Article Contents
Article Contents

# The General Non-Abelian Kuramoto Model on the 3-sphere

• * Corresponding author: Aladin Crnkić
• We introduce non-Abelian Kuramoto model on $S^3$ in the most general form. Following an analogy with the classical Kuramoto model (on the circle $S^1$), we study some interesting variations of the model on $S^3$ that are obtained for particular coupling functions. As a partial case, by choosing "standard" coupling function one obtains a previously known model, that is referred to as Kuramoto-Lohe model on $S^3$.

We briefly address two particular models: Kuramoto models on $S^3$ with frustration and with external forcing. These models on higher dimensional manifolds have not been studied so far. By choosing suitable values of parameters we observe different nontrivial dynamical regimes even in the simplest setup of globally coupled homogeneous population.

Although non-Abelian Kuramoto models can be introduced on various symmetric spaces, we restrict our analysis to the case when underlying manifold is the 3-sphere. Due to geometric and algebraic properties of this specific manifold, variations of this model are meaningful and geometrically well justified.

Mathematics Subject Classification: Primary: 34M45, 34C15, 92B25.

 Citation:

• Figure 1.  The population of $N = 50$ KL oscillators with intrinsic frequencies $w_l = (0.3,1.3,0.9)$, $u_l = (1.7,0.5,1.4)$ and phase shifts $\alpha = \frac{\pi}{3}, \, \beta = \frac{\pi}{4}, \, \gamma = \frac{\pi}{3}$ achieves coherent state: (a) evolution of the order parameters (thick line for the global order parameter $r$ and dashed and dotted lines for $r_\varphi$ and $r_\psi$ respectively) and (b) cosines of the angles between some pairs of KL oscillators. Initial conditions are sampled from the von Mises-Fisher on $S^{3}$ with mean direction $\mu = (0.5, 0.5, 0.5, 0.5)$ and the concentration $\kappa = 2.5$

Figure 2.  The population of $N = 50$ KL oscillators with intrinsic frequencies $w_l = (0.3,1.3,0.9)$, $u_l = (1.7,0.5,1.4)$ and phase shifts $\alpha = \frac{2\pi}{3}, \, \beta = \frac{\pi}{4}, \, \gamma = \frac{\pi}{3}$ achieves a fully incoherent state: (a) evolution of the order parameters (thick line for the global order parameter $r$ and dashed and dotted lines for $r_\varphi$ and $r_\psi$ respectively) and (b) cosines of the angles between some pairs of KL oscillators. Initial conditions are sampled from the von Mises-Fisher on $S^{3}$ with mean direction $\mu = (0.5, 0.5, 0.5, 0.5)$ and the concentration $\kappa = 2.5$

Figure 3.  Oscillations of the system with $N = 50$ KL oscillators with intrinsic frequencies $w_l = (0.3,1.3,0.9)$, $u_l = (1.7,0.5,1.4)$ and phase shifts $\alpha = \frac{\pi}{2}, \, \beta = \frac{\pi}{4}, \, \gamma = \frac{\pi}{3}$: (a) evolution of the order parameters (thick line for the global order parameter $r$ and dashed and dotted lines for $r_\varphi$ and $r_\psi$ respectively) and (b) cosines of the angles between some pairs of KL oscillators. Initial conditions are sampled from the von Mises-Fisher on $S^{3}$ with mean direction $\mu = (0.5, 0.5, 0.5, 0.5)$ and the concentration $\kappa = 2.5$

Figure 4.  Evolution of the global and angular order parameters (thick line for the global order parameter $r$ and dashed and dotted lines for $r_\varphi$ and $r_\psi$ respectively) in the population of $N = 50$ nonidentical oscillators with the external forcing. The global coupling strength is set at $K = 1$, the frequencies of the external signal are $p = 0$, $v = (1.2,2.3,3.7)$ and the intensity of external forcing is (a) $D = 0.5+0.5\mathrm{\textbf{j}}$; (b) $D = 0.9+0.5\mathrm{\textbf{j}}$ and (c) $D = 1.2+0.5\mathrm{\textbf{j}}$. The right intrinsic frequencies $u_l$ are zero, and the left intrinsic frequencies are sampled from the 3-dimensional Gaussian distribution with the expectation vector $\left(1,2,3\right)$ and covariance matrix $\left(\left(2,-1/4,1/3\right),\left(-1/4,2/3,1/5\right),\left(1/3,1/5,1/2\right)\right)$

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