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Article
Slow manifold of multiscale fast/slow SDEs or ODEs
By
Tony Roberts
Print
Via this web page you obtain a slow manifold of any supplied stochastic differential equation (
SDE
), or deterministic, autonomous or non-autonomous,
ODE
, when the
SDE
has fast and slow modes. The slow manifold supplies you with a faithful large time model of the stochastic dynamics. Being justified by a normal form coordinate transform you are assured that the dynamics are attractive over some finite domain and apply for all time. For example, this web page could help you analyse the stochastic bifurcation in the Stratonovich stochastic or determiistic non-autonomous system
dx/dt=epsilon*x-x*y ,
dy/dt=-y+x^2-2y^2+w(t) ,
where near the origin x(t) evolves slowly, y(t) decays quickly to some quasi-equilibrium, but the white noise or non-autonomous forcing w(t) `kicks' the system around. As parameter epsilon crosses zero, a stochastic bifurcation occurs. The stochastic or non-autonomous slow manifold, x=X(t)+..., contains the long term dynamics in the new variables X(t) so you are empowered to deduce the true slow dynamics near the bifurcation. Just click on the Submit button to see.
Keywords
Bifurcation analysis, Continuation, Control, Identification
Model
ODEs
Software Type
Other
Language
Other
Platform
Unix
Linux
Windows
MacOS
Availability
Available world wide via:
http://www.maths.adelaide.edu.au/anthony.roberts/sdesm.php
Contact Person
Tony Roberts, University of Adelaide,
[email protected]
References to Papers
A. J. Roberts. Normal form transforms separate slow and fast modes in stochastic dynamical systems. Physica A, 387:12--38, 2008.
http://dx.doi.org/10.1016/j.physa.2007.08.023
.
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