Normal form of stochastic or deterministic multiscale differential equations

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Part of the research
underlying this web service
was funded by the
Australian Research Council

Overview

Via this page you obtain a normal form of any supplied stochastic differential equation (SDE), or deterministic 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, this web page could help you analyse the stochastic bifurcation in the Stratonovich stochastic 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 w(t) `kicks' the system around. As parameter epsilon crosses zero, a stochastic bifurcation occurs. A stochastic, near identity, coordinate transform, x=X(t)+... and y=Y(t)+..., 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.

Submit your SDE for analysis

Fill in the fields below for your SDE system:

• your m slow variables must be denoted x(1),...,x(m);
• your ny fast stable variables must be denoted y(1),...,y(ny);
• your nz fast unstable variables must be denoted z(1),...,z(nz);
• the fast variables must be linearly decoupled, that is, the linear dynamics have been diagonalised; each of the linear decay/growth rates of the fast variables must be a positive number;
• any number of Stratonovich white noises (derivatives of Wiener processes) must be denoted w(any) where `any' denotes almost any label of your choice---numeric labels are usual;
• the noises w() must occur linearly in the RHSs of the SDEs, but may be multiplied by fast or slow variables;
• simply omit all w()'s to analyse a deterministic ODE;
• for the moment, the SDEs must be multinomial in form;
• Use the syntax of Reduce for the algebraic expressions (general examples)

Values of the fields for another example (adapted from Greg Pavliotis) are listed in the third column: dx1/dt=eps*y1, dx2/dt=eps*y2, dx3/d1=eps*(x1*y2-x2*y1), dy1/dt=-2*y1-a*y2+w1(t), dy2/dt=-3*y2+a*y1+w2(t).

Enter the magic word "a w h i l e" into

Description	Specify your SDE	Another example
Slow modes: the RHS of each of dx(1)/dt,...,dx(m)/dt separated by commas. Use w(any) to denote noise terms.	eps*x(1)-x(1)*y(1)	eps*y(1), eps*y(2), eps*(x(1)*y(2)-x(2)*y(1))
Fast stable modes the RHS of the ny SDEs dy(1)/dt,...,dy(ny)/dt each separated by commas. Use w(any) to denote noise terms.	-y(1)+x(1)^2-2*y(1)^2+w(1)	-2*y(1)-a*y(2)+w(1), -3*y(2)+a*y(1)+w(2)
Fast unstable modes the RHS of the nz SDEs dz(1)/dt,...,dz(nz)/dt each separated by commas. Use w(any) to denote noise terms.
Order of error of the analysis in the `nonlinear' terms on the RHS.	3 4 5	4
Print expressions with the following variables factored---this does not affect the analysis, but must not be empty. I auto-label all noise terms with variable sig.		sig,eps

Wait a minute or two

The analysis may take minutes after submitting. Be patient. Read the following. Please inform me of any problems.

In the results

• Each xx(i) denotes the new true slow variable X(i) where the original x(i)=X(i)+(nonlinear modifications), each yy(i) denotes the new fast stable variable Y(i) where the original y(i)=Y(i)+(nonlinear modifications), and each zz(i) denotes the new fast unstable variable Z(i) where the original z(i)=Z(i)+(nonlinear modifications),
• ou(w,tt,r) denotes Ornstein--Uhlenbeck convolution over the Stratonovich process w (not Ito): ou(w,tt,r)=exp(rt)*w where the asterisk * denotes convolution which is done over the past history of the process w when r<0, and done over the future (!) of the process w when r>0.
• For explanations and relevant theory, see my articles Normal form transforms separate slow and fast modes in stochastic dynamical systems [doi:10.1016/j.physa.2007.08.023] and Computer algebra derives normal forms of stochastic differential equations.

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