Synchronization of weakly perturbed Markov chain oscillators

Rate processes are simple and analytically tractable models for many dynamical systems which switch stochastically between a discrete set of quasi stationary states but they may also approximate continuous processes by coarse grained, symbolic dynamics. In contrast to limit cycle oscillators which are weakly perturbed by noise, the stochasticity in such systems may be strong and more complicated system topologies than the circle can be considered. Here we employ second order, time dependent perturbation theory to derive expressions for the mean frequency and phase diffusion constant of discrete state oscillators coupled or driven through weakly time dependent transition rates. We also describe a method of global control to optimize the response of the mean frequency in complex transition networks.Comment: 16 pages, 7 figure

Noise-Induced Synchronization of a Large Population of Globally Coupled Nonidentical Oscillators

We study a large population of globally coupled phase oscillators subject to common white Gaussian noise and find analytically that the critical coupling strength between oscillators for synchronization transition decreases with an increase in the intensity of common noise. Thus, common noise promotes the onset of synchronization. Our prediction is confirmed by numerical simulations of the phase oscillators as well as of limit-cycle oscillators

Disturbing synchronization: Propagation of perturbations in networks of coupled oscillators

We study the response of an ensemble of synchronized phase oscillators to an external harmonic perturbation applied to one of the oscillators. Our main goal is to relate the propagation of the perturbation signal to the structure of the interaction network underlying the ensemble. The overall response of the system is resonant, exhibiting a maximum when the perturbation frequency coincides with the natural frequency of the phase oscillators. The individual response, on the other hand, can strongly depend on the distance to the place where the perturbation is applied. For small distances on a random network, the system behaves as a linear dissipative medium: the perturbation propagates at constant speed, while its amplitude decreases exponentially with the distance. For larger distances, the response saturates to an almost constant level. These different regimes can be analytically explained in terms of the length distribution of the paths that propagate the perturbation signal. We study the extension of these results to other interaction patterns, and show that essentially the same phenomena are observed in networks of chaotic oscillators.Comment: To appear in Eur. Phys. J.