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Slow manifold of multiscale fast/slow SDEs or ODEs

By Tony Roberts
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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.
KeywordsBifurcation analysis, Continuation, Control, Identification
Model
  • ODEs
Software Type
  • Other
Language
  • Other
Platform
  • Unix
  • Linux
  • Windows
  • MacOS
Availability
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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