Suppose we are given an autonomous system of first order differential equations.

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

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