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Limit Cycle Bifurcations from an Order-3 Nilpotent Center of Cubic Hamiltonian Systems Perturbed by Cubic Polynomials

Li Zhang

School of Statistics and Mathematics, Shanghai Lixin University of Accounting and Finance, Shanghai 201209, P. R. China

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Chenchen Wang

Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China

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, and

Zhaoping Hu

http://orcid.org/0000-0001-9071-9374

Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China

E-mail Address: zhaopinghu@shu.edu.cn

Corresponding author.

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https://doi.org/10.1142/S0218127420501266Cited by:1 (Source: Crossref)

Abstract

From [Han et al., 2009a] we know that the highest order of the nilpotent center of cubic Hamiltonian system is $3$ . In this paper, perturbing the Hamiltonian system which has a nilpotent center of order $3$ at the origin by cubic polynomials, we study the number of limit cycles of the corresponding cubic near-Hamiltonian systems near the origin. We prove that we can find seven and at most seven limit cycles near the origin by the first-order Melnikov function.

The project was supported by National Nature Science Foundation of China (11401366, 11632008, 11572181).

Keywords:

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