## Lorenz System

A simplified system of equations describing the 2-D flow of fluid of uniform depth , with an imposed temperature difference , under gravity , with buoyancy , thermal diffusivity , and kinematic viscosity . The full equations are

 (1)

 (2)

Here, is the stream function,'' as usual defined such that
 (3)

In the early 1960s, Lorenz accidentally discovered the chaotic behavior of this system when he found that, for a simplified system, periodic solutions of the form

 (4)

 (5)

grew for Rayleigh numbers larger than the critical value, . Furthermore, vastly different results were obtained for very small changes in the initial values, representing one of the earliest discoveries of the so-called Butterfly Effect.

Lorenz included the following terms in his system of equations,

 (6) (7) (8)

and obtained the simplified equations
 (9) (10) (11)

where
 (12) (13) (14)

Lorenz took and .

The Critical Points at (0, 0, 0) correspond to no convection, and the Critical Points at

 (15)

and
 (16)

correspond to steady convection. This pair is stable only if
 (17)

which can hold only for Positive if . The Lorenz attractor has a Correlation Exponent of and Capacity Dimension (Grassberger and Procaccia 1983). For more details, see Lichtenberg and Lieberman (1983, p. 65) and Tabor (1989, p. 204).

References

Gleick, J. Chaos: Making a New Science. New York: Penguin Books, pp. 27-31, 1988.

Grassberger, P. and Procaccia, I. Measuring the Strangeness of Strange Attractors.'' Physica D 9, 189-208, 1983.

Lichtenberg, A. and Lieberman, M. Regular and Stochastic Motion. New York: Springer-Verlag, 1983.

Lorenz, E. N. Deterministic Nonperiodic Flow.'' J. Atmos. Sci. 20, 130-141, 1963.

Peitgen, H.-O.; Jürgens, H.; and Saupe, D. Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, pp. 697-708, 1992.

Tabor, M. Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, 1989.