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Normal form of multiscale fast/slow SDEs and ODEs

By Tony Roberts
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This web service algebraically derives a normal form of any in a wide class of stochastic differential equations (SDE), or deterministic, autonomous or non-autonomous, ODEs, when the SDE/ODE 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/autonomous/non-autonomous dynamics.

For example, consider the stochastic bifurcation, for small parameter \(\epsilon\), in the Stratonovich stochastic system \[\begin{array}{l} \frac{dx}{dt}=\epsilon x-xy, \\ \frac{dy}{dt}=-y+x^2-2y^2+w(t). \end{array}\] This movie shows some simulations from random initial conditions for different realisations of the random noise (at \(\epsilon=0.04\)): amongst the noise see that the solutions typically congregate somewhere near the curve \(y\approx x^2\). But is that correct? and what is the long term evolution amongst all the noise? The challenge is to tease all this, and more, from the equations themselves.

In this scenario, a stochastic or deterministic coordinate transform answers the challenge [1,2]. For the example system, and upon defining the history convolution \(e^{-t}\star w(t):=\int_0^te^{s-t}w(s)\,ds\), the near identity coordinate transform to \((X,Y)\) variables \[\begin{array}{l} x\approx X+XY+Xe^{-t}\star w,\\ y\approx Y+X^2+2Y^2+e^{-t}\star w, \end{array} \quad\Rightarrow\quad \begin{array}{l} \frac{dX}{dt}\approx \epsilon X-Xw,\\ \frac{dY}{dt}\approx -Y. \end{array}\] The movie, by displaying the time dependent coordinate curves, illustrates this stochastic coordinate transform for one realisation of the noise \(w(t)\): see how the different initial conditions converge onto and then track the fluctuating \(Y=0\) coordinate curve, as befits the exponential decay inherent in \(dY/dt\approx -Y\). Hence, from this coordinate transformation we know that the long term evolution of the system is \(dX/dt\approx \epsilon X-Xw\) on the stochastic slow manifold \(x\approx X+Xe^{-t}\star w\) and \(y\approx X^2+e^{-t}\star w\).

The computer algebra code underlying the web service [3] constructs analogous coordinate transforms for any system you enter. The web page shows how to enter the above example, and also lists a Levy area contraction example from Greg Pavliotis, and a travelling wave example from Potzsche and Rasmussen.

References
  1. Ludwig Arnold. Random Dynamical Systems. Springer Monographs in Mathematics. Springer, June 2003.
  2. A. J. Roberts. Normal form transforms separate slow and fast modes in stochastic dynamical systems. Physica A, 387:12–38, 2008. doi: 10.1016/j.physa.2007.08.023.
  3. A. J. Roberts. Normal form of stochastic or deterministic multiscale differential equations. Technical report, http://www.maths.adelaide.edu.au/anthony.roberts/sdenf.php, 2009. Revised April 2011, Feb 2012.
KeywordsBifurcation analysis, 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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