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.

  1. So download and install Reduce, and then download StoNormForm.zip.
  2. Start-up Reduce in the unzipped  StoNormForm  folder.
  3. Execute  in_tex "StoNormForm.tex"$  to load the procedure.
  4. Test by executing  examplenormalform();  and confirm the output is as in  stoNormForm.pdf 
  5. 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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