I'm reading a text about the Lorenz equations,

enter image description here

for $r>1$ and all the other parameters positive. At one point the author says

enter image description here

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).

up vote 1 down vote accepted

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.

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    I'm sorry, but I still don't understand a number of points: Regarding 1), if $L$ isn't a Lyapunov function, why does he that use the Lie derivative (which is denoted by $\dot L$) instead of the normal derivative? – temo Aug 6 at 18:41
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    Regarding 2), a) I think you mean $\dot L≤−δ<0$ outside of E1? Since on E_1 its impossible for the minimum to be negative, since $E_1$ contains $E$, where $\dot L$ is nonnegative. b) I can't see why $\{ x:L=M+1\}$ is compact (why should level sets of $L$ be bounded?), could you help me with that please? c) Even it this set is compact, so that we can use the maximum value theorem for $\dot L$ on it, I don't see how we can conclude that $\dot L$ is less than −δ outside $E_1$; wouldn't it be conceivable that if $\delta$ is the minimum of $\dot L$ on that set, [...] – temo Aug 6 at 18:54
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    [...] that $\dot L$ may take a value between $-\delta$ and $0$ outside of $E_1$? – temo Aug 6 at 18:58
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    P.S. I'm just realizing that it is not even clear to my why $\{ x: L=M+1\}$ is nonempty... ? – temo Aug 6 at 19:23
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    Even though it’s not a Lyapunov function, we still care what happens along trajectories (whether $L$ increases or decreases etc.) so that is why we use $\dot{L}$, this ultimately gives us a region where all trajectories must enter. – bobcliffe Aug 6 at 19:52

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