Synchronization in a Ring of Oscillators with Delayed Feedback

A ring of coupled oscillators with delayed feedback with various types of coupling between the oscillators is considered. For each type of coupling, the asymptotic behavior of the model solutions with respect to a large parameter is constructed for a wide variety of initial conditions. It is shown that the studying the behavior of solutions to the original infinite-dimensional models can be reduced to studying the dynamics of the constructed finite-dimensional mappings. High quality conclusions about the dynamics of the original systems are made. It is shown that the behavior of solutions significantly varies with variations in the type of coupling. Conditions on the system parameters are found under which the synchronization, two-cluster synchronization, and more complex modes are possible.