I would like to show that the quantity:
is negative on the surface:
for some sufficiently large value of $C$.
I was not able to massage the first quantity any more in order to make it look like the second. I also considered a change of coordinates, but had no luck. $\sigma, b, r$ are positive parameters.
This is a step in exercise 9.2.2 from Strogatz Nonlinear Dynamics and Chaos.