Slow passage through a transcritical bifurcation in piecewise linear differential systems: canard explosion and enhanced delay
In this paper we analyse the phenomenon of the slow passage through a transcritical bifurcation with special emphasis in the maximal delay ¿ as a function of the bifurcation parameter and the singular parameter ¿ . We quantify the maximal delay by constructing a piecewise linear (PWL) transcritical...
| Autores: | , |
|---|---|
| Tipo de recurso: | artículo |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/409561 |
| Acceso en línea: | https://hdl.handle.net/2117/409561 https://dx.doi.org/10.1016/j.cnsns.2024.108044 |
| Access Level: | acceso abierto |
| Palabra clave: | Dynamical systems Ergodic theory Piecewise linear systems Dynamic bifurcations Slow passage Transcritic bifurcation Enhanced delay Sistemes dinàmics diferenciables Teoria ergòdica Classificació AMS::37 Dynamical systems and ergodic theory Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics |
| Sumario: | In this paper we analyse the phenomenon of the slow passage through a transcritical bifurcation with special emphasis in the maximal delay ¿ as a function of the bifurcation parameter and the singular parameter ¿ . We quantify the maximal delay by constructing a piecewise linear (PWL) transcritical minimal model and studying the dynamics near the slow-manifolds. Our findings encompass all potential maximum delay behaviours within the range of parameters, allowing us to identify: (i) the trivial scenario where the maximal delay tends to zero with the singular parameter; (ii) the singular scenario where ¿ is not bounded, and also (iii) the transitional scenario where the maximal delay tends to a positive finite value as the singular parameter goes to zero. Moreover, building upon the concepts by Vidal and Françoise (2012), we construct a PWL system combining symmetrically two transcritical minimal models in such a way it shows periodic behaviour. As the parameter changes, the system presents a non-bounded canard explosion leading to an enhanced delay phenomenon at the critical value. Our understanding of the maximal delay ¿ of a single normal form, allows us to determine both, the amplitude of the canard cycles and, in the enhanced delay case, the increase of the amplitude for each passage. |
|---|