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# 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

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Reference

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## Mathematics Subject Classification

34-00*no label found*

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