Bifurcations in four-dimensional switched systems

In this paper, the focus is on a bifurcation of period- K orbit that can occur in a class of Filippov- type four- dimensional homogenous linear switched systems. We introduce a theoretical framework for analyzing the generalized Poincare map corresponding to switching manifold. This provides an approach to capturing the possible results concerning the existence of a period- K orbit, stability, a number of invariant cones, and related bifurcation phenomena. Moreover, the analysis identifies criteria for the existence of multi- sliding bifurcation depending on the sensitivity of the system behavior with respect to changes in parameters. Our results show that a period- two orbit involves multi- sliding bifurcation from a period- one orbit. Further, the existence of invariant torus, crossing- sliding, and grazing- sliding bifurcation is investigated. Numerical simulations are carried out to illustrate the results.