Overview
Via this page you obtain a normal form of any supplied
stochastic differential equation (
SDE), or
deterministic, autonomous or non-autonomous,
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
\[\begin{array}{l}
\frac{dx}{dt}=\epsilon x-xy, \\
\frac{dy}{dt}=-y+x^2-2y^2+w(t)
\end{array}\]
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, \(x=X(t)+\cdots\) and \(y=Y(t)+\cdots\),
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/ODE 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)
More interesting examples are listed at the end of this web page.
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 \(x_i=X_i+(\text{nonlinear modifications})\),
each yy(i) denotes the new fast stable variable \(Y_i\)
where the original \(y_i=Y_i+(\text{nonlinear modifications})\),
and each zz(i) denotes the new fast unstable variable \(Z_i\)
where the original \(z_i=Z_i+(\text{nonlinear modifications})\),
- ou(w,tt,r) denotes Ornstein--Uhlenbeck
convolution over the autonomous forcing/Stratonovich process \(w(t)\) (not Ito): ou(w,tt,r)\({}=\exp(rt)\star w(t)\) where the star
* denotes convolution which is done over the past
history of the process \(w(t)\) when \(r<0\), and
done over the future (!) of the process \(w(t)\) 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.
Levy area contraction
Another example system (adapted from
Greg Pavliotis) is
\[\begin{array}{l}
\frac{dx_1}{dt}=\epsilon y_1,\\
\frac{dx_2}{dt}=\epsilon y_2,\\
\frac{dx_3}{dt}=\epsilon (x_1y_2-x_2y_1),\\
\frac{dy_1}{dt}=-2y_1-ay_2+w_1(t),\\
\frac{dy_2}{dt}=-3y_2+ay_1+w_2(t).
\end{array}\]
Copy and paste the following into the above form.
| Description
| Specify your SDE/ODE
|
| Slow modes:
| small*y(1), small*y(2), small*(x(1)*y(2)-x(2)*y(1))
|
| Fast stable modes
| -2*y(1)-a*y(2)+w(1), -3*y(2)+a*y(1)+w(2)
|
| Fast unstable modes
|
|
| Order of error | 4
|
| Print | sig,small
|
To cope with the off-diagonal terms, the algorithm modifies the system to
\(
\dot x_{1}=\varepsilon ^{2} y_{1}\,,\quad\)
\(
\dot x_{2}=\varepsilon ^{2} y_{2}\,,\quad\)
\(
\dot x_{3}=\varepsilon ^{2} \big(-x_{2} y_{1}+x_{1} y_{2}\big)\,,\quad\)
\(
\dot y_{1}=\sigma w_{1}-\varepsilon y_{2} a-2 y_{1}\,,\quad\)
\(
\dot y_{2}=\sigma w_{2}+\varepsilon y_{1} a-3 y_{2}\,,\)
and computes the following results in which the new slow and fast variables have been nonlinearly separated by the time dependent coordinate transform.
Near identity, time dependent coordinate transform
\(
y_{1}\approx \sigma \varepsilon \big(-{e^{-2t}\star}w_{2}\, a+{e^{-3t}\star}w_{2}\, a\big)+\sigma {e^{-2t}\star}w_{1}\,+\varepsilon Y_{2} a+Y_{1}\,,\quad\)
\(
y_{2}\approx \sigma \varepsilon \big({e^{-2t}\star}w_{1}\, a-{e^{-3t}\star}w_{1}\, a\big)+\sigma {e^{-3t}\star}w_{2}\,+\varepsilon Y_{1} a+Y_{2}\,,\quad\)
\(
x_{1}\approx -1/2 \sigma \varepsilon ^{2} {e^{-2t}\star}w_{1}\,-1/2 \varepsilon ^{2} Y_{1}+X_{1}\,,\quad\)
\(
x_{2}\approx -1/3 \sigma \varepsilon ^{2} {e^{-3t}\star}w_{2}\,-1/3 \varepsilon ^{2} Y_{2}+X_{2}\,,\quad\)
\(
x_{3}\approx \sigma \varepsilon ^{2} \big(-1/3 {e^{-3t}\star}w_{2}\, X_{1}+1/2 {e^{-2t}\star}w_{1}\, X_{2}\big)+\varepsilon ^{2} \big(1/2 X_{2} Y_{1}-1/3 X_{1} Y_{2}\big)+X_{3}\,.\)
Resultant normal form S/ODEs
\(
\dot Y_{1}\approx -\varepsilon ^{2} Y_{1} a^{2}-2 Y_{1}\,,\quad\)
\(
\dot Y_{2}\approx \varepsilon ^{2} Y_{2} a^{2}-3 Y_{2}\,,\quad\)
\(
\dot X_{1}\approx -1/6 \sigma \varepsilon ^{3} w_{2} a+1/2 \sigma \varepsilon ^{2} w_{1}\,,\quad\)
\(
\dot X_{2}\approx 1/6 \sigma \varepsilon ^{3} w_{1} a+1/3 \sigma \varepsilon ^{2} w_{2}\,,\quad\)
\(
\dot X_{3}\approx \sigma \varepsilon ^{3} \big(1/6 w_{2} X_{2} a+1/6 w_{1} X_{1} a\big)+\sigma \varepsilon ^{2} \big(1/3 w_{2} X_{1}-1/2 w_{1} X_{2}\big).\)
Travelling waves in fluctuating kdV example
Potzsche and Rasmussen (2006) [Example 5.4] sought travelling wave solutions, \(u(x ? c^2t)\) with wave speed \(c^2\), of a modified KdV equation. This leads to the system \(\dot u_1=u_2\), \(\dot u_2=u_3\), \(\dot u_3=c^2u_2-a(t)u_1^2u_2\). For simplicity I set \(c^2=1\). A transform to diagonalise the linear part into slow variable \(x\), stable \(y\) and unstable \(z\) is then that \(u_1 = x+y+z\), \(u_2 = z - y\) and \(u_3 = z + y\).
Using
w(a) to denote the variable coefficient\(~a(t)\) (represented in the output by\(~\sigma w_a\)), copy and paste the following into the above form.
| Description
| Specify your SDE/ODE
|
| Slow modes:
| w(a)*(x(1)+y(1)+z(1))^2*(z(1)-y(1))
|
| Fast stable modes
| -y(1)-w(a)*(x(1)+y(1)+z(1))^2*(z(1)-y(1))/2
|
| Fast unstable modes
| +z(1)-w(a)*(x(1)+y(1)+z(1))^2*(z(1)-y(1))/2
|
| Order of error | 3
|
| Print | sig,small
|
Putting \(Z_1 = 0\) into the resulting coordinate transform gives the centre-stable manifold; conversely, putting \(Y_1 = 0\) gives the centre-unstable manifold. The expressions have the same convolutions as those of Potzsche and Rasmussen (2006).
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