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On limit cycles of the Liénard equation with Z2 symmetry

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  • Yu, P.
  • Han, M.

Abstract

This paper considers the limit cycles in the Liénard equation, described by x¨+f(x)x˙+g(x)=0, with Z2 symmetry (i.e., the vector filed is symmetric with the y-axis). Particular attention is given to the existence of small-amplitude (local) limit cycles around fine focus points when g(x) is a third-degree, odd polynomial function and f(x) is an even function. Such a system has three fixed points on the x-axis, with one saddle point at the origin and two linear centres which are symmetric with the origin. Based on normal form computation, it is shown that such a system can generate more limit cycles than the existing results for which only the origin is considered. In general, such a Liénard equation can have 2m small limit cycles, i.e., H(2m,3)⩾2m, where H denotes the Hilbert number of the system, 2m and 3 are the degrees of f and g, respectively.

Suggested Citation

  • Yu, P. & Han, M., 2007. "On limit cycles of the Liénard equation with Z2 symmetry," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 617-630.
    • Handle: RePEc:eee:chsofr:v:31:y:2007:i:3:p:617-630
    DOI: 10.1016/j.chaos.2005.10.013
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    References listed on IDEAS

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    1. Yu, P. & Han, M., 2005. "Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields," Chaos, Solitons & Fractals, Elsevier, vol. 24(1), pages 329-348.
    2. Wang, S. & Yu, P., 2005. "Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation," Chaos, Solitons & Fractals, Elsevier, vol. 26(5), pages 1317-1335.
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    Cited by:

    1. Sabatini, M., 2010. "Existence and uniqueness of limit cycles in a class of second order ODE’s with inseparable mixed terms," Chaos, Solitons & Fractals, Elsevier, vol. 43(1), pages 25-30.
    2. Zhang, Chunrui & Zheng, Baodong, 2009. "Bifurcation in Z2-symmetry quadratic polynomial systems with delay," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3078-3086.

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