2023/09/05 by Ting-Yang Hsiao, Hsiao, Ting-Yang, Yun-Feng Lo +3 · 1 citation
Computer Science · Mathematics · #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Nonlinear Dynamics and Pattern Formation
paper · pdf · doi:10.48550/arxiv.2309.01893
openalex publication_date 2023/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we propose an N oscillators Kuramoto model with quaternions 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 λc), 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<λc. 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 (N-1)/(2)\rfloor. Therefore, we conjecture that Lyapunov stable quaternionic synchronization always exists.