Symmetry breaking and recovering in a system of n hybridly coupled oscillators

We consider a ring of n Rayleigh oscillators coupled hybridly. Using the symmetrical property of the system we demonstrate the degeneracy of the Hopf bifurcation of the equilibrium at the origin. The degeneracy implies the existence and stability of the n-phase oscillation. We discuss some consequences of the perturbation of the symmetry. Then we study the case n = 3. We show the bifurcation diagram of the equilibria and of the periodic solutions. Especially, we analyze the mechanism for the symmetry breaking bifurcation of the fully symmetric solution. We report and explain the occurrence of both chaotic attractors and repellers and show two types of symmetry recovering crisis they undergo.