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The dynamics of a chemical system which is localized in two compartments between which mass transport occurs at a rate proportional to the difference in chemical concentrations between the two compartments have been studied. Pursuing a problem posed by Smale, we have tried to find a set of reactions such that when the cells are uncoupled there is a single unique stable steady state in each compartment, but when the cells are coupled there is a single unique global limit cycle oscillator in the combined system. We have analyzed open systems far from equilibrium in which there are two reacting chemicals. A linear stability analysis in the region of the homogeneous steady state has been performed for the case in which identical reactions are occurring in both compartments. We have also studied, using both local and global techniques, a class of systems in which different reactions are occuring in each compartment. For this latter class of systems we have been able to find evidence that there can be limit cycle oscillations which require transport in order to be sustained.
Glass et al. (Sun,) studied this question.