Super-exponential solutions: their numerical modelling and detection

Certain functional differential equations may have exact solutions that either grow or decay at a rate that is faster than exponential. This provides a challenge for conventional mathematical analysis of the equations because ideas based on characteristic functions need to be revisited and generalised. It turns out that it is not always straightforward to solve equations with super-exponential solutions, nor is it usually possible to detect in advance whether or not they are present. The purpose of this talk is to describe approaches for the numerical solution of delay differential equations whose solutions decay at a rate that is faster than exponential. We show that we can find good approximations to the exact solutions in this way and that we can also use our methods to predict when super-exponential solutions are present. In conclusion we are able to show that there are situations where we have been able to predict the existence of super-exponential solutions by our methods and that these predictions have been subsequently confirmed analytically.