The Markus-Yamabe conjecture for continuous and discontinuous piecewise linear differential systems
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 h...
| Autores: | , |
|---|---|
| Tipo de documento: | artigo |
| Data de publicação: | 2021 |
| País: | España |
| Recursos: | Universitat Autònoma de Barcelona |
| Repositório: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglês |
| OAI Identifier: | oai:ddd.uab.cat:257109 |
| Acesso em linha: | https://ddd.uab.cat/record/257109 https://dx.doi.org/urn:doi:10.1090/proc/15601 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Markus-Yamabe conjecture Kalman conjecture Hurwitz matrix Continuous piecewise linear differential system Discontinuous piecewise linear differential system |
| Resumo: | 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. |
|---|