October 2, 1997

(Revised May 27, 2005)

The Lorenz attractor is given by the following 3-dimensional system of ordinary differential equations:

d*x*/d*t* = *p*(*y* - *x*)

d*y*/d*t* = -*xz* + *rx* - *y*

d*z*/d*t* = *xy* - *bz*

In his orginal paper [E. N. Lorenz, J. Atmos. Sci. **20**, 130
(1963)],
Lorenz used the parameters *p* = 10, *r* = 28, and *b*
=
8/3 for which the trajectories produce a strange attractor.

For reasons unknown, published calculations of
the
largest Lyapunov exponent for the Lorenz attractor [see for example, A.
Wolf, J. B. Swift, H. L. Swinney, and J. A. Vasano, Physica **16D**,
285 (1985)] have usually used the values *p* = 16, *r* =
45.92,
and *b* = 4 for which the Lyapunov exponents (base-2) are (2.16,
0,
-32.4). It is more natural to express the exponents for a flow in
base-*e*, in which case the values are (1.50, 0, -22.46).
For
a flow, one of the exponents must be zero and the sum of the exponents
is -*p* - 1 - *b* = -21, which is approximately satisfied by
the quoted results.

Reported here is a numerical calculation of the
largest
Lyapunov exponent for the Lorenz attractor using Lorenz's original
parameters.
The calculation was performed in a several-day run on a 200-MHz Pentium
Pro using a PowerBASIC
program
available in both source and (DOS) executable
code. The program uses the fourth-order Runge-Kutta method with a
fixed step size of 0.001, and performed over 10^{9} iterations,
corresponding to a maximum time of over 10^{6}. More
details
on the numerical calculation of the Lyapunov exponent are available.
Output from the program is shown below:

The image above shows the Lorenz attractor as an
anaglyph that can be viewed in 3-D using red-blue glasses. The
values
of the Lyapunov exponents are **(0.906, 0, -14.572)**. From
these
exponents, the Kaplan-Yorke dimension can be calculated from
** D_{KY}
= 2 + l_{1} / |l_{3}|
= 2.062**. The program also calculates the capacity dimension

For a more recent and more accurate calculation see the Lyapunov
Exponent Spectrum Software.

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