Learning and phase tracking by frequency and weight adaptation for coupled networks of Kuramoto oscillators

Abstract

We consider synchronization between a pair of networks of Kuramoto oscillators. One network plays the role of a training network, the other a learning network. Our main result is an adaptive strategy which tunes the Kuramoto coupling strengths (weights) and the Kuramoto frequencies of the learning network to achieve tracking of the phases of the training network by the phases of the learning network. Tracking is proved using a Lyapunov stability approach. When the phases of the training network are not clustered, then the adaptive strategy also guarantees convergence of the learning weights and frequencies to those of the training network. When the phases of the training network are clustered, the adaptive strategy achieve partial convergence of the learning weights and frequencies characterized by the type of clustering of the training network. The results are illustrated by 4-node training and learning networks with a variety of clustering configurations.