Border collision bifurcations of chaotic attractors in discontinuous two-dimensional maps

Chaotic attractors in two-dimensional (2D) discontinuous piecewise-smooth maps can undergo border collision bifurcations, resulting in significant changes to their shape and structure. Through several examples, we demonstrate that in both fractal and full-measure chaotic attractors, these bifurcations can produce additional bands that retain the same structure as the original attractor. Moreover, we show that a border collision bifurcation can occur in combination with an expansion bifurcation, leading to a transformation of the attractor’s structure—for instance, a transition from a one-dimensional (1D) to a fractal or full-measure chaotic attractor.