Large-time limit of the general solution of an ODE is a fixed-point. Is the fixed-point stable?

Question

This question might well have an obvious affirmative answer (or an obvious counterexample!), which at present I cannot see. Suppose I have a first-order ODE $$ u'(t)=f(u) $$ whose general solution depends on one arbitrary constant $A$. Suppose that I take the large-$t$ limit of this general solution, and I find that $$ \lim_{t\to\infty}u_{gen}(t)=u^\star\ , $$ where $u^\star$ is a number (which does not depend on $A$ anymore). Furthermore, $u^\star$ is a fixed-point of my ODE, namely $f(u^\star)=0$. Can I immediately conclude that $u^\star$ is a stable fixed point? Or are there (sufficiently pathological, I guess) cases where the large-$t$ limit is an unstable fixed-point irrespective of the initial conditions? Thanks for your help, folks.

Answer 1

Consider the equation $$ \begin{cases} x'=x^2-y^2,\\ y'=2xy. \end{cases} $$ In order to draw the phase portrait just rewrite it in the form $z'=z^2$ on the complex plane.

This is not yet what you want (depends on what you want...) since the negative part of the horizontal axis tends to infinity. But you can see the differential equation on the $2$-sphere, in which case it has exactly one equilibrium point and all orbits tend to the equilibrium point (both when the time goes to $+\infty$ and to $-\infty$).

Answer 2

A somewhat more elementary example, where $r,\theta$ are polar coordinates: $$ \dot r=r(1-r)\\ \dot \theta=\sin^2\theta/2. $$ Point $(1,0)$ attracts all the orbits except the one starting at the origin. But is unstable.