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We propose a coupled system of fast and slow phase oscillators. We observetwo-step transitions to quasi-periodic motions by direct numerical simulationsof this coupled oscillator system. A low-dimensional equation for orderparameters is derived using the Ott-Antonsen ansatz. The applicability of theansatz is checked by the comparison of numerical results of the coupledoscillator system and the reduced low-dimensional equation. We investigatefurther several interesting phenomena in which mutual interactions between thefast and slow oscillators play an essential role. Fast oscillations appearintermittently as a result of excitatory interactions with slow oscillators ina certain parameter range. Slow oscillators experience an oscillator-deathphenomenon owing to their interaction with fast oscillators. This oscillatordeath is explained as a result of saddle-node bifurcation in a simple phaseequation obtained using the temporal average of the fast oscillations. Finallywe show macroscopic synchronization of the order 1:m between the slow and fastoscillators.

Cooperative dynamics in coupled systems of fast and slow phase oscillators