One Oscillator Subject to Periodic Force

Let the perturbation p represent a weak periodic force of period T while the period of the native oscillator is denoted by T, i.e., 

It is expected that the oscillator comes to oscillate with exactly the same frequency as the external one if zl is below some critical value Ac- More general entrainment in which some integer multiples of T and T become identical could be treated in a similar manner by assuming 

Note that the above y/ is not the same as the quantity we have so far worked with under the same notation. It is obvious that if y/ is constant in time, this implies that the oscillator is entrained to the external periodicity. The equation for now takes the form 

This shows that is a slowly varying function of t. On the other hand, the quantity Q, if viewed as a function of t and y/, is T-periodic in t (and T-periodic in y/). Since the slow variable y/ would hardly change during the period T one may safey time-average Q over this interval with y/ kept constant. In this way, we obtain (after setting = 1) 

Entrainment to the external periodicity occurs if (5.2.9) has a stable-equilibrium solution. In that case, we have at least one stable-unstable pair of equilibria (denoted as y/ and y/ respectively) in the interval (0, T) (Fig. 5.1a). A given equilibrium solution y o is stable if dr/dy/ y is negative, and unstable if it is positive. It may of course happen that more stable-unstable pairs of equilibrium points appear (Fig. 5.1b). When the condition for entrainment is not satisfied (Fig. 5.1 c), the oscillator gains a frequency different from the external one. Let this frequency difference be A co, or 

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