| 释义 |
Lorenz SystemA simplified system of equations describing the 2-D flow of fluid of uniform depth  , with an imposedtemperature  difference  , under gravity  , withbuoyancy  , thermal diffusivity  , andkinematic viscosity  . The full equations are
 | (1) |
 | (2) |
 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 simplifiedsystem, periodic solutions of the form
 | (4) |
 | (5) |
 . Furthermore, vastlydifferent results were obtained for very small changes in the initial values, representing one of the earliest discoveriesof the so-called Butterfly Effect.
Lorenz included the following terms in his system of equations,
and obtained the simplified equations
where
Lorenz took  and  .The Critical Points at (0, 0, 0) correspond to no convection, and the Critical Points at
 | (15) |
 | (16) |
 | (17) |
 if  . The Lorenz attractor has a Correlation Exponent of and Capacity Dimension  (Grassberger and Procaccia 1983). For more details, seeLichtenberg and Lieberman (1983, p. 65) and Tabor (1989, p. 204).See also Butterfly Effect, Rössler Model
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. |
