Lyapunov functions for Morse-Smale synchronisation diffeomorphisms
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912278513713152 |
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| author | Buescu, Jorge Oliveira, Henrique M. |
| author_facet | Buescu, Jorge Oliveira, Henrique M. |
| contents | This paper investigates the dynamical system governing the phase differences between three identical oscillators arranged symmetrically and coupled by burst interactions. By constructing a discrete Lyapunov function, we prove the existence of two asymptotically stable fixed points on the 2-torus T^2, which correspond to Huygens synchronisation of three clocks. The locked states have phase differences of (2 pi/3,4 pi/3) and (4 pi/3,2 pi/3). Each fixed point possesses an open basin of attraction. The closure of the union of the basins of attraction of the two asymptotically stable attractors is the torus T^2, implying that Huygens synchronisation occurs generically and with full Lebesgue measure with respect to initial conditions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13230 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lyapunov functions for Morse-Smale synchronisation diffeomorphisms Buescu, Jorge Oliveira, Henrique M. Dynamical Systems Mathematical Physics This paper investigates the dynamical system governing the phase differences between three identical oscillators arranged symmetrically and coupled by burst interactions. By constructing a discrete Lyapunov function, we prove the existence of two asymptotically stable fixed points on the 2-torus T^2, which correspond to Huygens synchronisation of three clocks. The locked states have phase differences of (2 pi/3,4 pi/3) and (4 pi/3,2 pi/3). Each fixed point possesses an open basin of attraction. The closure of the union of the basins of attraction of the two asymptotically stable attractors is the torus T^2, implying that Huygens synchronisation occurs generically and with full Lebesgue measure with respect to initial conditions. |
| title | Lyapunov functions for Morse-Smale synchronisation diffeomorphisms |
| topic | Dynamical Systems Mathematical Physics |
| url | https://arxiv.org/abs/2503.13230 |