A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painleve equations. The Lax pair associated to this equation is obtained, also by reduction.

The utility of mathematical models relies on their ability to predict the future from a known set of initial states. But there are non-linear systems, like the weather, where future behaviours are unpredictable unless their initial state is known to infinite precision. This is the butterfly effect. I will show how to analyse functions to overcome this problem for the classical Painleve equations, differential equations that provide archetypical non-linear models of modern physics.