Can an Ellipsoidal Region Contain All Trajectories of the Lorenz Equations?

[mliuzzolino]

Homework Statement

Show that there is a certain ellipsoidal region E of the form rx² + σy² + σ(z-2r)² ≤ C such that all trajectories of the Lorenz equations eventually enter E and stay in there forever.

Homework Equations

Lorenz Equations:

$\dot{x} = \sigma (y - x)$

$\dot{y} = rx - xz - y$

$\dot{z} = xy - bz$


V is the volume.

The Attempt at a Solution

$V = rx^2 + \sigma y^2 + \sigma(z-2r)^2$

$\dot{V} = 2rx\dot{x} + 2\sigma y \dot{y} + 2 \sigma(z - 2r) \dot{z}$

$\dfrac{1}{2} \dot{V} = rx(\sigma y - \sigma x) + \sigma y (rx - xz - y) + (\sigma z - 2 \sigma r)(xy - bz)$

$\Rightarrow \dfrac{1}{2 \sigma} \dot{V} = -rx^2 - y^2 -bz^2 + 2bz$


$\Rightarrow \dot{V} < 0$

Therefore, all trajectories of the Lorenz equations eventually enter E and stay in there forever.



I am not sure I approached this problem correctly, and the additional challenge problem states to try and obtain the smallest possible value of C with this attracting property. Did I miss something in my workthrough above involving C? If I subtract it from both sides of the original inequality and involve it in the workthrough involving the change in volume, isn't the value of C irrelevant given that its derivative is zero (being that it's a constant) ?

 

[christoff]

I remember not understanding the proof of this very well when I took my third year dynamical systems course. You seem to be on right track, however. See www.math.harvard.edu/archive/118r_spring_05/handouts/lorentz.pdf : it has a somewhat hand-wavey proof that may help you cement your ideas.

 

[mliuzzolino]

That seems to help quite a bit. Now I'm only going half as crazy as I was. Thanks a lot!