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I have been struggling with an ostensibly simple problem, that is how to apply perturbation analysis principles on a system of linear differential equations with linear perturbation of the following form:

$ \dot{x}=\frac{1}{\epsilon} a_0 x + a_1 z + bu\\ \dot{z} = c_0 x + c_1 z $

where $x,z,u\in\Re$ and $0<\epsilon<<1$. How can I decompose my system and write the solution $x(t),z(t)$ as an asymptotic expansion in $\epsilon$ ?

Update 1: Let us consider the case where $t=\mathcal{O}(\epsilon)$. Then, put $\tau=\frac{t}{\epsilon}=\mathcal{O}(1)$ and the above system becomes:

$ \frac{d x(\tau)}{d\tau}= a_0 x(\tau) + \epsilon a_1 z(\tau) + \epsilon bu\\ \frac{d z(\tau)}{d\tau} = \epsilon c_0 x + \epsilon c_1 z $

and let us now set $x(\tau)=x(\frac{t}{\epsilon})=x_0(\tau)+\epsilon x_1(\tau)+\mathcal{O}(\epsilon^2)$ and similarly $z(\tau)=z_0(\tau)+\epsilon z_1(\tau)+\mathcal{O}(\epsilon^2)$. Then we have the following inner system:

$ \frac{d x_0}{d\tau}=a_0 x_0(\tau)\\ \frac{d x_1}{d\tau}=a_0 x_1(\tau) + a_1 z_0(\tau) + bu(\tau)\\ \frac{d z_0}{d\tau}=0\\ \frac{d z_1}{d\tau}=c_0 x_0(\tau) + c_1 z_0(\tau) $

and then we may apply matching to both the outer (see @Jon's answer below) and the inner solution.

Question : Can it be considered expedient to consider an asymptotic expansion of the input variable $u$ like $u(t)=\frac{1}{\epsilon}\sum_{i\geq 0}\epsilon^i u_i(t)$?

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I would do the following. Just put $\lambda=\frac{1}{\epsilon}$ and now $\lambda\gg 1$. Now you will get

$ \lambda\dot{x}= a_0 x + \lambda a_1 z + \lambda bu\\ \dot{z} = c_0 x + c_1 z. $

Then put

$x(t)=x_0(t)+\frac{1}{\lambda}x_1(t)+\frac{1}{\lambda^2}x_2(t)+\ldots$

$z(t)=z_0(t)+\frac{1}{\lambda}z_1(t)+\frac{1}{\lambda^2}z_2(t)+\ldots$

that gives the first few equations for the perturbation series

$ \dot{x_0}= a_1 z_0 + bu\\ \dot{z}_0 = c_0 x_0 + c_1 z_0. $

$ \dot{x_1}= a_0 x_0 + a_1 z_1\\ \dot{z}_1 = c_0 x_1 + c_1 z_1. $

and so on.

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    @PantelisSopasakis: The assumption $u(t)=O(\epsilon)$ just holds if the external forcing is a small perturbation to your system. But this is just arbitrary and you can do as you like. On the other hand, you can add some dynamics to it and this should be in some way connected to the other variables of the system otherwise it remains an arbitrary forcing.2012-05-08