Phase waves in oscillatory chemical reactions

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Abstract
A theory is presented for the effect of heterogeneity on an oscillatory chemically reactive system in a stable limit cycle. We develop a perturbation technique free of secular behavior for the solution of the nonlinear partial differential equations. The solutions are obtained in terms of a phase function which obeys a diffusion equation. Due to this diffusion there exist phase waves which propagate down the phase gradient. We derive equations for the position of the phase front as a function of time, the departure (arrival) times of waves, the wave pattern and the lifetime of each wave emitted from a center of heterogeneous catalysis, the interference effects between neighboring heterogeneities (pacemakers), and local renormalization effects (frequency changes) from heterogeneities extending over regions of nonzero extent. The theory is shown to break down for systems with relaxation oscillations but is confirmed quantitatively for more smoothly varying limit cycles. The spatial range of dynamic control of heterogeneity grows with time in an oscillatory reacting system, but is limited by the chemical relaxation time for a monotonically reacting system.