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Limit cycles in a rotated family of generalized Liénard systems allowing for finitely many switching lines

Part of: Qualitative theory

Published online by Cambridge University Press:  21 March 2024

Hebai Chen Open the ORCID record for Hebai Chen [Opens in a new window] ,
Yilei Tang Open the ORCID record for Yilei Tang [Opens in a new window]  and
Weinian Zhang
Hebai Chen
Affiliation:
School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P. R. China (chen_hebai@csu.edu.cn)
Yilei Tang
Affiliation:
School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, P. R. China (mathtyl@sjtu.edu.cn)
Weinian Zhang
Affiliation:
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China (wnzhang@scu.edu.cn)
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Abstract

Analytic rotated vector fields have four significant properties: as the rotated parameter $\alpha$ changes, the amplitude of each stable (or unstable) limit cycle varies monotonically, each semi-stable limit cycle bifurcates at most two limit cycles, the isolated homoclinic loop (if exists) disappears while a unique limit cycle with the same stability arises or no closed orbits arise oppositely, and a unique limit cycle arises near the weak focus (if exists). In this paper, we prove that the four properties remain true for a rotated family of generalized Liénard systems having finitely many switching lines. Furthermore, we discuss variational exponent and use it to formulate multiplicity of limit cycles. Then we apply our results to give exact number of limit cycles to a continuous piecewise linear system with three zones and answer to a question on the maximum number of limit cycles in an SD oscillator.

Keywords

rotated vector fieldswitching generalized Liénard systemlimit cyclehomoclinic loop

MSC classification

Primary: 34C05: Location of integral curves, singular points, limit cycles
Secondary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) 34C23: Bifurcation 34C37: Homoclinic and heteroclinic solutions

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 155 , Issue 6 , December 2025 , pp. 2121 - 2149
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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