Shilnikov-type dynamics in three-dimensional piecewise smooth maps

Abstract

We show the existence of Shilnikov-type dynamics and bifurcation behaviour in general discrete three-dimensional piecewise smooth maps and give analytical results for the occurence of such dynamical behaviour. Our main example in fact shows a ‘two-sided’ Shilnikov dynamics, i.e. simultaneous looping and homoclinic intersection of the one-dimensional eigenmanifolds of fixed points on both sides of the border. We also present two complementary methods to analyse the return time of an orbit to the border: one based on recursion and another based on complex interpolation.