An equation-free approach for heterogeneous networks : this dissertation is submitted for the degree of Doctor of Philosophy, School of Mathematical and Computational Sciences, Massey University

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Massey University
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Oscillators exist in many biological, chemical and physical systems. Often when oscillators with different periods of oscillations are coupled, they synchronize and oscillate with the same period. Examples include groups of synchronously flashing fireflies or chirping crickets. There are two questions of interest in this work. (1) Under what conditions will a system of coupled oscillators synchronize? and (2) Can a large system of synchronized oscillators be represented by a smaller number of variables? We study these questions for Kuramoto like models which are coupled in different ways. Examples include spatially extended and all-all coupling. We study conditions under which synchronization occurs in small and large networks by varying the coupling strength, calculating stabilities of synchronized solutions and creating bifurcation diagrams of the steady state solution as a function of coupling strength in one and two-dimensions. We use an equation free approach to approximate the coarse scale behavior of a large, coupled network, for which the equations are known, by a low dimensional description of variables, for which no governing equations are available in closed form. Our results show that a small number of variables can reproduce the behavior of the stable solutions in the full systems. However, the equation-free approach did not work as well for the unstable solutions. Possible reasons for this are explored in the thesis.
Difference equations, Oscillation theory, Numerical solutions, Synchronization, Mathematical models, Bifurcation theory