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