Nonlinear system of differential equations

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Hi! I'm working with my PhD thesis at the moment, and I've stumbled upon a pretty involved problem. What I have is a system of equations like this:

[itex]\frac{dx}{dt} = A \cos(z)[/itex]
[itex]\frac{dy}{dt} = B x \frac{dx}{dt}[/itex]
[itex]\frac{dz}{dt} = y[/itex]

where [itex]A[/itex] and [itex]B[/itex] are constants. I also have a stochastic term to [itex]z[/itex] according to:

[itex]\delta z(t) = \lim_{N \rightarrow \infty}\pi\sqrt{\frac{t}{N\tau}}\sum_{i = 1}^{N}\zeta_i[/itex]

where [itex]\zeta_i[/itex] are random numbers of unit variance (normal distributed probability), and [itex]\tau[/itex] is the time scale for the decorrelation of [itex]z[/itex]. I wish to calculate the variance of [itex]x[/itex] as a result of the stochastic variation of [itex]z[/itex], i.e.,

[itex]\langle(\Delta x - \langle\Delta x\rangle)^2\rangle[/itex]

where [itex]\Delta x = x(\tau) - x(0)[/itex] and [itex]\langle ... \rangle[/itex] is the average of the expression within the brackets with respect to a variation of the values of [itex]\zeta_i[/itex], weighted according to their probability. I've already calculated the variance of [itex]x[/itex] for [itex]B = 0[/itex] for which [itex]z = y t + z_0[/itex] and [itex]dx/dt[/itex] can simply be integrated in time to obtain an analytical expression for [itex]x(t)[/itex]. How can I continue to get a more general solution to the problem? Can I e.g. use some perturbation theory for small values of [itex]B[/itex] to begin with?
 
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  • #2
hunt_mat
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If you want a numerical solution, then I would go for Newton's method, send me a message if you want further help.
 
  • #3
Mute
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Perhaps the following will be a bit helpful?

[tex]\dot{y}/\dot{x} = \frac{dy}{dx} = Bx.[/tex]

You can then solve for [itex]y(x)[/itex]. Similarly,

[tex]\dot{z} = \frac{dz}{dt} \frac{1}{\dot{x}} \Rightarrow \frac{dz}{dt} = \dot{x} y(x) = A \cos (z) y(x).[/tex]

This is a separable equation that you can use to solve for z(x). Once you solve for z(x), you can plug that into your equation for x to get

[tex]\frac{dx}{dt} = A\cos z(x),[/tex]

which is again separable (but you may not be able to express the integral in terms of elementary functions. I haven't tried but I'm guessing the integral to do won't be nice).

I'm not entirely sure how to add in the stochastic term, but hopefully this can get you started with something.
 
  • #4
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The system of 3 EDOs can be analytically solved, but the result has to be expressed on a parametric form, because the last integral cannot be expessed in terms of a finite number of elementary functions. Even in the simplest cases of constants C1 and C2 (nul for example), the integral involves some elliptic functions on a very complicated form.
 

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  • #5
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Oh, this is really great! Thanks everyone for your help :D
 

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