Editing Lyapunov function (section)

===Further discussion of the terms arising in the definition===
Lyapunov functions arise in the study of equilibrium points of dynamical systems. In <math>\R^n,</math> an arbitrary autonomous [[dynamical system]] can be written as

:<math>\dot{y} = g(y)</math>

for some smooth <math>g:\R^n \to \R^n.</math>

An equilibrium point is a point <math>y^*</math> such that <math>g\left(y^*\right) = 0.</math> Given an equilibrium point, <math>y^*,</math> there always exists a coordinate transformation <math>x = y - y^*,</math> such that:

:<math>\begin{cases} \dot{x} = \dot{y} = g(y) = g\left(x + y^*\right) = f(x) \\ f(0) = 0 \end{cases}</math>

Thus, in studying equilibrium points, it is sufficient to assume the equilibrium point occurs at <math>0</math>.

By the chain rule, for any function, <math>H:\R^n \to \R,</math> the time derivative of the function evaluated along a solution of the dynamical system is

:<math> \dot{H} = \frac{d}{dt} H(x(t)) = \frac{\partial H}{\partial x}\cdot \frac{dx}{dt} = \nabla H \cdot \dot{x} = \nabla H\cdot f(x).</math>

A function <math>H</math> is defined to be locally [[positive-definite function]] (in the sense of dynamical systems) if both <math>H(0) = 0</math> and there is a neighborhood of the origin, <math>\mathcal{B}</math>, such that:

:<math>H(x) > 0 \quad \forall x \in \mathcal{B} \setminus\{0\} .</math>