A note on the topological synchronisation of unimodal maps
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917850645528576 |
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| author | Gianfelice, Michele |
| author_facet | Gianfelice, Michele |
| contents | In this note we complete the analysis carried on in \cite{CGSV} about the topological synchronisation of unimodal maps of the interval coupled in a master-slave configuration, by answering to the questions raised in that paper. Namely, we compute the weak limits of the invariant measure of the coupled system as the coupling strength $k\in\left( 0,1\right) $ tends to $0$ and to $1$ and discuss the uniqueness of the invariant measure of its random dynamical system counterpart, proving that the convergence of the associated Markov chain to its unique stationary measure is geometric.
[CGSV] Caby Th., Gianfelice M., Saussol B., Vaienti S. "Topological synchronisation or a simple attractor?" Nonlinearity Vol. 36, no. 7, pp. 3603-3621 (2023). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_07661 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the topological synchronisation of unimodal maps Gianfelice, Michele Dynamical Systems Probability 37A10, 60J10 In this note we complete the analysis carried on in \cite{CGSV} about the topological synchronisation of unimodal maps of the interval coupled in a master-slave configuration, by answering to the questions raised in that paper. Namely, we compute the weak limits of the invariant measure of the coupled system as the coupling strength $k\in\left( 0,1\right) $ tends to $0$ and to $1$ and discuss the uniqueness of the invariant measure of its random dynamical system counterpart, proving that the convergence of the associated Markov chain to its unique stationary measure is geometric. [CGSV] Caby Th., Gianfelice M., Saussol B., Vaienti S. "Topological synchronisation or a simple attractor?" Nonlinearity Vol. 36, no. 7, pp. 3603-3621 (2023). |
| title | A note on the topological synchronisation of unimodal maps |
| topic | Dynamical Systems Probability 37A10, 60J10 |
| url | https://arxiv.org/abs/2405.07661 |