all solution of the Lorenz equation enter an ellipsoid
AllSolutionOfTheLorenzEquationEnterAnEllipsoid2
2005-05-11 19:38:52
2005-05-11 19:38:52
Result
Lorenz equation
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\newtheorem{thm}{Theorem}
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If $\sigma, \tau, \beta >0$ then all solutions of the Lorenz equation
\begin{eqnarray*}
\dot{x} & = & \sigma(y-x)\\
\dot{y} & = & x(\tau - z) -y\\
\dot{z} & = & xy - \beta z
\end{eqnarray*}
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
\begin{eqnarray*}
\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).
\end{eqnarray*}
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.