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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?

[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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