Maximum number of limit cycles of the discontinuous piecewise linear Liénard system with one switching line

Abstract

For the discontinuous piecewise linear Liénard system where is an piecewise linear function, determining the maximal number of limit cycles is a classical and long‐standing open problem in the theory of differential equations and dynamical systems. Despite significant advances over the years, this problem remains unresolved in its full generality. In this work, we study this question for the case and rigorously prove that a discontinuous piecewise linear Liénard system with a single discontinuity admits at most two limit cycles. To establish this result, we conduct a comprehensive analysis of the system's global phase portraits and bifurcation structures. Our study reveals a rich spectrum of dynamical phenomena, including the coexistence of crossing and sliding limit cycles, grazing limit cycle bifurcations, and double limit cycle bifurcations. These findings not only clarify the global dynamics of the system in such a discontinuous setting but also underscore the intricate behavior induced by nonsmoothness, suggesting potential relevance to applications across multiple disciplines.