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Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems

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  • Liang, Feng
  • Han, Maoan

Abstract

In this paper, we study the bifurcation of limit cycles in piecewise smooth systems by perturbing a piecewise Hamiltonian system with a generalized homoclinic or generalized double homoclinic loop. We first obtain the form of the expansion of the first Melnikov function. Then by using the first coefficients in the expansion, we give some new results on the number of limit cycles bifurcated from a periodic annulus near the generalized (double) homoclinic loop. As applications, we study the number of limit cycles of a piecewise near-Hamiltonian systems with a generalized homoclinic loop and a central symmetric piecewise smooth system with a generalized double homoclinic loop.

Suggested Citation

  • Liang, Feng & Han, Maoan, 2012. "Limit cycles near generalized homoclinic and double homoclinic loops in piecewise smooth systems," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 454-464.
    • Handle: RePEc:eee:chsofr:v:45:y:2012:i:4:p:454-464
    DOI: 10.1016/j.chaos.2011.09.013
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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.
    2. Wu, Yuhai & Gao, Yongxi & Han, Maoan, 2008. "Bifurcations of the limit cycles in a z3-equivariant quartic planar vector field," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1177-1186.
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    Cited by:

    1. Erli Zhang & Stanford Shateyi, 2023. "Exploring Limit Cycle Bifurcations in the Presence of a Generalized Heteroclinic Loop," Mathematics, MDPI, vol. 11(18), pages 1-12, September.
    2. Wang, Yanqin & Han, Maoan & Constantinescu, Dana, 2016. "On the limit cycles of perturbed discontinuous planar systems with 4 switching lines," Chaos, Solitons & Fractals, Elsevier, vol. 83(C), pages 158-177.
    3. Liu, Yuanyuan & Xiong, Yanqin, 2014. "Limit cycles for perturbing a piecewise linear Hamiltonian system with one or two saddles," Chaos, Solitons & Fractals, Elsevier, vol. 66(C), pages 86-95.

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