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Limit cycles near a homoclinic loop in two classes of piecewise smooth near-Hamiltonian systems

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  • Ma, Deyue
  • Yang, Junmin

Abstract

For two classes of piecewise smooth near-Hamiltonian systems, by studying some properties of the expansions of two Melnikov functions near a homoclinic loop, we give a simple relation between the coefficients of hj(j≥0,j∈Z) appearing in the two expansions. Based on this, we further give a general condition for each of the two systems to have as many as possible limit cycles near the homoclinic loop. Hence, by using the above main results and some techniques we obtain a lower bound of the maximum number of limit cycles near a homoclinic loop for each of two concrete systems with polynomial perturbations of degree n(n≥1).

Suggested Citation

  • Ma, Deyue & Yang, Junmin, 2025. "Limit cycles near a homoclinic loop in two classes of piecewise smooth near-Hamiltonian systems," Chaos, Solitons & Fractals, Elsevier, vol. 192(C).
    • Handle: RePEc:eee:chsofr:v:192:y:2025:i:c:s0960077925000402
    DOI: 10.1016/j.chaos.2025.116027
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    References listed on IDEAS

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    1. Yang, Junmin & Han, Maoan, 2024. "Limit cycles near a compound cycle in a near-Hamiltonian system with smooth perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
    2. Llibre, Jaume & Salhi, Tayeb, 2022. "On the limit cycles of the piecewise differential systems formed by a linear focus or center and a quadratic weak focus or center," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    3. 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.
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