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On the limit cycles of perturbed discontinuous planar systems with 4 switching lines

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  • Wang, Yanqin
  • Han, Maoan
  • Constantinescu, Dana

Abstract

Limit cycle bifurcations for a class of perturbed planar piecewise smooth systems with 4 switching lines are investigated. The expressions of the first order Melnikov function are established when the unperturbed system has a compound global center, a compound homoclinic loop, a compound 2-polycycle, a compound 3-polycycle or a compound 4-polycycle, respectively. Using Melnikov’s method, we obtain lower bounds of the maximal number of limit cycles for the above five different cases. Further, we derive upper bounds of the number of limit cycles for the later four different cases. Finally, we give a numerical example to verify the theory results.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:chsofr:v:83:y:2016:i:c:p:158-177
    DOI: 10.1016/j.chaos.2015.11.041
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    References listed on IDEAS

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    1. Buzzi, C.A. & Llibre, J. & Medrado, J.C., 2011. "On the limit cycles of a class of piecewise linear differential systems in ℝ4 with two zones," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 82(4), pages 533-539.
    2. 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.
    3. Huanhuan Tian & Maoan Han, 2014. "Limit Cycle Bifurcations by Perturbing a Compound Loop with a Cusp and a Nilpotent Saddle," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-14, July.
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    Cited by:

    1. Yang, Jihua, 2021. "On the number of limit cycles of a pendulum-like equation with two switching lines," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    2. Liang, Feng & Romanovski, Valery G. & Zhang, Daoxiang, 2018. "Limit cycles in small perturbations of a planar piecewise linear Hamiltonian system with a non-regular separation line," Chaos, Solitons & Fractals, Elsevier, vol. 111(C), pages 18-34.
    3. Yang, Jihua, 2020. "Limit cycles appearing from the perturbation of differential systems with multiple switching curves," Chaos, Solitons & Fractals, Elsevier, vol. 135(C).
    4. Chen, Ting & Huang, Lihong & Huang, Wentao & Li, Wenjie, 2017. "Bi-center conditions and local bifurcation of critical periods in a switching Z2 equivariant cubic system," Chaos, Solitons & Fractals, Elsevier, vol. 105(C), pages 157-168.

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