Functional Differential Equations

Researchers: Dr John Appleby, Dr David Reynolds
Ph.D. Students: Conall Kelly, Siobhan Devin

It is a well-established principle to model the evolution of physical, biological and economic systems using ordinary differential equations, in which the response of the system depends purely on the current state of the system. However, in many applications the response of the system can be delayed, or depend on the past history of the system in a more complicated way. Dynamical systems which respond in this way are called Functional Differential Equations (FDEs). Furthermore, in applications it is typical for the system to be perturbed by noise, be intrinsically random, or in which certain parameters in the model are unknown. In these cases, it is more appropriate to model the dynamics of the system using Stochastic Functional Differential Equations (SFDEs). The work of the Functional Differntial Equations Group involves the study of both FDEs and SFDEs, concentrating in particular on their long-time behaviour.

Areas of the sciences in which functional differential equations are applied include the study of materials with memory (viscoelastic materials); in mathematical demography and population dynamics, in the study of the dynamics of artificial neural networks in which there are transmission delays; and in problems in mathematical finance in which inefficient markets are modelled.

Some problems which are considered by the Group include

Asymptotic behaviour of Volterra integro-differential equations

The presence of a slowly fading memory in a functional differential equation has the potential to retard the speed at which solutions of such equations converge to their equilibrium states. Part of the group's activity centres on determining the rate of decay of solutions of scalar linear Volterra integro-differential equations. Recent work emphasises the robustness of such results, extending them to nonlinear and finite-dimsensional equations, as well as equations which are stochastically excited.

Asymptotic behaviour of stochastic differential- and functional differential-equations

In the Sciences, stochastic differential equations are used to model systems that are inherently random, or subject to random external perturbations. Furthermore, systems in Continuum Mechanics (such as those with viscoelastic materials) or in Financial Economics (in which agents form their decisions based on the market's past performance) have governing equations which involve integral terms representing the effect of the past. Stochastic functional-differential equations attempt to capture the effects of both randomness and memory. Current work includes explaining the slowed rate of convergence of solutions of such equations caused by a slowly decaying memory of the past, and determining conditions under which their solutions converge almost surely to equilibrium. The group also considers the slow convergence of highly nonlinear stochastic systems without delay as a means to understanding corresponding delay equations, and also as models of simulated annealing.

Qualitative behaviour of noise perturbed delay systems

Noisy perturbations of dynamical systems can lead to counter-intuitive effects, such as the stabilisation of deterministic systems to equilibrium states. The work of the group focusses on problems which involve the stabilisation and destabilisation of underlying deterministic delay systems by noisy perturbations, as well as the ability of noise to promote or suppress explosions of solutions of such equations in finite time. The joint presence of noise and delay in dynamical systems also can cause interesting oscillatory phenomena about equilibria which are not possible if either factor is absent.

Figure 1: Adding multiplicative noise to a 2-dimensional population dynamics model can suppress an explosion. The evolution of the deterministic model is given by blue and pink lines, that of the stochastic model is given by the red and green lines.

Abstract integro-differential systems modelling materials with memory