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 V(x,y) that is of class C¹ and satisfies V(0,0)=0 is called a Liapunov function if every open ball B_{d<\math><\sub>(0,0) contains at least one point where V>0. If there happens to exist d^* such that the function .{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.