Phase-locking for dynamical systems on the torus and perturbation theory for mathieu-type problems

Summary

When several oscillators are coupled together and the parameters of their coupling are varied, the oscillators pass through so-called phase-locked regimes. In physical terms this means that the oscillators tend to synchronize their motion. To describe this phenomenon, we frame the concepts ofpartial phase andphase-locking. A partial phase of a toral flow puts emphasis on how orbits of the flow drift around the torus in some fixed direction. The partial phase is locked if it grows in time along some orbit slower than any linear function. When a toral flow is given by a trigonometric polynomial, its phase-locked regions are quite narrow. With the coupling amplitude increasing, each region grows in width as some power of the amplitude. That power can be calculated in terms of both the partial phase and degree of the trigonometric polynomial.