Border collision bifurcations in a soft impact system

Abstract

The present work deals with the dynamics of a mechanical switching system in which the state variables are continuous at the switching events, but the first derivative of the vector field changes discontinuously across the switching boundary. Earlier works have shown that hard impacting systems yield discrete maps with a term of power ½, and stick-slip systems yield discrete maps with a term of power 3/2. Maps of the first kind exhibit square-root singularity while those of the second kind are smooth, and therefore no border collision bifurcation occur in them. In this Letter we consider an impacting system with a wall cushioned with spring-damper support. The spring is constrained such that the force on the mass changes discontinuously at a grazing contact. We focus our attention on the change in the Jacobian matrix of a fixed point caused by grazing. We show that a typical property of border collision in such systems is that the determinant remains invariant and the trace shows a singularity at the grazing point. We also explain the observed bifurcations based on the available theory of border collision bifurcations.