The Border Collision Normal Form with Stochastic Switching Surface

Abstract

In the study of discrete time piecewise smooth dynamical systems the deterministic border collision normal form describes the bifurcations as a fixed point moves across the switching surface with changing parameter. If the position of the switching surface varies randomly, we give conditions which imply that the attractor close to the bifurcation point is the attractor of an iterated function system. The proof uses an equivalent metric to the Euclidean metric because the functions involved are never contractions in the Euclidean metric. If the conditions do not hold, then a range of possibilities may be realized, including local instability, and some examples are investigated numerically.