reversing cases. This provides a more complete and unified picture of robust chaos by revealing its presence to be disassociated from the global topological properties of the map. We identify a region of parameter space in which the map exhibits robust chaos, and show that part of the boundary of this region consists of bifurcation points at which the chaotic attractor is destroyed.