A generalization of Françoise's algorithm for calculating higher order Melnikov functions
type: Item
instance of: scholarly article
title: A generalization of Françoise's algorithm for calculating higher order Melnikov functions
zbMATH Open document ID: 1029.34081
DOI: 10.1016/S0007-4497(02)01138-7
author: Ahmed Jebrane
author: Pavao Mardešić
author: Michèle Pelletier
published in: Bulletin des Sciences Mathématiques
publication date: 2003-03-26T00:00:00Z
review text: Let \(F(x,y)\in \mathbb{C}[x,y]\) be a complex polynomial. It is known that a finite set \(A\) of atypical values such that \(F\) defines a locally trivial fibration \(F: \mathbb{C}^2\setminus F^2(B)\to B\), \(B= \mathbb{C}\setminus A\), is called Milnor fibration. Taking the first homology groups \(H_1\) of the fibers, \(F_t= F^{-1}(t)\), \(t\in B\), gives the homology fibration, \(F_*: \bigcup_{t\in B}H_1(B)\). Let \(C\) be a section of the homology fibration defined on \(B\). Consider a polynomial one-form \(\omega\). We study the perturbation of the Hamiltonian form \(dF\), \(dF+\varepsilon\omega\). Choose a transveral parameterized by the values of the Hamiltonian, and denote \(P_\varepsilon(t)\) the holonomy of the foliation, \(dF+\varepsilon\omega\) along \(C\). It is analytic and can be written as \(P_\varepsilon(t)= t+\sum_{l\geq 1} \varepsilon^l M_l(t)\). If everything is real, then \(P_\varepsilon(t)\) is the Poincaré map. The functions \(M_\varepsilon(t)\) are called the Melnikov functions. Françoise gives an algorithm for calculating the first nonvanishing Melnikov function, \(M_l\), of a small polynomial perturbation of a Hamiltonian vector field and shows that \(M_l\) is given by an Abelian integral. This is done under the condition that vanishing of an Abelian integral of any polynomial form on the family of cycles implies that the form is algebraically relatively exact. The authors study a simple example where Françoise's condition is not verified. They generalize Françoise's algorithm to this case and show that \(M_l\) belongs to the \(C[\log t,t,1/t]\) module above the Abelian integrals. Wonderful illustrations are given when lifting the path corresponds to lifting four layers in the fiber. The paper is very well written and contains much of the necessary mathematics to make very understandable.
Mathematics Subject Classification ID: 34M35
Mathematics Subject Classification ID: 34M10
Mathematics Subject Classification ID: 14D05
zbMATH DE Number: 1888297
zbMATH Keywords: Françoise's algorithm
zbMATH Keywords: Fuchs system
zbMATH Keywords: Abelian integral
zbMATH Keywords: Melnikov functions
zbMATH Keywords: limit cycle
zbMATH Keywords: Milnor fibration
reviewed by: John Schmeelk
MaRDI profile type: Publication
cites work: Q3808900
cites work: Quelques calculs de cohomologie relative
cites work: Q3041331
cites work: Singularities and topology of hypersurfaces
cites work: The successive derivatives of the period function of a plane vector field
cites work: Q4850048
cites work: Petrov modules and zeros of abelian integrals
cites work: Affine upper bounds for the number of zeros of abelian integrals for quartic elliptic Hamiltonians
cites work: THE ORIGIN OF LIMIT CYCLES UNDER PERTURBATION OF THE EQUATIONdw/dz= -Rz/Rw, WHERER(z,w) IS A POLYNOMIAL
full work available at URL: s0007-4497(02)01138-7
OpenAlex ID: W2024476209
Recommended article: A note on higher order Melnikov functions
Recommended article: Q3595100
Recommended article: Q4299078
Recommended article: Q3356688
Recommended article: Numerical Algorithm for Calculating the Generalized Mittag-Leffler Function
Recommended article: Q4253956
Recommended article: Q4289002
Recommended article: High-precision computation of the confluent hypergeometric functions via Franklin-Friedman expansion
Recommended article: Higher-order Melnikov method
Recommended article: A new method to compute Mathieu functions
description: scientific article; zbMATH DE number 1888297