Lyapunov function

Suppose we are given an autonomous system of first order differential 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 $V>0.$ 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.

Title	Lyapunov function
Canonical name	LyapunovFunction
Date of creation	2013-03-22 13:42:29
Last modified on	2013-03-22 13:42:29
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	7
Author	CWoo (3771)
Entry type	Definition
Classification	msc 34-00
Synonym	Liapunov function