Poincare Maps of "<"-Shape Planar Piecewise Linear Dynamical Systems with a Saddle

It is important in the study of limit cycles to investigate the properties of Poincare maps of discontinuous dynamical systems. In this paper, we focus on a class of planar piecewise linear dynamical systems with "<"-shape regions and prove that the Poincare map of a subsystem with a saddle has at most one inflection point which can be reached. Furthermore, we show that one class of such systems with a saddle-center has at least three limit cycles; a class of such systems with saddle and center in the normal form has at most one limit cycle which can be reached; and a class of such systems with saddle and center at the origin has at most three limit cycles with a lower bound of two. We try to reveal the reasons for the increase of the number of limit cycles when the discontinuity happens to a system.