The Markus-Yamabe conjecture does not hold for discontinuous piecewise linear differential systems separated by one straight line
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Menezes, Lucyjane de A. S. (Universidade Federal de Goiás. Instituto de Matemática e Estatística (Brazil))

Date:	2020
Abstract:	The Markus-Yamabe conjecture is a conjecture on global asymptotic stability. The conjecture states that if a differentiable system x˙ = f(x) has a singularity and the Jacobian matrix Df(x) has everywhere eigenvalues with negative real part, then the singularity is a global attractor. In this paper we consider discontinuous piecewise linear differential systems in R2 separated by one straight line Σ such that the unique singularity of the system is at Σ and the Jacobian matrix of the system has everywhere eigenvalues with negative real part. We prove that these discontinuous piecewise linear differential systems can have one crossing limit cycle and consequently these systems do not satisfy the Markus-Yamabe conjecture.
Grants:	Ministerio de Economía y Competitividad MTM2016-77278-P Ministerio de Economía y Competitividad MTM2013-40998-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Discontinuous differential system ; Limit cycle ; Markus-Yamabe conjecture
Published in:	Journal of dynamics and differential equations, 2020 , ISSN 1572-9222

DOI: 10.1007/s10884-020-09825-8

Postprint 23 p, 2.6 MB

The record appears in these collections:
Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
Articles > Published articles