Systems of Weakly Coupled Oscillators

The neighborhood of the Hopf bifurcation point is itself an important asymptotic regime where the description of the dynamics is greatly simplified, as we saw in Chap. 2. Confining ourselves again to systems of oscillators which are coupled (but not necessarily through diffusion), there seems to exist at least one more physical situation for which an equally simplified description of the dynamics is expected. This is when the mutual coupling of the oscillators is weak if necessary, weak external forces may be included. The aim of this chapter is to present a simple perturbation treatment appropriate for such circumstances. 

Some peculiar features of the perturbation theory given here seem worth mentioning. The only assumption to be made in our theory is that the individual oscillators are weakly perturbed, and nothing is assumed about the specific nature of those oscillators. They may be quite general ones, except that they obey a system of n ordinary differential equations the system need not lie near some bifurcation point nor in any other extreme situation. Since limit cycle motion 

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In excitable reaction-diffusion systems, pulses can travel as a periodic wave train. In oscillatory reaction-diffusion systems, too, the existence of plane wave solutions has been theoretically established (Kopell and Howard, 1973 a). In this section we will be concerned with such periodic waves in one space dimension, particularly when the local wavenumber slowly and slightly varies with x. For these systems, the analogy to systems of weakly coupled oscillators might look even weaker. Actually, however, there exists a rather strong formalistic similarity between the two. 

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