# Lorenz (stability with Liapunov function)

1. Nov 30, 2008

### gop

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

Using the Liapunov function $$V=1/2(x^2+\sigma y^2 + \sigma z^2)$$, obtain conditions on sigma, rho, beta sufficient for global asymptotic stability of the origin in the Lorenz equation.

2. Relevant equations

The Lorenz equation

$$\dot{x}=\sigma (y-x); \dot{y}=\rho x-y-xz; \dot{z}=-\beta z + xy$$

3. The attempt at a solution

$$\dot{V}=-\sigma (x^2+y^2+\beta z^2-(\rho+1)xy)$$

Now i have to find conditions on beta and rho such the term in the brackets is positive , at least locally around (0,0,0). But I don't really know how to do that.