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On the number of limit cycles in small perturbations of a piecewise linear Hamiltonian system with a heteroclinic loop

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In this paper, the authors consider limit cycle bifurcations for a kind of nonsmooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with a center at the origin and a heteroclinic loop around the origin. When the degree of perturbing polynomial terms is n (n ≥ 1), it is obtained that n limit cycles can appear near the origin and the heteroclinic loop respectively by using the first Melnikov function of piecewise near-Hamiltonian systems, and that there are at most n + [n+1/2] limit cycles bifurcating from the periodic annulus between the center and the heteroclinic loop up to the first order in ε. Especially, for n = 1, 2, 3 and 4, a precise result on the maximal number of zeros of the first Melnikov function is derived.

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Author information

Correspondence to Feng Liang.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 11271261), the Natural Science Foundation of Anhui Province (No. 1308085MA08), and the Doctoral Program Foundation (2012) of Anhui Normal University.

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Liang, F., Han, M. On the number of limit cycles in small perturbations of a piecewise linear Hamiltonian system with a heteroclinic loop. Chin. Ann. Math. Ser. B 37, 267–280 (2016).

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  • Limit cycle
  • Heteroclinic loop
  • Melnikov function
  • Chebyshev system
  • Bifurcation
  • Piecewise smooth system

2000 MR Subject Classification

  • 34C05
  • 34C07
  • 37G15