Effects of time delay in rate processes

Abstract

Time delay processes in chemical kinetics occur naturally and can be imposed by means of a delayed feedback. We study the temporal evolution of homogeneous chemical reaction mechanisms (the autonomous system) describable by ordinary differential equations subjected to an imposed time delay, a delayed feedback. A natural delay process can be decomposed into analogous components. At short delays the delayed feedback is shown to stabilize different types of unstable stationary states of the autonomous systems, as well as induce bistability in a monostable system. At longer delays transitions into chaos are predicted. First we study the chaotic response of autonomous bistable systems which support only stationary states. A recipe for finding chaos is given. Chaos occurs for a small parameter range in which the delayed feedback repeatedly reinjects trajectories into the neighborhood of one of the marginally stable points of the autonomous system. Secondly, we consider the imposition of a delayed feedback to a system in which periodic oscillations occur and show this leads to both chaos and hyperchaos when memory effects in the delay cause the system to be reinjected into the neighborhood of a saddle-focus point. The examples studied in detail are thermo-illuminated chemical reactions, but we also consider the applicability of our results to reactions in continuously stirred tank reactors and biochemical processes. The results of this study show that natural time delays can regulate a wide range of dynamical behavior in biochemical processes.