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The cyclicity of the period annulus of a reversible quadratic system

Part of: Local and nonlocal bifurcation theory Qualitative theory

Published online by Cambridge University Press:  10 February 2021

Changjian Liu ,
Chengzhi Li  and
Jaume Llibre
Changjian Liu
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai519086, China (liuchangj@mail.sysu.edu.cn)
Chengzhi Li
Affiliation:
School of Mathematical Sciences, Peking University, Beijing100871, China (licz@math.pku.edu.cn)
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain (jllibre@mat.uab.cat)
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Abstract

We prove that perturbing the periodic annulus of the reversible quadratic polynomial differential system $\dot x=y+ax^2$, $\dot y=-x$ with a ≠ 0 inside the class of all quadratic polynomial differential systems we can obtain at most two limit cycles, including their multiplicities. Since the first integral of the unperturbed system contains an exponential function, the traditional methods cannot be applied, except in Figuerasa, Tucker and Villadelprat (2013, J. Diff. Equ., 254, 3647–3663) a computer-assisted method was used. In this paper, we provide a method for studying the problem. This is also the first purely mathematical proof of the conjecture formulated by Dumortier and Roussarie (2009, Discrete Contin. Dyn. Syst., 2, 723–781) for q ⩽ 2. The method may be used in other problems.

Keywords

Perturbation of quadratic reversible centreAbelian integrallimit cycle

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 152 , Issue 2 , April 2022 , pp. 281 - 290
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

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