Even in a linear second order equation like $x''+x'+\epsilon x=0$ the standard asymptotic expansion has a secular term already in the first order of $\epsilon$, namely $$x(t)=a_0+b_0e^{-t}+\epsilon(a_1+a_0t+b_1e^{-t}-b_0te^{-t})+O(\epsilon^2). $$ But the exact solution is obviously bounded uniformly in $t,\epsilon\geq0$, the expansion therefore is not valid for large times. The usual approach to eliminating secular terms is to use multiple time scales, but all perturbation theory texts I looked at (Holmes, Hunter, Kevorkian-Cole, Verhulst) only consider cases where the unperturbed equation is oscillatory. The Poincare-Lindstedt method or averaging that are used only make sense when there are oscillations. But the above equation is obviously non-oscillatory for small $\epsilon$. I am interested in non-linear equations that behave similarly to the linear example above, e.g. $$x''+x'\Big(1-\frac32 \frac{x'}{x}\Big)+\epsilon x^3=0.$$ They come up when approximately realizing mechanical constraints by viscous friction, which explains the absence of oscillations.

The issue seems to be that bounded solutions that are not exponentially stable produce secular terms even without oscillations, so one has all the pain of secular terms with no benefit of averaging. For instance, I suspect that for small $\epsilon$ solutions to the above nonlinear equation have a limit when $t\to\infty$, which depends on the initial values, it would be nice to approximate its dependence on them and $\epsilon$. Is there something of this sort done in the literature? References are appreciated.