# 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 |