Bifurcation of Collective Oscillations

From now on, the reference period T is adjusted to be 27t by adopting a suitable time scale. Thus, o = 1/2n. Since giy/, t) and r y/) are 2 jr-periodic functions of y/, they may be expemded as 

The factor 1/2 ti in th first expression is only for later convenience. It is clear that Qi= Q-i and Fi = or 

The stability and bifurcation may be discussed in essentially the same way as in Chap. 2. Let D be taken as a control parameter, all the other parameters being kept constant. As we decrease D, the sign of Re (7/ changes from negative to positive, and this corresponds to the instability of the constant solution to the /th mode. Thus the critical value of D is given by 

In the light of the discussion in Sect. 5.2.2, negative I Pi implies repulsive coupling. Thus the reason for the absence of a phase transition is clear. 

Right at the threshold, D, the only undamped modes are the critical mode pair, so that, except for initial transients, the small amplitude Q y/, t) is expected to contain only these components, or 

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Bifurcation of Collective Oscillations 87 The inhomogeneous terms up to v = 3 have the forms... 

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