Liapunov function: Difference between revisions

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#REDIRECT [[Lyapunov function]]

Suppose we are given an [[autonomous system]] of [[differential equation|first order differential equation]]s.

:<math> \frac{dx}{dt}=F(x,y)\quad\frac{dy}{dt}=G(x,y)</math>

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

A [[function (mathematics)|function]] <math> V(x,y)</math> that is of [[smooth function|class ]] <math>C^{1}</math> and satisfies <math>V(0,0)=0</math> is called a '''Liapunov function''' if every [[open ball]] <math> B_(\delta)(0,0)</math> contains at least one [[point]] where <math> V>0</math>. If there happens to exist <math> \delta^{*}</math> such that the function <math> \dot{V}</math>, given by

:<math>\dot{V}(x,y)=V_{x}(x,y)F(x,y)+V_{y}(x,y)G(x,y) </math>

is [[positive definite]] in <math> B_{\delta}^{*}(0,0) </math>, then the [[origin]] is an [[unstable]] critical point of the system.

{{planetmath|id=4386|title=Liapunov function}}

[[Category:Ordinary differential equations]]