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High order Melnikov functions and the problem of uniformity in global bifurcation

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In studying bifurcation of Hamiltonian system with small perturbation, it can not always be obtained the uniqueness of limit cycle uniformly for0 < ε ≪ 1 from monotonicity of I2(h)/I1(h), in case the first order Melnikov function M1(h)=μI1(h) + vI2(h) is degenerate at the end points of the interval of its definition. In this paper two examples have been constructed for showing the above conclusion.


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This work was partially supported by Natural Science Foundation of China.

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Zhi-Fen, Z., Bao-Yi, L. High order Melnikov functions and the problem of uniformity in global bifurcation. Annali di Matematica 161, 181–212 (1992). https://doi.org/10.1007/BF01759638

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  • Periodic Orbit
  • Hamiltonian System
  • Homoclinic Orbit
  • Closed Orbit
  • Lower Order Term