# Lorentz Equations - Chaos and Stability

Tags:
1. Nov 23, 2014

### unscientific

1. The problem statement, all variables and given/known data
The figure below shows the path of a particle governed by the Lorenz equations with r = 28, σ = 10, b = 8/3. The x'es and boxes show points where the path crosses the plane z = r − 2σ > 0.

(a) Which indicator shows a decreasing z and which shows an increasing z?

(b) Show the length of element $| \delta x |$ between the two paths either grows or decays exponentially if aligned with one of the eigenfunctions of jacobian $\frac{J + J^T}{2}$.

(c) Find $\frac{J + J^T}{2}$ and its eigenvalues at (0, 0, r-2σ). Hence deduce that of $\delta x$ grows is in the x-y plane while it decays is along the direction of z-axis.

(d) Show the volume decreases exponentially with $\delta V = \delta V_0 e^{−(σ+1+b)t}$
Since $\frac{(J + J^T)}{2}$ is symmetric, its eigenfunctions are orthogonal. Show that for a cubic element where the three displacement directions are along the eigenfunctions in section (c) decays at the same rate.

2. Relevant equations
Lorentz equations are given by:

$$\dot x = \sigma(y-x)$$
$$\dot y = rx - y - xz$$
$$\dot z = xy - bz$$

3. The attempt at a solution

Part (a)

For $\dot z < 0$, $xy < bz \approx 21$.

So the boxes represent decreasing z, the x'es represent increasing z.

Part (b)

How do I show it either grows or decays exponentially? Do I put in z = r − 2σ and find the eigenvalues? Wouldn't it be part (c)? I think this part is simpler than it seems.

Part (c)

The matrix $\frac{J + J^T}{2}$ becomes:

I found the eigenvalues to be:

$$\lambda_{1,2} = -\frac{ -(\sigma + 1) \pm \sqrt{ (\sigma+1)^2 - 4(\sigma - \frac{9}{4} \sigma^2) } }{2}$$
$$\lambda_3 = -b$$

Part (d)

$$\nabla \cdot \vec u = -(\sigma + 1 + b)$$

Then the result follows.

This is all that I have managed to do so far, would appreciate any input, thank you!

2. Nov 28, 2014