The size of the sync basin resolved
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911186553929728 |
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| author | Groisman, Pablo De Vita, Cecilia Bonder, Julián Fernández Zhang, Yuanzhao |
| author_facet | Groisman, Pablo De Vita, Cecilia Bonder, Julián Fernández Zhang, Yuanzhao |
| contents | Sparsely coupled Kuramoto oscillators offer a fertile playground for exploring high-dimensional basins of attraction due to their simple yet multistable dynamics. For $n$ identical Kuramoto oscillators on cycle graphs, it is well known that the only attractors are twisted states, whose phases wind around the circle with a constant gap between neighboring oscillators ($θ_j = 2πq j/n$). It was conjectured in 2006 that basin sizes of these twisted states scale as $e^{-kq^2}$ to the winding number $q$. Here, we provide new numerical and analytical evidence supporting the conjecture and uncover the dynamical mechanism behind the Gaussian scaling. The key idea is that, when starting with a random initial condition, the winding number of the solution stabilizes rapidly at $t \propto \log n$, before long-range correlation can develop among oscillators. This timescale separation allows us to calculate the winding number as a sum of weakly-dependent random variables, leading to a Central Limit Theorem derivation of the basin scaling. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_03419 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The size of the sync basin resolved Groisman, Pablo De Vita, Cecilia Bonder, Julián Fernández Zhang, Yuanzhao Mathematical Physics Dynamical Systems Adaptation and Self-Organizing Systems Pattern Formation and Solitons Sparsely coupled Kuramoto oscillators offer a fertile playground for exploring high-dimensional basins of attraction due to their simple yet multistable dynamics. For $n$ identical Kuramoto oscillators on cycle graphs, it is well known that the only attractors are twisted states, whose phases wind around the circle with a constant gap between neighboring oscillators ($θ_j = 2πq j/n$). It was conjectured in 2006 that basin sizes of these twisted states scale as $e^{-kq^2}$ to the winding number $q$. Here, we provide new numerical and analytical evidence supporting the conjecture and uncover the dynamical mechanism behind the Gaussian scaling. The key idea is that, when starting with a random initial condition, the winding number of the solution stabilizes rapidly at $t \propto \log n$, before long-range correlation can develop among oscillators. This timescale separation allows us to calculate the winding number as a sum of weakly-dependent random variables, leading to a Central Limit Theorem derivation of the basin scaling. |
| title | The size of the sync basin resolved |
| topic | Mathematical Physics Dynamical Systems Adaptation and Self-Organizing Systems Pattern Formation and Solitons |
| url | https://arxiv.org/abs/2506.03419 |