The Effects of Coupling Delay on the Dynamics and Synchronization of Small Oscillator Networks

To describe the dynamics of a complex system, it is often useful to separate that large system into interacting subsystems. Networks are an excellent way to portray these systems, where the nodes of the network are the subsystems, and the edges represent interactions between them. Researchers who seek to describe the broad characteristics of such types of systems often construct toy model representations, where (as simplification), the subsystem behavior is simply described by some type of non-linear oscillator. Due to finite propagation times of the coupling signals in real world settings, `delay' is often added to these modeled interactions between subsystems. While the addition of delay greatly complicates the analysis of the coupled system, an interesting effect is that the delay may actually result in greater simplification of the dynamics admitted on the network. Not only do these networks produce interesting mathematics, but they also work as models for many applications such as brain activity, laser arrays, social interactions, thermo-optical oscillators, coupled circuits, protein interaction, and more.;In this thesis, we study the effects of coupling delay on the dynamics of small oscillator networks. Specifically, we study the symmetries, synchronization behaviors, and dynamics on a set of four-node oscillator networks with chaotic node dynamics and diffusive delay-coupling. We prove what properties a finite-sized network of this type must have when it admits complete synchronization and discuss when phase and generalized synchronization may occur. We also perform master stability function analysis for general networks with diffusive delay-coupling and find master stability functions with Rössler and Lorenz system node dynamics when the synchronized solution is a fixed point. Lastly, we apply our work to solve an instability issue in dc microgrids, where we show how delay coupling can be used to stabilize the operating voltage and current of these electric circuits.