Dynamics of three coupled limit cycle oscillators with vastly different frequencies

A system of three coupled limit cycle oscillators with vastly different frequencies is studied. The three oscillators, when uncoupled, have the frequencies omega (1)=O(1), omega (2)=O(1/epsilon) and omega (3)=O(1/epsilon (2)), respectively, where epsilon a parts per thousand(a)1. The method of direct partition of motion (DPM) is extended to study the leading order dynamics of the considered autonomous system. It is shown that the limit cycles of oscillators 1 and 2, to leading order, take the form of a Jacobi elliptic function whose amplitude and frequency are modulated as the strength of coupling is varied. The dynamics of the fastest oscillator, to leading order, is unaffected by the coupling to the slower oscillator. It is also found that when the coupling strength between two of the oscillators is larger than a critical bifurcation value, the limit cycle of the slower oscillator disappears. The obtained analytical results are formal and are checked by comparison to solutions from numerical integration of the system.