If V=$\mathrm \Vert x \Vert_\mathrm{}^2$, then $\dot V=2V(V-1)$ according to a controls textbook making an off-hand mathematical observation.
I can possibly understand the 2V, but am not sure where the (V-1) comes from. Could anyone show this derivation or point me in the right direction?
That relationship only applies for particular systems, such as the one the textbook was using. The particular system was: $ \\ \dot x_1=x_1(x_1^2+x_2^2-2)-4x_1x_2^2 \\ \dot x_2=4x_1^2x_2+x_2(x_1^2+x_2^2-2) $