Synchronization of Network Coupled Chaotic and Oscillatory Dynamical Systems

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

Multiscale Dynamics in Communities of Phase Oscillators

We investigate the dynamics of systems of many coupled phase oscillators with het- erogeneous frequencies. We suppose that the oscillators occur in M groups. Each oscillator is connected to other oscillators in its group with "attractive" coupling, such that the coupling promotes synchronization within the group. The coupling between oscillators in different groups is "repulsive"; i.e., their oscillation phases repel. To address this problem, we reduce the governing equations to a lower-dimensional form via the ansatz of Ott and Antonsen . We first consider the symmetric case where all group parameters are the same, and the attractive and repulsive coupling are also the same for each of the M groups. We find a manifold L of neutrally stable equilibria, and we show that all other equilibria are unstable. For M \geq 3, L has dimension M - 2, and for M = 2 it has dimension 1. To address the general asymmetric case, we then introduce small deviations from symmetry in the group and coupling param- eters. Doing a slow/fast timescale analysis, we obtain slow time evolution equations for the motion of the M groups on the manifold L. We use these equations to study the dynamics of the groups and compare the results with numerical simulations.Comment: 29 pages, 6 figure

The stability of adaptive synchronization of chaotic systems

In past works, various schemes for adaptive synchronization of chaotic systems have been proposed. The stability of such schemes is central to their utilization. As an example addressing this issue, we consider a recently proposed adaptive scheme for maintaining the synchronized state of identical coupled chaotic systems in the presence of a priori unknown slow temporal drift in the couplings. For this illustrative example, we develop an extension of the master stability function technique to study synchronization stability with adaptive coupling. Using this formulation, we examine local stability of synchronization for typical chaotic orbits and for unstable periodic orbits within the synchronized chaotic attractor (bubbling). Numerical experiments illustrating the results are presented. We observe that the stable range of synchronism can be sensitively dependent on the adaption parameters, and we discuss the strong implication of bubbling for practically achievable adaptive synchronization.Comment: 21 pages, 6 figure