Symmetry-broken states on a spherical surface of coupled oscillators: From modulated coherence to spot and spiral chimeras

We consider nonlocally coupled oscillators on the surface of a sphere. We analyze the existence and stability of a variety of symmetry-broken states, including the well-known spot and spiral chimeras as well as a new type of symmetry-breaking, the modulated coherent state. The stabilities of different stationary solutions along the Ott-Antonsen invariant manifold are explored and a complete bifurcation diagram is presented. Our bifurcation analysis reveals that the modulated coherent state bifurcates from a uniformly coherent state and coexists with the spiral chimera. We find that the spiral chimera could appear via an inverse Hopf bifurcation and a transcritical bifurcation. Unlike the spot chimera that coexists with uniform coherence, the spiral chimera could occur exclusively for a large parameter region. It is demonstrated numerically that the modulated coherent and spiral chimera states occur even for a finite-size system.