The Markus-Yamabe conjecture for continuous and discontinuous piecewise linear differential systems
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Zhang, Xiang (Shanghai Jiao Tong University. School of Mathematical Sciences)

Date:	2021
Abstract:	In 1960 Markus and Yamabe made the following conjecture: If a C1 differential system x˙ = F(x) in Rn has a unique equilibrium point and the Jacobian matrix of F(x) for all x ∈ Rn has all its eigenvalues with negative real part, then the equilibrium point is a global attractor. Until 1997 we do not have the complete answer to this conjecture. It is true in R2, but it is false in Rn for all n > 2. Here we extend the conjecture of Markus and Yamabe to continuous and discontinuous piecewise linear differential systems in Rn separated by a hyperplane, and we prove that for the continuous piecewise linear differential systems it is true in R2, but it is false in Rn for all n > 2. But for discontinuous piecewise linear differential systems it is false in Rn for all n ≥ 2.
Grants:	Agencia Estatal de Investigación MTM2016-77278-P Ministerio de Economía y Competitividad MDM-2014-0445 Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617 European Commission 777911
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Markus-Yamabe conjecture ; Kalman conjecture ; Hurwitz matrix ; Continuous piecewise linear differential system ; Discontinuous piecewise linear differential system
Published in:	Proceedings of the American Mathematical Society, Vol. 149, Issue 12 (December 2021) , p. 5267-5274, ISSN 1088-6826

DOI: 10.1090/proc/15601

Postprint 10 p, 280.5 KB

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