Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order \(m\)
type: Item
instance of: scholarly article
title: Limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order \(m\)
zbMATH Open document ID: 1406.34068
author: Junmin Yang
author: P. Yu
author: Maoan Han
published in: Journal of Differential Equations
publication date: 2018-11-06T00:00:00Z
review text: The authors study the explicit expansion of the first order Melnikov function near a double homoclinic loop passing through a nilpotent saddle of order \(m\) in a near-Hamiltonian system. For \(m \geq 1\), they derive the formulas of the coefficients in the expansion, which can be used to study limit cycle bifurcations. In particular, for \(m = 2\), these coefficients are used to consider the limit cycle bifurcations of general near-Hamiltonian systems. Further, they give the existence conditions and the distributions for \(10, 11, 13, 15\) and \(16\) (respectively \(11, 13\) and \(16\)) limit cycles when the homoclinic loop is of cuspidal type (respectively smooth type). As an application, they consider a near-Hamiltonian system with a nilpotent saddle of order \(2\) and obtain the lower bounds of the maximal number of limit cycles.
reviewed by: Stathis Antoniou
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 6973898
zbMATH Keywords: limit cycle
zbMATH Keywords: bifurcation
zbMATH Keywords: nilpotent saddle
zbMATH Keywords: near-Hamiltonian system
zbMATH Keywords: Melnikov function
MaRDI profile type: Publication
full work available at URL: j.jde.2018.07.042
OpenAlex ID: W2883913766
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Wikidata QID: Q129474554
DOI: 10.1016/J.JDE.2018.07.042
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description: scientific article; zbMATH DE number 6973898