Detecting alien limit cycles near a Hamiltonian 2-saddle cycle
type: Item
instance of: scholarly article
title: Detecting alien limit cycles near a Hamiltonian 2-saddle cycle
zbMATH Open document ID: 1194.34084
author: Freddy Dumortier
author: M. Caubergh
author: Robert Roussarie
author: Stijn Luca
published in: Discrete and Continuous Dynamical Systems
publication date: 2009-12-16T00:00:00Z
review text: From the authors' abstract and text: The paper aims at providing an example of a cubic Hamiltonian 2-saddle cycle which after bifurcation can give rise to an alien limit cycle; this is a limit cycle which is not controlled by a zero of the related abelian integral. To guarantee the existence of an alien limit cycle one can verify generic conditions on the abelian integral and on the transition map associated to the connections of the 2-saddle cycle. In this paper, a general method is developed to compute the first and the second derivative of the transition map along a connection between two saddles. Next, a concrete generic Hamiltonian 2-saddle cycle is analyzed using these formulas to verify the generic relation between the second-order derivative of both transition maps, and a calculation of the Abelian integral. The principal result in this paper establishes the presence of this bifurcation phenomenon of alien limit cycles in the unfolding of the quadratic Hamiltonian system with two centers and two heteroclinic loops \[ \begin{matrix}\dot{x}=-H_y+\varepsilon[\bar{\mu}_3xy +\bar{\mu}_4y^2x+H(x-\frac{\sqrt3\pi}{8}xy)],\\ \dot{y}=H_x+\varepsilon y(\bar{\mu}_1+\bar{\mu}_2x),\end{matrix} \] where the Hamiltonian \(H\) is given by \(H(x,y)=y(x^2+\frac{1}{12}y^2-1)\). Verification of the generic conditions which guarantee the presence of alien limit cycle that bifurcates from the 2-saddle cycle lying in \(\{y\leq 0\}\) is quite involved.
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 5648548
zbMATH Keywords: Planar vector field
zbMATH Keywords: two-saddle cycle
zbMATH Keywords: limit cycle
zbMATH Keywords: alien limit cycle perturbation of Hamiltonian system
zbMATH Keywords: Abelian integral
zbMATH Keywords: transition map
reviewed by: Iliya D. Iliev
MaRDI profile type: Publication
full work available at URL: dcds.2009.25.1081
OpenAlex ID: W2030479805
DOI: 10.3934/DCDS.2009.25.1081
description: scientific article; zbMATH DE number 5648548