Constructed limit cycles in a discontinuous system with multiple vector fields

Abstract

In this paper, limit cycles in a discontinuous dynamical system with different vector fields in different domains are constructed. Such limit cycles are obtained through algebraic equations based on specific mapping structures. The stability and bifurcations of limit cycles are studied through eigenvalue analysis. The grazing and sliding bifurcations on the bifurcation trees are presented. Once a grazing or sliding bifurcation occurs, a limit cycle switches to a new limit cycle or vanishes. Such bifurcations (i.e., saddle-node bifurcation, Neimark bifurcation, period-doubling bifurcation, grazing bifurcation, and sliding bifurcation) are for onset and vanishing of limit cycles at specific parameters. This study presents how to construct limit cycles in discontinuous dynamical systems and how to develop the corresponding mathematical conditions of motion switchability at boundaries.