Synchronization rates in classes of relaxation oscillators

Relaxation oscillators arise frequently in physics, electronics, mathematics, and biology. Their mathematical definitions possess a high degree of flexibility in the sense that through appropriate parameter choices relaxation oscillators can be made to exhibit qualitatively different kinds of oscillations. We study numerically four different classes of relaxation oscillators through their synchronization rates in one-dimensional chains with a Heaviside step function interaction and obtain the following results. Relaxation oscillators in the sinusoidal and relaxation regime both exhibit an average time to synchrony, S>∼n, where n is the chain length. Relaxation oscillators in the singular limit exhibit S>∼n^p, where p is a numerically obtained value less than 0.5. Relaxation oscillators in the singular limit with parameters modified so that they resemble spike oscillations exhibit S>∼log(n) in chains and S>∼log(L) in two-dimensional square networks of length L. Finally, using a sigmoid interaction results in S>∼n², for relaxation oscillators in the sinusoidal and relaxation regimes, indicating that the form of the coupling is a controlling factor in the synchronization rate.