PlanetMath: Lyapunov function

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Lyapunov function

(Definition)

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 that is of class $C^{1}$ and satisfies is called a Lyapunov function if every open ball $B_{\delta}(0,0)$ contains at least one point where If there happens to exist $\delta^{*}$ such that the function $\dot{V}$ , given by

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

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

"Lyapunov function" is owned by CWoo. [ full author list (2) | owner history (1) ]

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Other names:

Liapunov function

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Cross-references: unstable, positive definite, point, contains, open ball, class, function, critical point, isolated, origin, differential equations, first order, autonomous system
There is 1 reference to this entry.

This is version 4 of Lyapunov function, born on 2003-06-23, modified 2008-02-16.
Object id is 4386, canonical name is LiapunovFunction.
Accessed 5605 times total.

Classification:

AMS MSC:

34-00 (Ordinary differential equations :: General reference works )

Pending Errata and Addenda

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