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The Successive Derivatives of the Period Function of a Plane Vector Field

By JEAN-PIERRE FRANCOISE

Abstract

International audiencePreviously, we provided an expression which generalized the classical Melnikov function to any order, for the first nonzero derivative of a return mapping. Our method relied on the decomposition of a 1-form associated to the relative cohomology of the perturbed Hamiltonian. With the same techniques, we give a formula for the first nonzero derivative of a period function. We focus on the particular example ofH=(1/2)(x2+y2) and then we define a class of Hamiltonians for which the same computation remains valid. Finally, we investigate relations with Birkhoff normal form

Topics: bifurcation theory, [ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS]
Publisher: Elsevier
Year: 1998
DOI identifier: 10.1006/jdeq.1998.3437
OAI identifier: oai:HAL:hal-01401585v1
Provided by: Hal-Diderot
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