# all solution of the Lorenz equation enter an ellipsoid

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

 $\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle\sigma(y-x)$ $\displaystyle\dot{y}$ $\displaystyle=$ $\displaystyle x(\tau-z)-y$ $\displaystyle\dot{z}$ $\displaystyle=$ $\displaystyle xy-\beta z$

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

 $\displaystyle\dot{V}$ $\displaystyle=$ $\displaystyle 2\tau x\dot{x}+2\sigma y\dot{y}+2\sigma(z-2\tau)\dot{z}$ $\displaystyle=$ $\displaystyle 2\tau x\sigma(y-x)+2\sigma y(x(\tau-z)-y)+2\sigma(z-2\tau)(xy-% \beta z)$ $\displaystyle=$ $\displaystyle-2\sigma(\tau x^{2}+y^{2}+\beta(z-r)^{2}-b\tau^{2}).$

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 $\dot{V}<0$ when a solution is located at the exterior of the ellipsoid.

Title all solution of the Lorenz equation enter an ellipsoid AllSolutionOfTheLorenzEquationEnterAnEllipsoid 2013-03-22 15:15:28 2013-03-22 15:15:28 Daume (40) Daume (40) 4 Daume (40) Result msc 34-00 msc 65P20 msc 65P30 msc 65P40 msc 65P99