## You are here

HomeLyapunov function

## Primary tabs

# Lyapunov function

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

$\frac{dx}{dt}=F(x,y)\quad\frac{dy}{dt}=G(x,y)$ |

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

A function $V(x,y)$ that is of class $C^{{1}}$ 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.

Synonym:

Liapunov function

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

34-00*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Apr 15

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier

new question: Prove a formula is part of the Gentzen System by LadyAnne

Mar 30

new question: A problem about Euler's totient function by mbhatia

new problem: Problem: Show that phi(a^n-1), (where phi is the Euler totient function), is divisible by n for any natural number n and any natural number a >1. by mbhatia

new problem: MSC browser just displays "No articles found. Up to ." by jaimeglz

Mar 26

new correction: Misspelled name by DavidSteinsaltz

Mar 21

new correction: underline-typo by Filipe

Mar 19

new correction: cocycle pro cocyle by pahio

Mar 7

new image: plot W(t) = P(waiting time <= t) (2nd attempt) by robert_dodier

new image: expected waiting time by robert_dodier

new image: plot W(t) = P(waiting time <= t) by robert_dodier