Limit Cycles and Bifurcations in a Class of Discontinuous Piecewise Linear Systems

Abstract

In this paper, we consider planar piecewise linear differential systems with a line of discontinuity sharing a linear part. We study not only the number of crossing limit cycles, but also the number of sliding ones, and the coexistence of two configurations of limit cycles. In particular, we proved that both numbers of crossing limit cycles and sliding ones are at most [Formula: see text], but the total number of limit cycles is at most [Formula: see text]. Finally, by complete analysis on the number of limit cycles, we show some bifurcations which exist in generic Filippov systems, revealing also two nongeneric bifurcations.