Lyapunov function

Suppose we are given an autonomous system of first orderdifferential equations.

$$\frac{dx}{dt}=F(x,y)\mathit{\hspace{1em}}\frac{dy}{dt}=G(x,y)$$

Let the origin be an isolated critical point of the above system.

A function $V(x,y)$ that is of class $C^1$ and satisfies$V(0,0)=0$ is called a Lyapunov function if every open ball$B_{\delta }(0,0)$ contains at least one point where $V>0.$ Ifthere happens to exist ${\delta }^{*}$ such that the function$\dot{V}$, given by

$$\dot{V}(x,y)=V_x(x,y)F(x,y)+V_y(x,y)G(x,y)$$

is positive definite in $B_{\delta }^{*}(0,0)$. Then the origin isan unstable critical point of the system.