Overview
I provide both a web service or underlying code, 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 either
the web service or the code 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.
To use the web service
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.
To use the underlying code The code 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.
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