Invariant cones for non-smooth dynamical systems

In this paper we study the dynamics of piecewise linear systems with regard to generalisations of the familiar notion of Hopf bifurcation for smooth systems. It is shown that the concept of invariant planes with a change from a stable focus to an unstable focus through a center should be replaced in the case of non-smooth systems by an invariant cone with similar dynamical behaviour. In that way results for planar systems are extended into higher dimensions. To determine the invariant cone and its stability properties various extended systems are derived.