all solution of the Lorenz equation enter an ellipsoid

If σ,τ,β>0 then all solutions of the Lorenz equationMathworldPlanetmath

x˙ = σ(y-x)
y˙ = x(τ-z)-y
z˙ = xy-βz

will enter an ellipsoid centered at (0,0,2τ) in finite time. In addition the solution will remain inside the ellipsoid once it has entered. To observe this we define a Lyapunov functionMathworldPlanetmath


It then follows that

V˙ = 2τxx˙+2σyy˙+2σ(z-2τ)z˙
= 2τxσ(y-x)+2σy(x(τ-z)-y)+2σ(z-2τ)(xy-βz)
= -2σ(τx2+y2+β(z-r)2-bτ2).

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


is strictly contained in the ellipsoid


Therefore all solution will eventually enter and remain inside the above ellipsoid since V˙<0 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