Full question:
Consider the Lorenz model with $\dot{x} = \frac{3}{2}y-x$, $\dot{y} = \frac{1}{2}x-y-xz$, and $\dot{z} = xy-z$.
(a) Show that the Lorenz model has a unique equilibrium point. (b) Show that the equilibrium point is globally stable. (HINT: Use a Liapunov function.)
To clarify, I know part (a) and have found our equilibrium point. This can be done through some fairly simple algebra. However I'm not so skilled at using Liapunov functions yet, and would like to see how to handle part (b) while using one. My first attempt was using the function $V(x,y,z) = x^2+y^2+z^2$, but it didn't go so well.
Thanks in advance for any and all advice!
Best Answer
$V(x,y,z) = x^2 + y^2 + z^2$ will almost work. You should get $$ \dfrac{d}{dt} V(x,y,z) = 2 x \dot{x} + 2 y \dot{y} + 2 z \dot{z} = -(x-y)^2 - z^2 \le 0$$ Try instead $V(x,y,z) = x^2 + 2 y^2 + 2 z^2$.