or instance, for a simple linear system we have $$\dot{q} = a q
    \quad \rightarrow \quad q(t) = q(0) e^{at},$$ and we can immediately
    understand that the long-term behavior of the system is a (stable) decaying
    exponential if $a<0$, an (unstable) growing exponential if $a>0$, and that
    the system does nothing if $a=0$. Here we are with certainly one of the
    simplest nonlinear systems we can imagine, and we can't even solve this
    system? </p>

    <p> All is not lost.  If what we care abou