Synchronization, Learning and Classification for a System of Kuramoto Models

Abstract

Kuramoto systems or Kuramoto networks model the behaviour of large sets of coupled oscillators. Arising initially in the context of systems of chemical and biological oscillators, they now find applications in various areas of science, engineering and medicine, including neuroscience. A key property of Kuramoto networks is their synchronization behaviour: for a network with N oscillators, it is possible that all N oscillators synchronize, that several clusters of synchronized oscillators emerge, or that there is no synchronization of the oscillators. The behaviour is a function of the network parameters, namely coupling strengths and natural frequencies for the oscillators, as well as their initial conditions. In this thesis, we consider control systems theory approaches for Kuramoto networks that focus on adaptive learning of system parameters and phase tracking, observation-based classification of synchrony, and a combination of both. We first consider synchronization and learning approaches for pairs of Kuramoto networks. One network plays the role of a training network, the other is a learning network. We consider synchronization and system parameter learning based on phase information. Our main result is an adaptive learning strategy that tunes the system parameters of the learning oscillator – the Kuramoto coupling strengths and the natural frequencies – to achieve phase tracking, i.e. synchronization, between the training and learning phases. Tracking is proved using a Lyapunov stability approach. The adaptive strategy also guarantees partial convergence of the learning weights and frequencies to those of the training oscillator. Partial convergence is characterized by the linear dependence of the phase differences of the states of the training oscillator. The results are illustrated by a Kuramoto network with N = 4 oscillators. Secondly, and generalizing the synchronization and learning result, we consider networks where only output information is available and not all phases of the network i may be measured independently. A crucial aspect of this approach is the concept of observability and observer design for dynamical systems, i.e. how to make use of output information to recreate phase information. This is an unsolved problem for Kuramoto networks where a training system is not necessarily in an equilibrium state. To overcome this problem we develop a machine learning-based approach using so-called “fingerprints” of the networks output signals, i.e. spectrogram images that represent the possible synchronization behaviours. We use a simple artificial neural network architecture to develop a pattern recognition tools that classifies the “fingerprints” and thus the types synchrony as observed by outputs of Kuramoto networks of a fixed size. The approach is illustrated by simulation and classification results for Kuramoto networks with N = 4 and N = 7 oscillators. Using the classifier approach we then develop a switched systems adaptive control framework to determine the type of Kuramoto network responsible or able to create a given “fingerprint” that matches the “fingerprint” of the training system. Our second main result is an adaptive algorithm that can learn the behaviour of a Kuramoto network, from a set or family of possible networks, to match the output-based “fingerprint” of the training system. The results are illustrated for networks of N = 4 and N = 7 oscillators with a variety of synchrony outputs, respectively