## Overview

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

- So download and install Reduce, and then download StoNormForm.zip.
- Start-up Reduce in the unzipped
`StoNormForm`folder. - Execute
`in_tex "StoNormForm.tex"$`to load the procedure. - Test by executing
`examplenormalform();`and confirm the output is as in`stoNormForm.pdf` - See examples in
`manyExamples.pdf`and then try for systems of your interest.

**Maybe use the web service**Alternatively, click this link to expand this page. Then via the web form below you obtain a normal form of any supplied stochastic differential equation (SDE), or deterministic, autonomous or non-autonomous, ODE, when the SDE has fast and slow modes. The normal form decouples the slow modes from the fast and so supplies you with a faithful large time model of the stochastic dynamics. Being a coordinate transform you are assured that the dynamics are attractive over some finite domain and apply for all time.

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.

The source code is now available for collaborative
development via the folder `StochasticNormalForm` of
a
Github repository

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