Sinai–Ruelle–Bowen measure for normal form map of grazing bifurcations of impact oscillators

Abstract

Grazing bifurcations are typical non-smooth bifurcations that occur in impact oscillators. They are described by a discrete map with square-root singularities. In this paper we study the structure of a strange attractor which is the closure of the unstable manifold at hyperbolic fixed point of the normal form map for grazing bifurcations of one-degree-of-freedom impact oscillators from the ergodic theoretic point of view. We prove that for some set of values of the parameters this map has an Sinai–Ruelle–Bowen measure which is supported on the strange attractor and is ergodic.