# all solution of the Lorenz equation enter an ellipsoid

If $\sigma ,\tau ,\beta >0$ then all solutions of the Lorenz equation^{}

$\dot{x}$ | $=$ | $\sigma (y-x)$ | ||

$\dot{y}$ | $=$ | $x(\tau -z)-y$ | ||

$\dot{z}$ | $=$ | $xy-\beta z$ |

will enter an ellipsoid centered at $(0,0,2\tau )$ in finite time.
In addition the solution will remain inside the ellipsoid once it
has entered. To observe this we define a Lyapunov function^{}

$$V(x,y,z)=\tau {x}^{2}+\sigma {y}^{2}+\sigma {(z-2\tau )}^{2}.$$ |

It then follows that

$\dot{V}$ | $=$ | $2\tau x\dot{x}+2\sigma y\dot{y}+2\sigma (z-2\tau )\dot{z}$ | ||

$=$ | $2\tau x\sigma (y-x)+2\sigma y(x(\tau -z)-y)+2\sigma (z-2\tau )(xy-\beta z)$ | |||

$=$ | $-2\sigma (\tau {x}^{2}+{y}^{2}+\beta {(z-r)}^{2}-b{\tau}^{2}).$ |

We then choose an ellipsoid which all the solutions will enter and remain inside. This is done by choosing a constant $C>0$ such that the ellipsoid

$$\tau {x}^{2}+{y}^{2}+\beta {(z-r)}^{2}=b{\tau}^{2}$$ |

is strictly contained in the ellipsoid

$$\tau {x}^{2}+\sigma {y}^{2}+\sigma {(z-2\tau )}^{2}=C.$$ |

Therefore all solution will eventually enter and remain inside the above ellipsoid since $$ when a solution is located at the exterior of the ellipsoid.

Title | all solution of the Lorenz equation enter an ellipsoid |
---|---|

Canonical name | AllSolutionOfTheLorenzEquationEnterAnEllipsoid |

Date of creation | 2013-03-22 15:15:28 |

Last modified on | 2013-03-22 15:15:28 |

Owner | Daume (40) |

Last modified by | Daume (40) |

Numerical id | 4 |

Author | Daume (40) |

Entry type | Result |

Classification | msc 34-00 |

Classification | msc 65P20 |

Classification | msc 65P30 |

Classification | msc 65P40 |

Classification | msc 65P99 |