One-Oscillator Problem

Although the system with which we are ultimately concerned comprises the continuous fields or discrete populations of coupled oscillators, it seems appropriate to begin with a simpler system whose study fully illustrates the perturbation idea. Once the method has been formulated for it, its extension to more complicated systems will turn out to be extremely easy. 

Let ATo(0 denote a linearly stable T-periodic solution of an n-dimensional system of ordinary differential equations (2.1.1), or 

Let C denote the closed orbit corresponding to Since C is supposed to 

The quantity 0 may be called the phase defined on C, and its value is only determined to an integer multiple of T, It would be very inconvenient, however, if the definition of phase were restricted to C. This is because arbitrarily small perturbations could generally kick the state point out of C, so that without definition of 0 outside C we could no longer say anything about the phase of perturbed oscillations. It is therefore desirable to extend the definition of 0 so that we could say, e.g., how the phase, or its rate of change, is influenced by the perturbation. Since the perturbation is assumed to be weak, 0 need only be defined in the vicinity of C. 

To make the picture clearer, we imagine a circular tube through which the orbit C threads (Fig. 3.1). We want to define I X) for each A inside the tube. We use here a language appropriate to a three-dimensional state space, but actually we are working with an ( 2)-dimensional system. Let G denote this w-dimen-sional tubular region containing all neighborhoods of C. The domain of attraction of C is assumed to contain G inside it. The tube may be thin to the extent that the perturbation is weak. 

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