The Markus-Yamabe conjecture for discontinuous piecewise linear differential systems in Rn separated by a conic × Rn-2
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...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:257120 |
| Acceso en línea: | https://ddd.uab.cat/record/257120 https://dx.doi.org/urn:doi:10.1007/s10884-021-10110-5 |
| Access Level: | acceso abierto |
| Palabra clave: | Markus-Yamabe conjecture Hurwitz matrix Discontinuous piecewise linear differential systems |
| Sumario: | 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. |
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