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Cyclicity of period annulus for a class of quadratic reversible systems with a nonrational first integral

Part of: Qualitative theory Local and nonlocal bifurcation theory

Published online by Cambridge University Press:  09 November 2022

Xiuli Cen ,
Changjian Liu ,
Yangjian Sun  and
Jihua Wang
Xiuli Cen
Affiliation:
School of Mathematics and Statistics, HNP-LAMA, Central South University, Changsha, Hunan 410083, P.R. China (cenxiuli2010@163.com)
Changjian Liu
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, Guangdong 519082, P.R. China (liuchangj@mail.sysu.edu.cn)
Yangjian Sun
Affiliation:
School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, P.R. China (syj1508556017@163.com)
Jihua Wang
Affiliation:
School of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong 512075, P.R. China (wangjh78@mail.sysu.edu.cn)
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Abstract

In this paper, we study the quadratic perturbations of a one-parameter family of reversible quadratic systems whose first integral contains the logarithmic function. By the criterion function for determining the lowest upper bound of the number of zeros of Abelian integrals, we obtain that the cyclicity of either period annulus is two. To the best of our knowledge, this is the first result for the cyclicity of period annulus of the one-parameter family of reversible quadratic systems whose first integral contains the logarithmic function. Moreover, the simultaneous bifurcation and distribution of limit cycles from two-period annuli are considered.

Keywords

Abelian integrallimit cycleChebyshev systemquadratic reversible systemsimultaneous bifurcation and distribution

MSC classification

Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications)
Secondary: 34C08: Connections with real algebraic geometry (fewnomials, desingularization, zeros of Abelian integrals, etc.) 37G15: Bifurcations of limit cycles and periodic orbits
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 5 , October 2023 , pp. 1706 - 1728
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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