Liapunov function

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

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

A function ${\displaystyle V(x,y)}$ that is of class ${\displaystyle C^{1}}$ and satisfies ${\displaystyle V(0,0)=0/math>iscalleda'''Liapunovfunction'''ifevery[[openball]]<math>B_{(}delta)(0,0)}$ contains at least one point where ${\displaystyle V>0}$. If there happens to exist ${\displaystyle \delta ^{*}}$ such that the function ${\displaystyle {\dot {V}}}$, given by

${\displaystyle \displaystyle {\dot {V}}(x,y)=V_{x}(x,y)F(x,y)+V_{y}(x,y)G(x,y)}$

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