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We know perturbation theory express the desired solution of differential equations in terms of a formal power series in some "small" perturbation parameters:

$y=y_0+\epsilon ^1 y_1+\epsilon ^2 y_2+\cdots$

Where $y_0$ is the solution to the exactly solvable system obtained by dropping out terms with $\epsilon$ in the original equation.

On the other hand, the method of dominant balance seems don't require the existence of small parameter $\epsilon$. It substitutes $y=e^{s(x)}$ into the equation and drop the negligible terms to get the dominant part $s_0(x)$. And does it iteratively with the substitution like $y(x)\sim e^{s_0(x)+c(x)}$ to get sub-dominant parts. Unlike perturbation theory, at every step we must re-evaluate the equation to determine the dominant terms (since the substitutions are different, we get different equations every time).

Here comes my question$-$is there any way to view the method of dominant balance as part of the perturbation theory? Or what is the connection between these two methods?

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    $\begingroup$ In my opinion the perturbation theory approach is part of the method of dominant balance. You're simply letting the powers of $\epsilon$ in your Poincaré expansion tell you which terms are balanced and which aren't. $\endgroup$ – Antonio Vargas Mar 16 '14 at 14:53

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