Overview
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
- 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)
Wait a minute or two
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 |
sig,small |
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 |
sig,small |
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