The number of limit cycles of cubic Hamiltonian system with perturbation
type: Item
instance of: scholarly article
title: The number of limit cycles of cubic Hamiltonian system with perturbation
zbMATH Open document ID: 1117.34037
author: Chengqiang Wu
author: Yonghui Xia
published in: Nonlinear Analysis. Real World Applications
publication date: 2007-02-19T00:00:00Z
review text: The following perturbed Hamiltonian system is considered: \[ \frac{dx}{dt} = y + \varepsilon \sum_{j=0}^{l} a_j x^j | y| ^{2m-1}, \quad \frac{dy}{dt} = -x - x^3, \tag{1} \] where \(0<\varepsilon<<1\), \(l=2n+1\) or \(2n+2\), \(n\) and \(m\) are arbitrary positive integers and \(a_0\), \(a_1, \dots, a_l\) are real. The main result of the work is that this perturbed Hamiltonian system has at most \(n+m\) limit cycles. The particular case \(m=1\) is studied and an upper bound of \(n\) limit cycles is given, which is sharp by a suitable choice of the parameters \(a_i\), \(i=0,1,2, \dots, l\). Moreover, the particular cases \((m,n)=(1,1)\) and \((m,n)=(1,2)\) are extensively studied and general conditions for the number and stability of existing limit cycles are described in terms of the coefficients of the perturbation. The main tool to prove these results is the following well-known result. Let us consider a perturbed Hamiltonian system of the form \[ \frac{dx}{dt} = -\frac{\partial H}{\partial y} + \varepsilon P(x,y), \quad \frac{dy}{dt} = \frac{\partial H}{\partial x} + \varepsilon Q(x,y), \] where \(0<\varepsilon<<1\), \(H\), \(P\) and \(Q\) are of class \(\mathcal{C}^1\) in \(\mathbb{R}^2\) and \(\Gamma_h\) is a closed curve defined by \(H(x,y)=h\). We assume that the closed orbit \(\Gamma_{h_1}\) surrounds \(\Gamma_{h_0}\) if \(h_1>h_0\). Let us define the Abelian integral: \[ A(h) = \int_{\Gamma_h} Q(x,y) dx - P(x,y) dy. \] Then, the following facts hold: 1) If the perturbed Hamiltonian system has a limit cycle near \(\Gamma_{h^{*}}\), then \(A(h^{*})=0\). 2) If \(A(h^*)=0\) and \(A'(h^*) \neq 0\), then the perturbed Hamiltonian system has a unique limit cycle near \(\Gamma_{h^{*}}\), which is stable (resp. unstable) when \(A'(h^*)<0\) (resp. \(A'(h^*)>0\)). We refer the reader to the following two classical references [\textit{R. Roussarie}, Bifurcation of planar vector fields and Hilbert's sixteenth problem. Progress in Mathematics 164. Basel: Birkhäuser (1998;…
reviewed by: Maite Grau
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 5128066
zbMATH Keywords: Hamiltonian system
zbMATH Keywords: perturbed system
zbMATH Keywords: limit cycle
zbMATH Keywords: stability
MaRDI profile type: Publication
full work available at URL: j.nonrwa.2006.02.002
OpenAlex ID: W2029728292
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DOI: 10.1016/J.NONRWA.2006.02.002
description: scientific article; zbMATH DE number 5128066