Synchronization in the quaternionic Kuramoto model

Abstract

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.

Figures (12)

• Figure 1: This is the time evolution of $\{|q_n(t)-q_m(t)|\}_{1\leq m<n\leq N}$ for 5 oscillators ($N=5$) in the case of strong coupling strength. The parameters and initial conditions are given in the subsection \ref{['subsec:numerical']}. We can observe full phase-locking synchronization in this case, which is consistent with Theorem \ref{['q p locking']}.
• Figure 2: This is the time evolution of $\{|\dot{q}_n(t)-\dot{q}_m(t)|\}_{1\leq m<n\leq N}$ for 5 oscillators ($N=5$) in the case of strong coupling strength. The parameters and initial conditions are given in the subsection \ref{['subsec:numerical']}. We can observe frequency synchronization in this case, which is consistent with Theorem \ref{['F sync']}.
• Figure 3: This is the time evolution of $\{|q_n(t)-q_m(t)|\}_{1\leq m<n\leq N}$ for 5 oscillators ($N=5$) with identical natural frequencies ($\boldsymbol{\omega}=\boldsymbol{0}$) in the case of strong coupling strength. The parameters and initial conditions are given in the subsection \ref{['subsec:numerical']}. We can observe phase synchronization in this case, which is consistent with Theorem \ref{['3.5']}.
• Figure 4: This plot shows the "deceleration strip" (red) and "acceleration strip" (green) of the Lyapunov function $L$ in a neighborhood of the equilibrium point $(\pi/2,\cosh^{-1}(\omega/\lambda))$.
• Figure 5: The trajectory (blue) with initial condition $(w(0),v(0))$ will hit the curve $\dot{w}=0$ (cyan), and then hit the vertical line $w=\pi/2$. Then by a symmetry argument, the trajectory will eventually return to $(w(0),v(0))$ and form a closed loop around the equilibrium point $(\pi/2,\cosh^{-1}(\omega/\lambda))$.

Theorems & Definitions (27)

• Definition 1: Exponential, Sine, Cosine functions on $\mathbb{H}$
• Theorem 2.1
• proof
• Definition 2: Full phase-locking synchronization
• Definition 3: Frequency synchronization
• Definition 4: Phase synchronization
• Definition 5: Critical coupling strength
• Lemma 3.1: Real part phase-locking
• proof
• Lemma 3.2: Imaginary parts phase and frequency synchronization