Conformists and contrarians on spheres

Abstract

We investigate the conformists–contrarians model of identical Kuramoto oscillators evolving on a sphere. Using group-theoretic and geometric approach, we reduce the model to the dynamical system on extended Ott–Antonsen manifold. Further reduction yields the system of three scalar ODE’s for global variables. This three-dimensional dynamical system is studied analytically in order to investigate an interplay between conformists and contrarians on spheres. Our study demonstrates that conformists–contrarians models on spheres display the same types of equilibria and dynamical phenomena in all dimensions. However, critical combination of parameters, for which particular equilibrium states arise, does depend on the dimension. In particular, models on spheres exhibit traveling waves consisting of contrarians. We derive an exact formula for the relation between parameter values for which such waves arise in different dimensions. Finally, we take a closer look at trajectories of traveling waves on spheres, demonstrating subtleties of this dynamical phenomenon.