Transitions in active rotator systems: invariant hyperbolic manifold approach
Résumé
Our main focus is on a general class of active rotators with mean field interactions, that is, globally coupled large families of dynamical systems on the unit circle with nontrivial stochastic dynamics. The dynamics of each isolated system is d psi(t) = -delta V'(psi(t))dt + dw(t), where V' is a periodic function, w is a Brownian motion, and delta is an intensity parameter. It is well known that the interacting dynamics is accurately described, in the limit of infinitely many interacting components, by a Fokker-Planck PDE and the model reduces for delta = 0 to a particular case of the Kuramoto synchronization model, for which one can show the existence of a stable normally hyperbolic manifold of stationary solutions for the corresponding Fokker-Planck equation. (We are interested in the case in which this manifold is nontrivial, which happens when the interaction is sufficiently strong, that is, in the synchronized regime of the Kuramoto model.) We use the robustness of normally hyperbolic structures to infer qualitative and quantitative results on the vertical bar delta vertical bar <= delta(0) cases with delta(0) a suitable threshold; in fact, we obtain an accurate description of the dynamics on the invariant manifold for delta not equal 0 and we link it explicitly to the potential V. This approach allows us to have a complete description of the phase diagram of the active rotator model, at least for vertical bar delta vertical bar <= delta(0), thus identifying for which values of the parameters (notably, noise intensity and/or coupling strength) the system exhibits periodic pulse waves or stabilizes at a quiescent resting state. Moreover, some of our results are very explicit and this brings a new insight into the combined effect of active rotator dynamics, noise, and interaction. The links with the literature on specific systems, notably neuronal models, are discussed in detail.