Small Solutions: Problems, Consequences and Detection

The existence of (super-exponentially decaying)} solutions to an equation is a potential problem to the Mathematical modeller. In the light of recent work on Mixed Type Functional Equations (MFDEs), also known as Forward-Backward Equations, we discuss research relating to Delay Differential Equations (DDES) that might enhance our insight into MFDEs. We will:

1. discuss the concept of a small solution;
2. outline some of the potential problems associated with the existence of small solutions;
3. give details of the methodology underpinning our numerical approach to the detection of small solutions (to Delay Differential Equations).

We include a summary of known analytical results and of our numerical approach. We consider the characteristic shapes of the resulting eigenspectra and hence justify our decision to pursue a statistical analysis of the eigenvalue data. We consider the following statistical parameters as possible ways of determining whether or not an equation admits small solutions.

1. Standard deviation.
2. Skewness (Is the distribution symmetrical?)
3. Kurtosis (Is the distribution ‘peaked’?)
4. Spearman’s rank correlation (Can we identify a monotonic relationship?)

We justify our conclusion that, based on the statistical analysis carried out to date, this method of ‘decision making’ was not reliable and indicate how this part of our project informed the next stage of our work.