Synchronization and Transient Dynamics in the Chains of Electrically Coupled Fitzhugh--Nagumo Oscillators

Chains of N FitzHugh–Nagumo oscillators with a gradient in natural frequencies and strong diffusive coupling are analyzed in this paper. We study the system’s dynamics in the limit of infinitely large coupling and then treat the case when the coupling is large but finite as a perturbation of the former case. In the large coupling limit, the 2N -dimensional phase space has an unexpected structure: there is an (N − 1)-dimensional cylinder foliated by periodic orbits with an integral that is constant on each orbit. When the coupling is large but finite, this cylinder becomes an analog of an inertial manifold. The phase trajectories approach the cylinder on the fast time scale and then slowly drift along it toward a unique limit cycle. We analyze these dynamics using geometric theory for singularly perturbed dynamical systems, asymptotic expansions of solutions (rigorously justified), and Lyapunov’s method.