Limit cycles and bifurcations in a class of planar piecewise linear systems with a nonregular separation line

Abstract

In this paper, we consider a class of planar piecewise linear differential systems with a nonregular separation line, which can be transformed to a normal form with only 5 parameters under some conditions. We study the coexistence of crossing limit cycles and sliding limit cycles by establishing Poincaré maps for two subsystems that have the same Jacobi matrix. Using expressions and properties of Poincaré maps, we can clarify the relationship between trajectories of two subsystems. Our main results reveal that (1) the numbers of crossing limit cycles and sliding limit cycles can be 2; (2) the coexistence number of limit cycles can be 3. Moreover, by a complete discussion of classification and analysis, we show saddle-node bifurcation and critical crossing cycle bifurcation existing in generic Filippov systems. Finally, four numerical examples are given to verify the correctness of the obtained theoretical results.