Dynamics and stability of chimera states in two coupled populations of oscillators

We consider networks formed from two populations of identical oscillators, with uniform strength all-to-all coupling within populations, and also between populations, with a different strength. Such systems are known to support chimera states in which oscillators within one population are perfectly synchronised while in the other the oscillators are incoherent, and have a different mean frequency from those in the synchronous population. Assuming that the oscillators in the incoherent population always lie on a closed smooth curve $\mathcal{C}$, we derive and analyse the dynamics of the shape of $\mathcal{C}$ and the probability density on $\mathcal{C}$, for four different types of oscillators. We put some previously derived results on a rigorous footing, and analyse two new systems.