Hunt for chimeras in fully coupled networks of nonlinear oscillators

Abstract

The purpose of this work is to study the dynamic properties of solutions to special systems of ordinary differential equations, called fully connected networks of nonlinear oscillators. Methods. A new approach to obtain periodic regimes of the chimeric type in these systems is proposed, the essence of which is as follows. First, in the case of a symmetric network, a simpler problem is solved of the existence and stability of quasi-chimeric solutions — periodic regimes of two-cluster synchronization. For each of these modes, the set of oscillators falls into two disjoint classes. Within these classes, full synchronization of oscillations is observed, and every two oscillators from different classes oscillate asynchronously. Results. On the basis of the proposed methods, it is separately established that in the transition from a symmetric system to a general network, the periodic regimes of two-cluster synchronization can be transformed into chimeras. Conclusion. The main statements of the work concerning the emergence of chimeras were obtained analytically on the basis of an asymptotic study of a model example. For this example, the notion of a canonical chimera is introduced and the statement about the existence and stability of solutions of chimeric type in the case of asymmetry of the network is proved. All the results presented are extended to a continuous analogue of the corresponding system. The obtained results are illustrated numerically.