Consider the ODE $$y''+y'+f(y)=0,\quad (*)$$ with $f(y)=y+y^3$.
I'm interested in the asymptotic behavior of solutions $y(x)$ as $x\to\infty$. From Wolfram Alpha's example plots (and from what I've managed), I suspect that $\lim y=\lim y'=0.$ Since a priori, no term seems to be small compared to the others, my approach so far was to distinguish cases of different balances between the summands, but some cases are missing. The ones I've done are:
Firstly, if $y\to 0$ is indeed true, then $y^3<<y$, hence the “limit equation” $$y''+y'+y=0 $$ is linear and has explicit solutions, so it's easy to check that this case is consistent with the claim.
The next cases are the ones where we can neglect one of the derivatives. $$y'+f(y)=0$$ can be solved by integrating. Similarly straightforward, $$y''+f(y)=0$$ has an integrating factor, namely $y'$. In both cases, I can check the consistency of the solution with the assumptions.
Question: How can I treat the cases, $y<<y^3,$ and the case where no term becomes negligible with respect to the others? I don't see an a priori reason to discard the cases and proceeding as above seems to require my to solve $(*)$ explicitly or the not much easier $y''+y'+y^3=0.$ (But then I wouldn't need to do the asymptotics…) Any help is welcome, and I would be happy with taking a completely different approach.
Let $u = y$ and $v = y'$. We then have a first order system of ODEs $$ \begin{aligned} u' &= v \\ v' &= - u - v - u^3. \end{aligned} $$ Now consider the function $E(u,v) = \frac{1}{2}v^2 + \frac{1}{2}u^2 + \frac{1}{4} u^4$. From the chain rule, we have $$ \frac{d}{dt}E = E_uu' + E_vv' = -v^2 \leq 0, $$ so this function is nonincreasing along trajectories of the system. Also notice that $E = 0 \iff u,v=0$ and $E>0 \iff u,v \neq 0$. This function $E$ satisfies the properties of a Lyapunov function, so we can apply the Lyapunov stability theorem to conclude that the origin $(u,v)=(0,0)$ is stable, so all solutions tend to $y=y'=0$ as $t\to\infty$.
More detailed asymptotics can be obtained via linearization or dominant balance as you have stated.