I'm reading a text about the Lorenz equations,
for $r>1$ and all the other parameters positive. At one point the author says
My questions are:
1) Why is $L$ a Liapunov function? There are three fixed point $$(0,0,0), (\pm \sqrt{b(r-1)}, \pm \sqrt{b(r-1)}, r-1),$$and for none of them does $L$ become $0$.
2) Why didn't he just simply use the complement $\mathbb{R^3}\setminus E$ to obtain a set where the Lie derivative $\dot L$ of the Liapunov function is strictly decreasing? I don't see why he needs to have that that the Lie derivative is actually $\delta$ away from becoming positive, $\dot L\leqslant \delta$.
(And I don't understand his argument using $E_1$ either, since it's not at all clear to me why that gives us such a $\delta$.)
3) $\mathbb{R^3}\setminus E$ or $\mathbb{R^3}\setminus E_1$ seems to be a trapping region, but not $E_1$. Is that a typo?
BTW, the original text from which I took these screenshots was taken from here, see pp. 236 in the book (resp. 247 in your pdf reader).
1) The author does not state that $L$ is a Lyapunov function, it isn't, as you pointed out. They use it as a bounding function.
2) He introduces $E_{1}$ precisely to bound $\dot{L}$ strictly away from zero. If he didn't do this, then $\dot{L}$ might become arbitrarily small, and the argument that the trajectory will enter $E_{1}$ after finite time does not hold.
The fact that he can bound $\dot{L} \leq -\delta < 0$ outside of $E_{1}$ follows from the maximum value theorem of a continuous function ($\dot{L}$) on a compact domain ($\{x\,:\,L = M+1\}$), i.e. that it obtains its maximum value somewhere.
3) $E_{1}$ is a trapping region because as he argues, the trajectories will enter it after some finite time, and they won't be able to leave.