Melnikov function and limit cycle bifurcation from a nilpotent center
type: Item
instance of: scholarly article
title: Melnikov function and limit cycle bifurcation from a nilpotent center
zbMATH Open document ID: 1143.37043
author: Jiao Jiang
author: Maoan Han
published in: Bulletin des Sciences Mathématiques
publication date: 2008-05-15T00:00:00Z
review text: The near-Hamiltonian planar system \(\dot{x}=H_y+\varepsilon f(x,y)\), \(\dot{y}=-H_x+\varepsilon g(x,y)\) is considered, where \(H\), \(f\) and \(g\) are polynomials in \(x,y\) and \(\varepsilon>0\) is a small parameter when the unperturbed system has a nilpotent center. It is assumed that the equation \(H(x,y)=h\) defines a closed curve \(L_h\) surrounding the origin for \(h\) in an interval \(I\subset\mathbb{R}\), which intersects the positive \(x\)-axis at the point \(A(h)=(a(h),0)\). Let \(B(h,\varepsilon)\) be the first intersection point of the positive orbit of the perturbed system starting at \(A(h)\) with the positive \(x\)-axis. Then \(H(B)-H(A)=\int_{AB}dH=\varepsilon[M(h)+O(\varepsilon)]\), where the Abel integral \(M(h)=\oint_{L_h}(g\,dx-f\,dy)\) is the first-order Melnikov function of the system. The aim of the article is to give an algorithm for computation the first coefficients of the expansion of \(M(h)\). Application of this algorithm is given and the number of limit cycles for a cubic system is obtained.
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37J20
Mathematics Subject Classification ID: 37G15
Mathematics Subject Classification ID: 37G40
zbMATH DE Number: 5275728
zbMATH Keywords: near-Hamiltonian systems
zbMATH Keywords: nilpotent center
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: Melnikov function
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: j.bulsci.2006.11.006
OpenAlex ID: W2027696719
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DOI: 10.1016/J.BULSCI.2006.11.006
description: scientific article; zbMATH DE number 5275728