Cyclicity 1 and 2 conditions for a 2-polycycle of integrable systems on the plane
type: Item
instance of: scholarly article
title: Cyclicity 1 and 2 conditions for a 2-polycycle of integrable systems on the plane
zbMATH Open document ID: 0935.34022
published in: Journal of Differential Equations
publication date: 2000-04-25T00:00:00Z
review text: The authors study the cyclicity for a 2-polycycle of the \(C^{\infty}\) planar system \[ \dot x=P_0(x,y)+\varepsilon P(x,y,\varepsilon ,\delta),\quad \dot x=Q_0(x,y)+\varepsilon Q(x,y,\varepsilon ,\delta), \tag{1} \] where \(\varepsilon >0\) is small and \(\delta\in U\subset \mathbb{R}^n\) with \(U\) bounded. For \(\varepsilon =0\) system (1) is supposed to have a 2-polycycle \(L=L_1\cup L_2\cup S_1\cup S_2\), where \(L_1\) and \(L_2\) are heteroclinic orbits connecting hyperbolic saddles \(S_1\) and \(S_2\). Assuming that for \(\varepsilon =0\) system (1) is integrable (has a \(C^1\) first integral), the authors find 1) conditions necessary for \(L\) to generate a limit cycle for some \(\delta =\delta_0\in U\) and \(\varepsilon\) small; 2) generic conditions under which system (1) has at most one or two limit cycles near \(L\) for \(\varepsilon +|\delta - \delta _0|\) small, i.e. conditions for \(L\) to have cyclicity 1 or 2.
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 1333006
zbMATH Keywords: polycycle
zbMATH Keywords: limit cycle
zbMATH Keywords: bifurcation
author: Maoan Han
author: Zhifen Zhang
reviewed by: Eugene Ershov
MaRDI profile type: Publication
full work available at URL: jdeq.1998.3585
OpenAlex ID: W1976071188
cites work: Elementary graphics of cyclicity 1 and 2
cites work: Degenerate and non-trivial hyperbolic polycycles with two vertices
cites work: Degenerate and non-trivial hyperbolic 2-polycycles: appearance of two independent Ecalle-Roussarie compensators and Khovanskii's theory
cites work: Q5688224
cites work: Bifurcations of limit cycles from a heteroclinic cycle of Hamiltonian systems
cites work: The cyclicity of polycycle \(S^{(2)}\) of planar integrable systems with small perturbations.
cites work: Q4333074
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Finitely-smooth normal forms of local families of diffeomorphisms and vector fields
cites work: Bifurcations of infinite polycycles for polynomic vector fields on the plane
cites work: Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics
cites work: Q4514329
DOI: 10.1006/JDEQ.1998.3585
description: scientific article; zbMATH DE number 1333006