Parameter identification on abelian integrals to achieve Chebyshev property
type: Item
instance of: scholarly article
title: Parameter identification on abelian integrals to achieve Chebyshev property
zbMATH Open document ID: 1484.34094
author: Zhanbo Chen
author: Xianbo Sun
author: P. Yu
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2021-06-18T00:00:00Z
review text: The authors consider the Hamiltonian system: \[ \dot{x} \, = \, H_y, \qquad \dot{y} \, = \, -H_x, \] where the subindexes denote partial derivatives, and where \[ H(x,y) \, = \, \frac{y^2}{2} + \frac{ \vartheta \alpha \beta}{2} x^2 + \frac{\vartheta (\alpha+\beta-\alpha\beta)}{2} x^4+ \frac{\vartheta(1-\alpha-\beta)}{4} x^6- \frac{\vartheta}{2} x^8, \] with \(\alpha\), \(\beta\) and \(\vartheta\) are real parameters. For several values of these parameters the system exhibits a period annulus surrounding the origin of coordinates and whose outer boundary is a heteroclinic loop. The aim of the paper is to study the bifurcation of limit cycles from this period annulus under perturbations of the form: \[ \dot{x} \, = \, y, \quad \dot{y} \, = \, \vartheta x(x^2-1)(x^2+\alpha)(x^2+\beta)+\varepsilon(a_0+a_1x^2+a_2x^4+a_3x^6)y, \] where \(a_0\), \(a_1\), \(a_2\) and \(a_3\) are real parameters and \(\varepsilon\) is the perturbation parameter. \par Let \(\Upsilon^\dagger\) be the heteroclinic loop of the unperturbed system satisfying \(H(\Upsilon^\dagger)=H(-1,0)\) and let \(\Upsilon_h\) be the closed orbit of the unperturbed system satisfying \(\Upsilon_h \, = \, \{ (x,y): x \in (-1,1), H(x,y)=h \}\) where \(0<h<H(\Upsilon^\dagger)\). It is well-known that the following abelian integral controls the bifurcation of limit cycles at first order of \(\varepsilon\) in the sense that its simple zeroes ensure the existence of a hyperbolic limit cycle when \(\varepsilon\) is small enough. This abelian integral is defined in the following way, \[ I(h,a_0,a_1,a_2,a_3) \, = \, a_0 I_0(h) + a_1 I_1(h) + a_2 I_2(h) +a_3 I_3(h), \] with \[ I_i(h) \, = \, \oint_{\Upsilon_h} x^{2i}ydx, \quad i=0,1,2,3. \] The study of the zeroes of this abelian integral is difficult and several criterions have been developed in the previous years. One seeks to find a Chebyshev property on the set of functions \(I_0(h), I_1(h), I_2(h), I_3(h)\). The particular problem tackled in this paper is specially di…
reviewed by: Maite Grau
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 7360894
zbMATH Keywords: Hilbert's 16th problem
zbMATH Keywords: Lienard system
zbMATH Keywords: limit cycle heteroclinic loops
zbMATH Keywords: bifurcation
describes a project that uses: DISCOVERER
MaRDI profile type: Publication
full work available at URL: dcdsb.2020375
OpenAlex ID: W3114653250
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DOI: 10.3934/DCDSB.2020375
description: scientific article; zbMATH DE number 7360894