Lorenz system for $s=10$, $r=28$ and $b=8/3$ has unstable critical points, but they never go out of the butterfly shape

Question

I'm studying the Lorenz dynamical system, and I'm asking myself if the critical points are unstable critical points.

Considering the theory they are unstable - one eigenvalue $\in \mathbb{R}$ which is negative and 2 complex eigenvalues with a negative real part. But when I look at the critical points the results seem to oscillate around the critical value and never go to $\infty$.

It appears to me it's not an unstable critical point then? Can someone clarify which type of critical points these are?

Answer 1

The Lorenz system has a strange attractor: this means that trajectories are chaotic but nevertheless stay contained in a set. For the Lorenz system, this attractor has indeed the butterfly shape you notice.

Proving the Lorenz system indeed admits an attractor is a research question that was solved relatively recently. You can see this Nature paper which treats this issue, and which summarizes the research article by Tucker (1999) which first proves the existence of such an attractor.

Answer 2

That a critical point is unstable doesn't mean that neighbouring trajectories have to diverge to infinity.

The definition of stability (in the sense of Lyapunov) is that for every neighbourhood $U$ of the critical point there is a neighbourhood $V \subseteq U$ of the critical point, such that trajectories starting in $V$ at $t=0$ remain in $U$ for all $t \ge 0$. And "unstable" just means "not stable".

In this case, if you take an open set $U$ that contains the critical point but is small enough to stay clear of the "butterfly", then there is no such $V$. So the critical point is indeed unstable.