Suppose we are given an autonomous system of first order differential equations. $$ \frac{dx}{dt}=F(x,y)\quad\frac{dy}{dt}=G(x,y) $$

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 $V>0.$ If there 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 is an unstable critical point of the system.