Insight into the Qualitative Behaviour of Numerical Solutions to Some Delay Differential Equations

We are interested in equations of the form y 0 (t) = f (y(t); y(t ?)): (z) In recent work we have considered, in particular, equations which exhibit supercritical Hopf bifurcations. We have investigated numerical methods applied to such equations and we have developed theoretical results that give conditions under which Hopf bifurcations arise in the discrete scheme. The theory developed to date applies to a limited class of problems and methods. In the present paper, we consider whether an alternative approach can yield new and improved results in this area. We consider a naturally corresponding ordinary diierential equation and explore how far existing theory for numerical solutions of ODEs can be adapted to the numerical solution of (z). We present our initial results which provide new insights for a simple class of numerical methods and we indicate how these results can be extended to provide analysis of qualitative behaviour of solutions for a range of problems and methods.