Chimera states through invariant manifold theory

We establish the existence of chimera states, simultaneously supporting synchronous and asynchronous dynamics, in a network consisting of two symmetrically linked star subnetworks consisting of identical oscillators with shear and Kuramoto--Sakaguchi coupling. We show that the chimera states may be metastable or asymptotically stable. If the intra-star coupling strength is of order \(\varepsilon\), the chimera states persist on time scales at least of order \(1/\varepsilon\) in general, and on time-scales at least of order \(1/\varepsilon^2\) if the intra-star coupling is of Kuramoto--Sakaguchi type. If the intra-star coupling configuration is sparse, the chimeras are asymptotically stable. The analysis relies on a combination of dimensional reduction using a M\"obius symmetry group and techniques from averaging theory and normal hyperbolicity.