Robust Chaos in Orientation-Reversing and Non-Invertible Two-Dimensional Piecewise-Linear Maps

The two-dimensional border-collision normal form is a four-parameter family of piecewise-linear maps generalizing the Lozi family and relevant to diverse applications. It was recently shown to exhibit a chaotic attractor throughout an open region of parameter space, but only for parameter combinations where the map is invertible and orientation-preserving. In this paper we formally identify analogous regions for the orientation-reversing and non-invertible cases, the latter being particularly relevant to control systems. This is achieved by constructing a trapping region in phase space and an invariant expanding cone in tangent space, and provides a more complete picture of robust chaos by showing its presence is disassociated from the global topological properties of the map. We also show some boundaries of the regions are optimal in that they correspond to bifurcations where the chaotic attractor is destroyed.