Limit cycles near homoclinic and heteroclinic loops
type: Item
instance of: scholarly article
title: Limit cycles near homoclinic and heteroclinic loops
zbMATH Open document ID: 1165.34016
author: Junmin Yang
author: Alexandrina-Alina Tarţa
author: Yang Gao
author: Maoan Han
published in: Journal of Dynamics and Differential Equations
publication date: 2008-12-01T00:00:00Z
review text: The paper deals with a near-Hamiltonian system in the form of \[ \dot{x}=H_y + \varepsilon p(x,y,\varepsilon,\delta), \quad \dot{y}=-H_x + \varepsilon q(x,y,\varepsilon,\delta), \] where \(H(x,y)\), \(p\) and \(q\) are analytic functions in \((x,y)\in \mathbb{R}^2\), and \(p\) and \(q\) being \(C^1\) in a small real parameter \(\varepsilon \geq 0\), \(\delta \in D \subset \mathbb{R}^m\) is a vector parameter with \(D\) compact. The main purpose is to give an explicit formula to compute the first four coefficients appeared in the expansion of the first order Melnikov function for both homoclinic and heteroclinic cases and to present a way to find limit cycles near the loops by using these coefficients. An application of the obtained results to a class of polynomial systems with three limit cycles is presented.
reviewed by: Alexander Grin
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37G20
zbMATH DE Number: 5375873
zbMATH Keywords: near-Hamiltonian system
zbMATH Keywords: limit cycle
zbMATH Keywords: Melnikov function
zbMATH Keywords: bifurcation
zbMATH Keywords: 16-th Hilbert's problem
zbMATH Keywords: homoclinic loop
zbMATH Keywords: heteroclinic loop
MaRDI profile type: Publication
full work available at URL: s10884-008-9108-3
OpenAlex ID: W1979832219
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DOI: 10.1007/S10884-008-9108-3
description: scientific article; zbMATH DE number 5375873