Chapter 7.6, Problem 4P

Consider the ellipsoid

$V\left(x,y,z\right)=2r{x}^2+\sigma y^2+\sigma {\left(z-3r\right)}^2=c>0$.

(a) Calculate

$\frac{dV}{dt}=\frac{\partial V}{\partial x}\frac{dx}{dt}+\frac{\partial V}{\partial y}\frac{dy}{dt}+\frac{\partial V}{\partial z}\frac{dz}{dt}$

along trajectories of the Lorenz equations (1).

(b) Determine a sufficient condition on $c$ so that every trajectory crossing $V\left(x,y,z\right)=c$ is directed inward.

(c) Evaluate the condition found in part (b) for the case $\sigma =10,b=\frac{8}{3},r=28$.