Slow manifold of stochastic or deterministic multiscale differential equations

Via this 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 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. The stochastic 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.

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) or non-autonomous time dependent factors 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 an autonomous system of ODEs;
• for the moment, the SDEs must be either multinomial in form or a rational multinomial---if the latter, then multiply through by a common denominator transfer to the RHS the nonlinear terms involving time derivatives, to end up with a multinomial RHS that includes nonlinear terms with a time derivative factor;
• Use the syntax of Reduce for the algebraic expressions

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).

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 true slow variable X(i) where the original x(i)=X(i)+(nonlinear modifications).
• ou(w,tt,r) denotes 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 autonomous forcing/Stratonovich process w as 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.