Reduction to invariant cones for non-smooth systems

The reduction of smooth dynamical systems to lower dimensional center manifolds containing the essential bifurcation dynamics is a very useful approach both for theoretical investigations as well as for numerical computation. Since this approach relies on smoothness properties of the system and on the existence of a basic linearization the question arises if this approach can be carried over to non-smooth systems. Extending previous works we show that such a reduction is indeed possible by using an appropriate Poincaré map: the linearization will be replaced by a basic piecewise linear system; a fixed point of the Poincaré map generates an invariant cone which takes the role of the center manifold. The occurrence of nonlinear higher order terms will change this invariant “manifold” to a cone-like surface in Rn containing the essential dynamics of the original problem. In that way the bifurcation analysis can be reduced to the study of one-dimensional maps.