Abstract
Delay-differential equations belong to the class of infinite-dimensional dynamical systems. However, it is often observed that the solutions are rapidly attracted to smooth manifolds embedded in the finite-dimensional state space, called inertial manifolds. The computation of an inertial manifold yields an ordinary differential equation (ODE) model representing the long-term dynamics of the system. Note in particular that any attractors must be embedded in the inertial manifold when one exists, therefore reducing the study of these attractors to the ODE context, for which methods of analysis are well developed. This contribution presents a study of a previously developed method for constructing inertial manifolds based on an expansion of the delayed term in small powers of the delay, and subsequent solution of the invariance equation by the Fraser functional iteration method. The combined perturbative-iterative method is applied to several variations of a model for the expression of an inducible enzyme, where the delay represents the time required to transcribe messenger RNA and to translate that RNA into the protein. It is shown that inertial manifolds of different dimensions can be computed. Qualitatively correct inertial manifolds are obtained. Among other things, the dynamics confined to computed inertial manifolds display Andronov–Hopf bifurcations at similar parameter values as the original DDE model.
Highlights
Differential-equation models for the dynamics of chemical or biochemical systems often display a hierarchy of relaxation times [1,2,3]
If we are only interested in processes occurring on longer time scales, and not in the initial fast transients, a model constrained to a low-dimensional manifold allows us to study the important dynamics in a reduced model
A method for computing inertial manifolds of systems of Delay-differential equations (DDEs) has been developed for models of the form ẋ = f ( x (θ ), y(θ )), (42a)
Summary
Differential-equation models for the dynamics of chemical or biochemical systems often display a hierarchy of relaxation times [1,2,3] This hierarchy of relaxation times corresponds to a progressive dimensional collapse, in which the solutions are confined to manifolds in phase space of lower and lower dimensions [1,2]. If we are only interested in processes occurring on longer time scales, and not in the initial fast transients, a model constrained to a low-dimensional manifold allows us to study the important dynamics in a reduced model. These slow-timescale models are less stiff than their parent models [4]. The simplest case of an invariant manifold is a complete trajectory integrated forward and backward in time from some initial condition
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