School of Mathematical & Statistical Sciences Faculty Publications

Maximum number of limit cycles of the piecewise linear Liénard system with three zones

Document Type

Article

Publication Date

1-2026

Abstract

For the piecewise linear Liénard system ẋ=F(x)−yx˙=F(x)−y, ẏ=xy˙=x, where F(x)F(x) is a continuous piecewise linear function with nn fold points, the question of how many limit cycles that such a system can have has been a classical and open problem in differential equations and dynamical systems. Nowadays, we only know the answer for the case n=1n=1. For the cases n⩾2n⩾2, the problem still remains open. This paper aims to give an affirmative answer for the case n=2n=2 of this open problem, that is, the maximum number of limit cycles of the continuous piecewise linear function with 2 fold points is 2. This system exhibits abundantly interesting and rich dynamics, including the generalized Hopf bifurcation, boundary equilibrium bifurcation, grazing limit cycle bifurcation, and double limit cycle bifurcation, which may have potential interdisciplinary applications.

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© 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/I9PPSAPWGR7VTRXNHQXD?target=10.1112/jlms.70430

Publication Title

Journal of the London Mathematical Society

DOI

10.1112/jlms.70430

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