## Overview

I provide a procedure, download StoNormForm.zip, to construct a normal form system for a specified system of slow/fast/hyperbolic non-autonomous ordinary differential equations or stochastic differential equations. Thus the procedure may be used to construct and prove stochastic invariant manifolds, and their foliation, and thus analyse bifurcations or complex behaviour in the stochastic/non-autonomous system.

The procedure uses computer algebra, the package Reduce, to construct approximations to the invariant manifolds.

For example, the web form could help you analyse the stochastic bifurcation in the Stratonovich stochastic system $\begin{array}{l} \frac{dx}{dt}=a x-xy, \\ \frac{dy}{dt}=-y+x^2-2y^2+w(t) \end{array}$ where near the origin $$x(t)$$ evolves slowly, $$y(t)$$ decays quickly to some quasi-equilibrium, but the white noise$$~w(t)$$ `kicks' the system around. As parameter$$~a$$ crosses zero, a stochastic bifurcation occurs. A stochastic, near identity, coordinate transform, $$x=X(t)+\cdots$$ and $$y=Y(t)+\cdots$$, decouples the fast/slow dynamics in the new variables$$~X(t)$$ and$$~Y(t)$$ so you are empowered to deduce the true slow/fast dynamics in the bifurcation. Just click on the Submit button to see.