PlanetMath: natural symmetry of the Lorenz equation

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natural symmetry of the Lorenz equation

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The Lorenz equation has a natural symmetry defined by

$\displaystyle (x,y,z) \mapsto (-x,-y,z).$

(1)

To verify that (1) is a symmetry of an ordinary differential equation (Lorenz equation) there must exist a $3\times3$ matrix which commutes with the differential equation. This can be easily verified by observing that the symmetry is associated with the matrix

defined as

$\displaystyle R = \begin{bmatrix}-1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & 1 \end{bmatrix}.$

(2)

Let

$\displaystyle \dot{\textbf{x}} = f(\textbf{x}) = \begin{bmatrix}\sigma(y-x) \\ x(\tau - z) -y \\ xy - \beta z \end{bmatrix}$

(3)

where $f(\textbf{x})$ is the Lorenz equation and $\textbf{x}^T = (x,y,z)$ . We proceed by showing that $Rf(\textbf{x}) = f(R\textbf{x})$ . Looking at the left hand side

$\displaystyle Rf(\textbf{x})$	$\displaystyle =$	$\displaystyle \begin{bmatrix} -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} \sigma(y-x) \ x(\tau - z) -y \ xy - \beta z \end{bmatrix}$
	$\displaystyle =$	$\displaystyle \begin{bmatrix} \sigma(x-y) \ x(z - \tau ) + y \ xy - \beta z \end{bmatrix}$

and now looking at the right hand side

$\displaystyle f(R\textbf{x})$	$\displaystyle =$	$\displaystyle f(\begin{bmatrix} -1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} x \ y \ z \end{bmatrix})$
	$\displaystyle =$	$\displaystyle f(\begin{bmatrix} -x \ -y \ z \end{bmatrix})$
	$\displaystyle =$	$\displaystyle \begin{bmatrix} \sigma(x-y) \ x(z - \tau ) + y \ xy - \beta z \end{bmatrix}.$

Since the left hand side is equal to the right hand side then (1) is a symmetry of the Lorenz equation.

"natural symmetry of the Lorenz equation" is owned by Daume.

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Cross-references: right hand side, left hand side, differential equation, matrix, symmetry of an ordinary differential equation, symmetry, Lorenz equation
There are 2 references to this entry.

This is version 2 of natural symmetry of the Lorenz equation, born on 2003-07-06, modified 2003-07-27.
Object id is 4428, canonical name is NaturalSymmetryOfTheLorenzEquation.
Accessed 4118 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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