Calculation of Ffor a Simple Model

Unfortunately, it is generally impossible to treat (5.2.21) analytically. We therefore consider in Sect. 5.4 a soluble model where the coupling and the distribution of the natural frequency oja are given by some simple functions. An interesting feature revealed, then, is that a critical condition for the appearance of collective oscillations exists. Before going into such discussions, however, we make a few remarks on a simple coupled-oscillator model for which r may be calculated explicitly. 

Consider a pair of Ginzburg-Landau oscillators which are interacting through a discretized version of diffusion. The evolution equations for the complex amplitudes W = X+ Y and W - X are then given by 

It is interesting to examine whether the pair coupling in the model is attractive or repulsive. We saw that the coupling is attractive if dF/d0 o 0, and repulsive if dr/dy/ Q Q. But (5.3.6) shows that 

4 Soluble Many-Osdllator Model Showing Synchronization-Desynchronization Transitions 

The population models of limit cycle oscillators which we obtained in Sect. 5.2 (c) seem to have been seldom investigated in the past. Although a general analytical treatment of (5.2.21) would be difficult, there certainly exists, in the limit of large N, a special subclass of systems for which a number of interesting analytical results are available. 

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