Cooperative dynamics in coupled systems of fast and slow phase oscillators

We propose a coupled system of fast and slow phase oscillators. We observe two-step transitions to quasi-periodic motions by direct numerical simulations of this coupled oscillator system. A low-dimensional equation for order parameters is derived using the Ott-Antonsen ansatz. The applicability of the ansatz is checked by the comparison of numerical results of the coupled oscillator system and the reduced low-dimensional equation. We investigate further several interesting phenomena in which mutual interactions between the fast and slow oscillators play an essential role. Fast oscillations appear intermittently as a result of excitatory interactions with slow oscillators in a certain parameter range. Slow oscillators experience an oscillator-death phenomenon owing to their interaction with fast oscillators. This oscillator death is explained as a result of saddle-node bifurcation in a simple phase equation obtained using the temporal average of the fast oscillations. Finally we show macroscopic synchronization of the order 1:m between the slow and fast oscillators.