The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Valls, Clàudia 1973- (Universidade de Lisboa. Instituto Superior Técnico. Departamento de Matemática)

Data:	2023
Resum:	In 1960 Markus and Yamabe made the conjecture that if a C1 differential system x˙=F(x) in Rn has a unique equilibrium point and DF(x) is Hurwitz for all x∈Rn, then the equilibrium point is a global attractor. This conjecture was completely solved in 1997 and it turned out to be true in R2 and false in Rn for all n≥3. In (The Markus-Yamabe conjecture for continuous and discontinuous piecewise linear differential systems, 2020) the authors extended the Markus-Yamabe conjecture to continuous and discontinuous piecewise linear differential systems in Rn separated by a hyperplane, they proved for the continuous systems that the extended conjecture is true in R2 and false in Rn for all n≥3, but for discontinuous systems they proved that the conjecture is false in Rn for all n≥2. In this paper first we show that there are no continuous piecewise linear differential systems separated by a conic×Rn-2 except the linear differential systems in Rn. And after we prove that the extended Markus-Yamabe conjecture to discontinuous piecewise linear differential systems in Rn separated by a conic×Rn-2 is false in Rn for all n≥2.
Ajuts:	Agencia Estatal de Investigación PID2019-104658GB-I00 European Commission 777911
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Llengua:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Matèria:	Markus-Yamabe conjecture ; Hurwitz matrix ; Discontinuous piecewise linear differential systems
Publicat a:	Journal of dynamics and differential equations, Vol. 35 (September 2023) , p.2579-2588, ISSN 1572-9222

DOI: 10.1007/s10884-021-10110-5

Postprint 10 p, 761.2 KB

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