Editing Lyapunov function (section)

==Example==
Consider the following differential equation on <math>\R</math>:

:<math>\dot x = -x.</math>

Considering that <math>x^2</math> is always positive around the origin it is a natural candidate to be a Lyapunov function to help us study <math>x</math>. So let <math>V(x)=x^2</math> on <math>\R </math>. Then,

:<math>\dot V(x) = V'(x) \dot x = 2x\cdot (-x) = -2x^2< 0.</math>

This correctly shows that the above differential equation, <math>x,</math> is asymptotically stable about the origin. Note that using the same Lyapunov candidate one can show that the equilibrium is also globally asymptotically stable.