On the limit cycles of planar discontinuous piecewise linear differential systems with a unique equilibrium

Abstract

This paper deals with planar discontinuous piecewise linear differential systems with two zones separated by a vertical straight line $ x = k $. We assume that the left linear differential system ($ x<k $) and the right linear differential system ($ x>k $) share the same equilibrium, which is located at the origin $ O(0, 0) $ without loss of generality.Our results show that if $ k = 0 $, that is when the unique equilibrium $ O(0, 0) $ is located on the line of discontinuity, then the discontinuous piecewise linear differential systems have no crossing limit cycles. While for the case $ k\neq 0 $ we provide lower and upper bounds for the number of limit cycles of these planar discontinuous piecewise linear differential systems depending on the type of their linear differential systems, i.e. if those systems have foci, centers, saddles or nodes, see Table 2.