The successive derivatives of the period function of a plane vector field
type: Item
instance of: scholarly article
title: The successive derivatives of the period function of a plane vector field
zbMATH Open document ID: 0943.34021
author: Jean-Pierre Francoise
published in: Journal of Differential Equations
publication date: 1999-10-07T00:00:00Z
review text: Previously, the author provided an expression which generalized the classical Melnikov function to any order, for the first nonzero derivative of a return mapping [the author, Publ. Mat., Barc. 41, No. 1, 127-134 (1997; Zbl 0892.34023)]. This method relied on the decomposition of a 1-form associated to the relative cohomology of the perturbed Hamiltonian. With the same techniques, he gives a formula for the first nonzero derivative of a period function. He focuses on the particular example of \(H= (1/2)(x^2+ y^2)\) and then he defines a class of Hamiltonians for which the same computation remains valid. Finally, he investigates relations with the Birkhoff normal form.
reviewed by: C. Chicone
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 37K05
zbMATH DE Number: 1212364
zbMATH Keywords: period function
zbMATH Keywords: differential forms
zbMATH Keywords: planar system
zbMATH Keywords: relative cohomology
zbMATH Keywords: perturbed Hamiltonian
zbMATH Keywords: Birkhoff normal form
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3437
OpenAlex ID: W1985051324
cites work: Bifurcation of Critical Periods for Plane Vector Fields
cites work: Successive derivatives of a first return map, application to the study of quadratic vector fields
cites work: Q4352419
cites work: The first derivative of the period function of a plane vector field
cites work: On the period spectrum of a symplectic mapping
cites work: Q5687391
cites work: An algorithmic approach to the center problem for homogeneous perturbations
cites work: Q4394773
cites work: Higher-order Melnikov functions for degenerate cubic Hamiltonians
cites work: Q3338368
cites work: Quadratic Isochronous centers commute
cites work: Preuve d'une conjecture de Brieskorn
cites work: regularity properties of the period function near a center of a planar vector field
DOI: 10.1006/JDEQ.1998.3437
description: scientific article; zbMATH DE number 1212364