Bifurcation of three-dimensional discontinuous cycles

We consider three-dimensional discontinuous dynamical systems with non-fixed moments of impacts. Existence of the center manifold is proved for the system. The result is applied for the extension of the planar Hopf bifurcation theorem [M.U. Akhmet, Perturbations and Hopf bifurcation of the planar discontinuous dynamical system, Nonlinear Analysis 60 (2005) 163-178]. Illustrative examples are constructed for the theory. (C) 2009 Elsevier Ltd. All rights reserved.