Robust chaos and border-collision bifurcations in non-invertible piecewise-linear maps

Abstract

This paper investigates border-collision bifurcations in piecewise-linear planar maps that are non-invertible in one region. Maps of this type arise as normal forms for grazing–sliding bifurcations in three-dimensional Filippov-type systems. A possible strategy is presented for classifying fixed and period-2 points, that are involved in such bifurcations. This allows one to determine a region of parameter space where a bifurcation leading to chaos might occur. The main part of the paper contains a careful proof of the onset of attractors which are robust to small parameter changes. An intricate structure is revealed of the limiting set on which the attractor lives, consisting of distinct continuous line segments. As parameters are varied, the attractor on the segments can change from being chaotic to periodic. Also, the mechanism by which the number of line segments can change is uncovered.