Integrability and zero-Hopf bifurcation in the Sprott A system
type: Item
instance of: scholarly article
title: Integrability and zero-Hopf bifurcation in the Sprott A system
zbMATH Open document ID: 1450.34018
author: Luís Barreira
author: Jaume Llibre
author: Claudia Valls
published in: Bulletin des Sciences Mathématiques
publication date: 2020-08-06T00:00:00Z
review text: Consider the polynomial system \[ \frac{dx}{dt}= y, \quad \frac{dy}{dt}=-x-yz, \quad \frac{dz}{dt}=y^2-a ,\tag{1} \] where \( a \) is a real parameter. It is known that system (1) has no equilibrium for \( a \neq 0 \), but it exhibits chaotic behavior for \( a=1 \). The authors contribute to the understanding of the behavior of system (1) by proving (i). (1) has no Darboux first integral for \( a \neq 0 \). (ii). In case \( a=0 \), the unique Darboux first integrals are functions of \( x^2+y^2+z^2 \). (iii). Hopf bifurcation occurs from the non-isolated equilibrium \( (0,0,0) \) for increasing \( a \) and crossing \( a=1 \).
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C29
zbMATH DE Number: 7229227
zbMATH Keywords: Darboux integrability
zbMATH Keywords: Sprott A system
zbMATH Keywords: zero-Hopf bifurcation
zbMATH Keywords: averaging theory
MaRDI profile type: Publication
full work available at URL: j.bulsci.2020.102874
OpenAlex ID: W3031532852
cites work: Q4023157
cites work: The entwined wiggling of homoclinic curves emerging from saddle-node/Hopf instabilities
cites work: Multiplicity of invariant algebraic curves in polynomial vector fields
cites work: Qualitative theory of planar differential systems
cites work: Q3940514
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Equations de Pfaff algébriques
cites work: Elements of applied bifurcation theory
cites work: Darboux theory of integrability in \(\mathbb C^n\) taking into account the multiplicity
cites work: Darboux theory of integrability for polynomial vector fields in taking into account the multiplicity at infinity
cites work: On the formation of hidden chaotic attractors and nested invariant tori in the Sprott A system
cites work: Averaging methods in nonlinear dynamical systems
cites work: Bifurcation to Quasi-Periodic Tori in the Interaction of Steady State and Hopf Bifurcations
cites work: Q3997917
DOI: 10.1016/J.BULSCI.2020.102874
description: scientific article; zbMATH DE number 7229227