Bifurcation theory of limit cycles by higher order Melnikov functions and applications
type: Item
instance of: scholarly article
title: Bifurcation theory of limit cycles by higher order Melnikov functions and applications
author: Shanshan Liu
author: Maoan Han
published in: Journal of Differential Equations
publication date: 2024-06-21T00:00:00Z
review text: For planar near Hamiltonian systems, the authors studied the maximum number of limit cycles that can emerge from Hopf and homoclinic bifurcations using higher order Melnikov functions. They applied the theoretical results to investigate a symmetric 8-loop Hamiltonian with perturbations up to the cubic nonlinearity. For the Hopf bifurcation, they found that the maximum number of limit cycles in the neighbourhood of two centers is six. This result is not new. For the homoclinic bifurcation, they found that the maximum number of limit cycles in the neighbourhood of the double homoclinic loop or one of the two symmetric homoclinic loops is eight. This result is new. As the total cyclicity of period annuli surrounding centers is known to be at most nine, it remains open whether nine limit cycles can appear in the symmetric 8-loop system considered in this paper.
reviewed by: Kwok-wai Chung
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 7869051
zbMATH Keywords: higher order Melnikov function
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: homoclinic bifurcation
zbMATH Keywords: limit cycle
MaRDI profile type: Publication
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DOI: 10.1016/J.JDE.2024.04.036
description: scientific article; zbMATH DE number 7869051