Usually, multiple scales analysis (e.g. Poincaré-Lindstedt method or other multi-scale expansions) is applied on an ODE. Suppose we start from the exact solution of the ODE, how do we obtain the approximate solution from the exact solution directly using multiple scales, without starting from the underlying ODE?
For example, I have an oscillatory function $$x(t) = \frac{1+b^2 \cos(f(t)) \cos(at)}{\sqrt{(1+b^2)(1+b^2 \cos^2(a t))}}$$
where $f(t) = \int_{0}^{t} dt^\prime \sqrt{1+b^2 \cos^2(a t^\prime)} $, and $0 < a \ll b \ll 1$ so we have three scales. It can be shown that $x(t)$ exhibits a bounded oscillation between $-1$ and $1$, but the Taylor expansion gives secular terms linear in $t$. So regular perturbation theory fails here. Is there a way to obtain a reasonable approximation of $x(t)$ using the idea of multiple scales?