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Limit Cycle Bifurcations of a Planar Near-Integrable System with Two Small Parameters

Abstract

In this paper we consider a class of polynomial planar system with two small parameters, ε and λ, satisfying 0 < ελ ≪ 1. The corresponding first order Melnikov function M1 with respect to ε depends on λ so that it has an expansion of the form \({M_1}(h,\lambda ) = \sum\limits_{k = 0}^\infty {{M_{1k}}(h){\lambda ^k}} \). Assume that M1k′ (h) is the first non-zero coefficient in the expansion. Then by estimating the number of zeros of M1k′ (h), we give a lower bound of the maximal number of limit cycles emerging from the period annulus of the unperturbed system for 0 < ελ ≪ 1, when k′ = 0 or 1. In addition, for each k ∈ ℕ, an upper bound of the maximal number of zeros of M1k(h), taking into account their multiplicities, is presented.

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Correspondence to Maoan Han.

Additional information

The first author is supported by the National Natural Science Foundation of China (11671013); the second author is supported by the National Natural Science Foundation of China (11771296).

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Liang, F., Han, M. & Jiang, C. Limit Cycle Bifurcations of a Planar Near-Integrable System with Two Small Parameters. Acta Math Sci 41, 1034–1056 (2021). https://doi.org/10.1007/s10473-021-0402-z

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  • DOI: https://doi.org/10.1007/s10473-021-0402-z

Key words

  • Limit cycle
  • Melnikov function
  • integrable system

2010 MR Subject Classification

  • 34C05
  • 37C10