The discontinuous planar piecewise linear systems with two improper nodes have at most one limit cycle

Abstract

The existence and number of limit cycles of planar piecewise linear systems with two improper nodes are studied. By constructing the Poincaré half maps and the successor function, we prove that such systems have at most one limit cycle, and when the limit cycle exists, it must be hyperbolic. Furthermore, we explicitly give the parameter regions where the limit cycle exists.