Theory of transient chimeras in finite Sakaguchi-Kuramoto networks

Abstract: Chimera states are a phenomenon in which order and disorder can co-exist within a network that is fully homogeneous. Precisely how transient chimeras emerge in finite networks of Kuramoto oscillators with phase-lag remains unclear. Utilizing an operator-based framework to study nonlinear oscillator networks at finite scale, we reveal the spatiotemporal impact of the adjacency matrix eigenvectors on the Sakaguchi-Kuramoto dynamics. We identify a specific condition for the emergence of transient chimeras in these finite networks: the eigenvectors of the network adjacency matrix create a combination of a zero phase-offset mode and low spatial frequency waves traveling in opposite directions. This combination of eigenvectors leads directly to the coherent and incoherent clusters in the chimera. This approach provides two specific analytical predictions: (1) a precise formula predicting the combination of connectivity and phase-lag that creates transient chimeras, (2) a mathematical procedure for rewiring arbitrary networks to produce transient chimeras.