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

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) $