Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (4): 1034-1056.doi: 10.1007/s10473-021-0402-z

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LIMIT CYCLE BIFURCATIONS OF A PLANAR NEAR-INTEGRABLE SYSTEM WITH TWO SMALL PARAMETERS

Feng LIANG1, Maoan HAN2, Chaoyuan JIANG1   

  1. 1. The Institute of Mathematics, Anhui Normal University, Wuhu 241000, China;
    2. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
  • Received:2019-11-18 Revised:2020-09-13 Online:2021-08-25 Published:2021-09-01
  • Contact: Maoan HAN E-mail:mahan@shnu.edu.cn
  • Supported by:
    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).

Abstract: In this paper we consider a class of polynomial planar system with two small parameters, $\varepsilon$ and $\lambda$, satisfying $0<\varepsilon\ll\ lambda\ll1$. The corresponding first order Melnikov function $M_1$ with respect to $\varepsilon$ depends on $\lambda$ 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 $M_{1k'}(h)$ is the first non-zero coefficient in the expansion. Then by estimating the number of zeros of $M_{1k'}(h)$, we give a lower bound of the maximal number of limit cycles emerging from the period annulus of the unperturbed system for $0<\varepsilon\ll\lambda\ll1$, when $k'=0$ or $1$. In addition, for each $k\in \mathbb{N}$, an upper bound of the maximal number of zeros of $M_{1k}(h)$, taking into account their multiplicities, is presented.

Key words: Limit cycle, Melnikov function, integrable system

CLC Number: 

  • 34C05
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