Proof of Artés-Llibre-Valls's conjectures for the Higgins-Selkov and the Selkov systems
type: Item
instance of: scholarly article
title: Proof of Artés-Llibre-Valls's conjectures for the Higgins-Selkov and the Selkov systems
zbMATH Open document ID: 1459.34116
author: Hebai Chen
author: Yilei Tang
published in: Journal of Differential Equations
publication date: 2019-03-22T00:00:00Z
review text: Concerning the Higgins-Selkow system \[ \dot x=(y-x)x^2-x,\quad \dot y=\frac{1}{\sqrt\alpha}-x\tag{\(*\)} \] the authors prove, that \begin{itemize} \item[(i)] (\(*\)) has no limit cycle for \(\alpha\in(0,1]\), (\(*\)) has at most one limit cycle for \(\alpha\in(1,3)\). If the limit cycle exists, it is hyperbolic and stable. \item[(ii)] There exist \(\alpha^*\in(1,3)\) such that (\(*\)) has a unique limit cycle for \(\alpha\in(1,\alpha^*)\) and no limit cycle for \(\alpha\in(\alpha^*,3)\). The amplitude of the limit cycle increases with \(\alpha\). \end{itemize} Concerning the Selkov system \[ \dot x=-x+ay+x^2y,\quad \dot y=b-ay-x^2y\tag{\(**\)} \] they prove \begin{itemize} \item[(i)] (\(**\)) has no limit cycle for \(a=-b^2\). \item[(ii)] (\(**\)) has no limit cycle for \(a\ge\sigma_0\), where \(\sigma_0\) is the unique root of \(1-4a(1+a)b^2-4a(1+a)^2b^4=0\). \item[(iii)] (\(**\)) has no limit cycle for \(b=1\), \(a\ge 0\). \item[(iv)] In case \(b=1\), (\(**\)) has no limit cycle for \(-1<a\le a_0\) and at most one limit cycle for \(a_0<a<0\), where \(a_0\approx-0.11584\). \end{itemize} The proofs are based on the transformation of (\(*\)) and (\(**\)) into Liénard systems.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 92C45
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 7040869
zbMATH Keywords: limit cycle
zbMATH Keywords: Higgins-Selkov system
zbMATH Keywords: Selkov system
zbMATH Keywords: Liénard system
MaRDI profile type: Publication
full work available at URL: j.jde.2018.12.011
OpenAlex ID: W2903758936
Wikidata QID: Q122983036
cites work: Dynamics of the Higgins-Selkov and Selkov systems
cites work: Dynamical analysis of a cubic Liénard system with global parameters
cites work: Global dynamics of a SD oscillator
cites work: Cubic Lienard equations with linear damping
cites work: Q3871201
cites work: Centennial History of Hilbert's 16th Problem
cites work: HILBERT'S 16TH PROBLEM AND BIFURCATIONS OF PLANAR POLYNOMIAL VECTOR FIELDS
cites work: Rotated vector fields and the global behavior of limit cycles for a class of quadratic systems in the plane
cites work: Q4004269
DOI: 10.1016/J.JDE.2018.12.011
Recommended article: A proof of Perko's conjectures for the Bogdanov-Takens system
Recommended article: On a theorem of Halphen and its application to integrable systems
Recommended article: The Kirillov-Reshetikhin conjecture and solutions of T-systems
Recommended article: A proof of Pełczyński's conjecture for the Haar system
Recommended article: A symplectic case of Artin's conjecture
Recommended article: On a conjecture on the integrability of Liénard systems
Recommended article: Proof of a conjecture of Kulakova et al. related to the \(\mathfrak{s} \mathfrak{l}_2\) weight system
Recommended article: Artin's conjecture and systems of diagonal equations
Recommended article: A note on a system theoretic approach to a conjecture by Peller- Khrushchev
Recommended article: Totik's conjecture for Chrestenson and Walsh systems
description: scientific article; zbMATH DE number 7040869