all solution of the Lorenz equation enter an ellipsoid
If then all solutions of the Lorenz equation
will enter an ellipsoid centered at in finite time.In addition the solution will remain inside the ellipsoid once ithas entered. To observe this we define a Lyapunov function
It thenfollows that
We then choose an ellipsoid which all the solutions will enter andremain inside. This is done by choosing a constant such thatthe ellipsoid
is strictly contained in the ellipsoid
Therefore all solution will eventually enter and remain inside the above ellipsoid since when a solution is located at the exterior of theellipsoid.