Synchronization in the quaternionic Kuramoto model
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917571210510336 |
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| author | Hsiao, Ting-Yang Lo, Yun-Feng Wang, Winnie |
| author_facet | Hsiao, Ting-Yang Lo, Yun-Feng Wang, Winnie |
| contents | In this paper, we propose an $N$ oscillators Kuramoto model with quaternions $\mathbb{H}$. In case the coupling strength is strong, a sufficient condition of synchronization is established for general $N\geqslant 2$. On the other hand, we analyze the case when the coupling strength is weak. For $N=2$, when coupling strength is weak (below the critical coupling strength $λ_c$), we show that new periodic orbits emerge near each equilibrium point, and hence phase-locking state exists. This phenomenon is different from the real Kuramoto system since it is impossible to arrive at any synchronization when $λ<λ_c$. We prove a theorem that states a set of closed and dense contour forms near each equilibrium point, resembling a tree's growth rings. In other words, the trajectory of phase difference lies on a $4D$-torus surface. Therefore, this implies that the phase-locking state is Lyapunov stable but not asymptotically stable. The proof uses a new infinite buffer method (``$δ/n$ criterion") and a Lyapunov function argument. This has been studied both analytically and numerically. For $N=3$, we consider the ``Lion Dance flow", the analog of Cherry flow for our model, to demonstrate that the quaternionic synchronization exists even when the coupling strength is ``super weak" (when $λ/ω<0.85218915...$). Also, numerical evaluation reveals that when $N>3$, the stable manifold of Lion Dance flow exists, and the number of these equilibria is $\lfloor \frac{N-1}{2}\rfloor$. Therefore, we conjecture that Lyapunov stable quaternionic synchronization always exists. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_01893 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Synchronization in the quaternionic Kuramoto model Hsiao, Ting-Yang Lo, Yun-Feng Wang, Winnie Dynamical Systems Mathematical Physics In this paper, we propose an $N$ oscillators Kuramoto model with quaternions $\mathbb{H}$. In case the coupling strength is strong, a sufficient condition of synchronization is established for general $N\geqslant 2$. On the other hand, we analyze the case when the coupling strength is weak. For $N=2$, when coupling strength is weak (below the critical coupling strength $λ_c$), we show that new periodic orbits emerge near each equilibrium point, and hence phase-locking state exists. This phenomenon is different from the real Kuramoto system since it is impossible to arrive at any synchronization when $λ<λ_c$. We prove a theorem that states a set of closed and dense contour forms near each equilibrium point, resembling a tree's growth rings. In other words, the trajectory of phase difference lies on a $4D$-torus surface. Therefore, this implies that the phase-locking state is Lyapunov stable but not asymptotically stable. The proof uses a new infinite buffer method (``$δ/n$ criterion") and a Lyapunov function argument. This has been studied both analytically and numerically. For $N=3$, we consider the ``Lion Dance flow", the analog of Cherry flow for our model, to demonstrate that the quaternionic synchronization exists even when the coupling strength is ``super weak" (when $λ/ω<0.85218915...$). Also, numerical evaluation reveals that when $N>3$, the stable manifold of Lion Dance flow exists, and the number of these equilibria is $\lfloor \frac{N-1}{2}\rfloor$. Therefore, we conjecture that Lyapunov stable quaternionic synchronization always exists. |
| title | Synchronization in the quaternionic Kuramoto model |
| topic | Dynamical Systems Mathematical Physics |
| url | https://arxiv.org/abs/2309.01893 |