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.
- 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
If you like this web page, please link to it
so others can find it more easily.
