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| Format: | Preprint |
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2023
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| Online-Zugang: | https://arxiv.org/abs/2309.10119 |
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| _version_ | 1866913332266532864 |
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| author | Pérez-Cervera, Alberto Teruel, Antonio E. |
| author_facet | Pérez-Cervera, Alberto Teruel, Antonio E. |
| contents | In this paper we analyse the phenomenon of the slow passage through a transcritical bifurcation with special emphasis in the maximal delay $z_d(λ,\varepsilon)$ as a function of the bifurcation parameter $λ$ and the singular parameter $\varepsilon$. 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 $z_d(λ, \varepsilon)$ 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 A. Vidal and J.P. Françoise (Int. J. Bifurc. Chaos Appl. 2012), we construct a PWL system combining symmetrically two transcritical minimal models in such a way it shows periodic behaviour. As the parameter $λ$ 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 $z_d(λ, \varepsilon)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_10119 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Slow passage through a transcritical bifurcation in piecewise linear differential systems: canard explosion and enhanced delay Pérez-Cervera, Alberto Teruel, Antonio E. Dynamical Systems In this paper we analyse the phenomenon of the slow passage through a transcritical bifurcation with special emphasis in the maximal delay $z_d(λ,\varepsilon)$ as a function of the bifurcation parameter $λ$ and the singular parameter $\varepsilon$. 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 $z_d(λ, \varepsilon)$ 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 A. Vidal and J.P. Françoise (Int. J. Bifurc. Chaos Appl. 2012), we construct a PWL system combining symmetrically two transcritical minimal models in such a way it shows periodic behaviour. As the parameter $λ$ 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 $z_d(λ, \varepsilon)$ 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. |
| title | Slow passage through a transcritical bifurcation in piecewise linear differential systems: canard explosion and enhanced delay |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2309.10119 |