# Lorenz Equations and Chaos

1. Apr 29, 2013

### mliuzzolino

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

Lorenz Equations:

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

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

$\dot{z} = xy - bz$

V is the volume.

3. 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) ?

2. Apr 29, 2013

### christoff

3. Apr 29, 2013

### mliuzzolino

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