We consider various problems relating to synchronization in networks of coupled oscillators. In Chapter 2 we extend a recent exact solution technique developed for all-to-all connected Kuramoto oscillators to certain types of networks by considering large ensembles of system realizations. For certain network types, this description allows for a reduction to a low dimensional system of equations. In Chapter 3 we compute the Lyapunov spectrum of the Kuramoto model and contrast our results both with the results of other papers which studied similar systems and with those we would expect to arise from a low dimensional description of the macroscopic system state, demonstrating that the microscopic dynamics arise from single oscillators interacting with the mean field. Finally, Chapter 4 considers an adaptive coupling scheme for chaotic oscillators and explores under which conditions the scheme is stable, as well as the quality of the stability.
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