School of Mathematical & Statistical Sciences Faculty Publications

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

Document Type

Article

Publication Date

6-2026

Abstract

For the discontinuous piecewise linear Liénard system ̇𝑥=𝐹⁡(𝑥)−𝑦,̇𝑦=𝑥,

where 𝐹⁡(𝑥) is an 𝑛 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 𝑛 =1 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.

Comments

© 2026 The Author(s). The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

https://londmathsoc.onlinelibrary.wiley.com/share/ZSMM94BD7Y9NEXBRBPZ4?target=10.1112/jlms.70612

Publication Title

Journal of the London Mathematical Society

DOI

10.1112/jlms.70612

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