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Limit cycle bifurcations by perturbing a cubic Hamiltonian system with a heteroclinic loop

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  • Gao, Yiwen
  • Xiong, Yanqin

Abstract

This paper investigates a cubic Hamiltonian system described by the equations ẋ=y−ax3,ẏ=−bx+3ax2y+cx3,a≠0under polynomial perturbations of degree n. Initially, we present a detailed classification of all possible phase portraits of the unperturbed system in the phase plane. Through rigorous analysis, we derive the necessary and sufficient conditions under which heteroclinic loops can emerge in the system. By utilizing the first-order Melnikov function, this research explores limit cycle bifurcations in perturbed systems that exhibit heteroclinic loops in their unperturbed forms. An upper bound is established for the number of limit cycles that can bifurcate from periodic orbits of the unperturbed system, which is given by 7[n−12]+1.

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  • Gao, Yiwen & Xiong, Yanqin, 2025. "Limit cycle bifurcations by perturbing a cubic Hamiltonian system with a heteroclinic loop," Chaos, Solitons & Fractals, Elsevier, vol. 199(P3).
    • Handle: RePEc:eee:chsofr:v:199:y:2025:i:p3:s0960077925008781
    DOI: 10.1016/j.chaos.2025.116865
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