The general non-abelian Kuramoto model on the 3-sphere (Q2173162)

The authors present the non-abelian Kuramoto model on \(S^3\) in its most general form. This is a generalisation of the Kuramoto model of coupled phase oscillators to oscillators whose state is described by 3, not 1, phase variables. The model is presented as a system of unit quaternion-valued ordinary differential equations of the form \[\dot{q}_l=q_lfq_l+w_lq_l+q_lu_l-\bar{f}\] for \(l=1,2\dots N\) where \(f\) is a quaternionic function, overbar indicates complex conjugate, and \(w_l\) and \(u_l\) are pure quaternions, interpreted as intrinsic frequencies of the oscillator \(l\). \(f\) describes the form of coupling within the network, and for mean-field coupling and \(u_l=0\) the standard non-abelian Kuramoto model (the Kuramoto-Lohe model) on the group \(\mathrm{SU}(2)\) is recovered. The authors introduce and numerically investigate the non-abelian Kuramoto-Sakaguchi model (involving the analogue of a phase shift in the sinusoidal coupling for the Kuramoto model) and the periodically forced non-abelian Kuramoto model.