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On the number of zeros of Abelian integral for some Liénard system of type (4,3)

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  • Sun, Xianbo
  • Su, Jing
  • Han, Maoan

Abstract

In this article, we study the Abelian integral M(h) corresponding to the following Liénard system,x˙=y,y˙=x3(x-1)+ε(a+bx+cx2+x3)y,where 0<ε≪1, a, b and c are real bounded parameters. Using the expansion of M(h) and a new algebraic criterion developed in Maeñosas and Villadelprat (2011) [6], we found that the lower and upper bounds of the maximal number of zeros of M are respectively 4 and 5. Hence, the above system can have 4 limit cycles and has at most 5 limit cycles bifurcating from the corresponding period annulus. The results obtained are new for this kind of Liénard system as we known.

Suggested Citation

  • Sun, Xianbo & Su, Jing & Han, Maoan, 2013. "On the number of zeros of Abelian integral for some Liénard system of type (4,3)," Chaos, Solitons & Fractals, Elsevier, vol. 51(C), pages 1-12.
    • Handle: RePEc:eee:chsofr:v:51:y:2013:i:c:p:1-12
    DOI: 10.1016/j.chaos.2013.02.003
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    References listed on IDEAS

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    1. Yang, Junmin & Han, Maoan, 2011. "Limit cycle bifurcations of some Liénard systems with a cuspidal loop and a homoclinic loop," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 269-289.
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    Cited by:

    1. Sun, Xianbo & Zhao, Liqin, 2016. "Perturbations of a class of hyper-elliptic Hamiltonian systems of degree seven with nilpotent singular points," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 194-203.
    2. Wang, Jihua, 2016. "Bound the number of limit cycles bifurcating from center of polynomial Hamiltonian system via interval analysis," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 30-38.

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