Date of Award

5-9-2016

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Dr. Igor Belykh

Second Advisor

Dr. Alexandra Smirnova

Third Advisor

Dr. Vladimir Bondarenko

Fourth Advisor

Dr. Yaroslav Molkov

Abstract

Synchronization is an important phenomenon which plays a central role in the function or dysfunction of a wide spectrum of biological and technological networks. Despite the vast literature on network synchronization, the majority of research activities have been focused on oscillators connected through one network. However, in many realistic biological and engineering systems the units can be coupled via multiple, independent networks. This thesis contributes toward the rigorous understanding of the emergence of stable synchronization in dynamical networks with mixed coupling. A mixed network is composed of subgraphs connecting a subnetwork of oscillators via one of the individual oscillator's variables. An illustrative example is a network of Lorenz systems with mixed couplings where some of the oscillators are coupled through the x-variable, some through the y-variable and some through both. This thesis presents a new general synchronization method called the Mixed Connection Graph method, which removes a long-standing obstacle in studying synchronization in mixed dynamical networks of different nature. This method links the stability theory, including the Lyapunov function approach with graph theoretical quantities. The application of the method to specific networks reveals surprising, counterintuitive effects, not seen in networks with one connection graph.

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