APPROXIMATING STATE-SPACE MANIFOLDS WHICH ATTRACT SOLUTIONS OF SYSTEMS OF DELAY-DIFFERENTIAL EQUATIONS

Although the theory of delay-differential equations (DDEs) is generall y best set in a function space, some systems of DDEs have solutions wh ich, after the decay of transients, lie on a low-dimensional manifold in their state space. When the delay is small, highly accurate approxi mations to the state-space manifold which attracts the solutions can b e constructed by a simple functional equation treatment. This allows t he reduction of the original system of DDEs to a smaller system of ord inary differential equations. The simplified model obtained may be use d to facilitate bifurcation analysis. The method is applied to two bio chemical models, namely to a delay-differential version of Michaelis-M enten kinetics (the Brown model) and to a simple inducible operon mode l. (C) 1998 American Institute of Physics. [S0021-9606(98)01343-9].