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Limit cycles near a nilpotent center and a homoclinic loop to a nilpotent singularity of Hamiltonian systems

Part of: Qualitative theory Local and nonlocal bifurcation theory

Published online by Cambridge University Press:  24 October 2025

Lijun Wei ,
Maoan Han ,
Jilong He  and
Xiang Zhang
Lijun Wei*
Affiliation:
School of Mathematics, Hangzhou Normal University , Hangzhou 310036, P. R. China
Maoan Han
Affiliation:
School of Mathematical Sciences, Zhejiang Normal University , Jinhua 321004, P. R. China e-mail: mahan@shnu.edu.cn
Jilong He
Affiliation:
Department of Mathematics, Huzhou University , Huzhou 313000, P. R. China e-mail: hjl_csu@163.com
Xiang Zhang
Affiliation:
School of Mathematical Sciences, MOE–LSC, and CAM-Shanghai, Shanghai Jiao Tong University , Shanghai 200240, P. R. China e-mail: xzhang@sjtu.edu.cn
*
e-mail: weilijun39@126.com
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Abstract

For a planar analytic Hamiltonian system, which has a period annulus limited by a nilpotent center and a homoclinic loop to a nilpotent singularity, we study its analytic perturbation to obtain the number of limit cycles bifurcated from the periodic orbits inside the period annulus. By characterizing the coefficients and their properties of the high-order terms in the expansion of the first-order Melnikov function near the loop, we provide a new way to find more limit cycles. Moreover, we apply these general results to concrete systems, for instance, an $(m+1)$th-order generalized Liénard system, and an mth-order near-Hamiltonian system with a hyperelliptic Hamiltonian of degree $6$.

Keywords

Melnikov functionlimit cyclenilpotent centerhomoclinic loopnilpotent singularity

MSC classification

Primary: 34C07: Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert's 16th problem and ramifications) 37G15: Bifurcations of limit cycles and periodic orbits 34C05: Location of integral curves, singular points, limit cycles

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 36
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work is partially supported by the National Key R&D Program of China (Grant No. 2022YFA1005900). The first author is partially supported by the NSF of Zhejiang Province (Grant No. LY24A010 009), by the NNSF of China (Grant Nos. 12471156 and 12371167), and by the Shanghai Frontier Research Center of Modern Analysis. The second author is partially supported by the NNSF of China (Grant No. 11931016). The fourth author is partially supported by the NNSF of China (Grant No. 12471169).

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