PlanetMath: all solution of the Lorenz equation enter an ellipsoid

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all solution of the Lorenz equation enter an ellipsoid

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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

$\displaystyle V(x,y,z)=\tau x^2 + \sigma y^2 + \sigma (z-2\tau )^2.$

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

such that the ellipsoid

$\displaystyle \tau x^2 + y^2 + \beta(z -r)^2 = b\tau^2$

is strictly contained in the ellipsoid

$\displaystyle \tau x^2 + \sigma y^2 + \sigma (z-2\tau )^2=C.$

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.

"all solution of the Lorenz equation enter an ellipsoid" is owned by Daume.

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Cross-references: exterior, eventually, contained, strictly, Lyapunov function, addition, finite, ellipsoid, Lorenz equation, solutions
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This is version 1 of all solution of the Lorenz equation enter an ellipsoid, born on 2005-05-11.
Object id is 7042, canonical name is AllSolutionOfTheLorenzEquationEnterAnEllipsoid2.
Accessed 3904 times total.

Classification:

AMS MSC:	34-00 (Ordinary differential equations :: General reference works )
	65P20 (Numerical analysis :: Numerical problems in dynamical systems :: Numerical chaos)
	65P30 (Numerical analysis :: Numerical problems in dynamical systems :: Bifurcation problems)
	65P40 (Numerical analysis :: Numerical problems in dynamical systems :: Nonlinear stabilities)
	65P99 (Numerical analysis :: Numerical problems in dynamical systems :: Miscellaneous)

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