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THE NUMBER OF LIMIT CYCLES BIFURCATING FROM THE PERIOD ANNULUS OF QUASI-HOMOGENEOUS HAMILTONIAN SYSTEMS AT ANY ORDER

Abstract : A necessary and sufficient condition is given for quasi-homogeneous polynomial Hamiltonian systems having a center. Then it is shown that there exists a bound on the number of limit cycles bifurcating from the period annulus of quasi-homogeneous Hamiltonian systems at any order of Melnikov functions; and the explicit expression of this bound is given in terms of , where n is the degree of perturbation polynomials, k is the order of the first nonzero higher order Melnikov function, and is the weight exponent of quasi-homogeneous Hamiltonian with center. This extends some known results and solves the Arnol'd-Hilbert's 16th problem for the perturbations of homogeneous or quasi-homogeneous polynomial Hamiltonian systems.
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Submitted on : Friday, February 19, 2021 - 12:34:12 PM
Last modification on : Sunday, February 21, 2021 - 3:21:55 AM

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Jean-Pierre Françoise, Hongjin He, Dongmei Xiao. THE NUMBER OF LIMIT CYCLES BIFURCATING FROM THE PERIOD ANNULUS OF QUASI-HOMOGENEOUS HAMILTONIAN SYSTEMS AT ANY ORDER. Journal of Differential Equations, Elsevier, 2021, 276, pp.318-341. ⟨10.1016/j.jde.2020.12.015⟩. ⟨hal-03146819⟩

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