In Part I, a method is developed for treating reacting-diffusingsystems whose kinetics lead to a limit cycle behavior with frequencyvarying slowly with position. This is accomplished by including theeffects of the spatial variation in a phase variable that is introducedto parametrize the limit cycles at each point. The normalization ofthis phase variable is chosen such that the limit cycles at each pointhave the same frequency with respect to this variable. The method ismotivated by treating the weakly coupled behavior of two oscillatorsthat, unlike previous works, are not nearly identical. The analogybetween discrete oscillator systems and continuous chemically reactingmixtures is explored by treating a chain of coupled limit cycleoscillators and considering various limits as the number of oscillatorstends to infinity.
In Part II a different effect of spatial nonuniformities is studied:the modification of the simple diffusion approximation to treat situationsof large deviations from equilibrium distributions in a mixture orproximity to some critical temperature. A technique for treatingbifurcation from the continuous spectrum by introducing a slow spacescale is developed. Using this, the onset of a spatially nonuniformstate in a model chemically reacting system is studied, occurringwhen some reaction rate crosses the values at which linearized theorypredicts destabilization of the uniform steady state solutions admittedby the system. Subs critical bifurcation is also found possible.
