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, ε and λ, satisfying 0<ε lambda1. The corresponding first order Melnikov function M1 with respect to ε depends on λ so that it has an expansion of the form M1(h,λ)=k=0M1k(h)λk. 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<ελ1, when k=0 or 1. In addition, for each kN, an upper bound of the maximal number of zeros of M1k(h), taking into account their multiplicities, is presented.

Key words: Limit cycle, Melnikov function, integrable system

CLC Number: 

  • 34C05
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