The Lorenz attractor is an attractor that arises in a simplified system of equations describing the two-dimensional 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 a stream function, defined such that the velocity components of the fluid motion are
(3)
(4)
(Tabor 1989, p. 205).
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
(5)
(6)
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 terms
(7)
(8)
(9)
where
is proportional to convective intensity, to the temperature difference between descending and ascending
currents, and
to the difference in vertical temperature profile from linearity in his system of
equations. From these, he obtained the simplified equations
where
is the Prandtl number, Ra is the Rayleigh number, is the critical Rayleigh number, and is a geometric factor (Tabor 1989, p. 206). Lorenz took
and .
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). As
one of his list of challenging problems for mathematics (Smale's
problems), Smale (1998, 2000) posed the open question of whether the Lorenz attractor
is a strange attractor. This question was answered
in the affirmative by Tucker (2002), whose technical proof makes use of a combination
of normal form theory and validated interval arithmetic.
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,
with an imposed temperature difference 
, under gravity 
, with buoyancy 
, thermal diffusivity 
, and kinematic viscosity 
. The full equations are
is a stream function, defined such that the velocity components 
of the fluid motion are
. 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.
is proportional to convective intensity, 
to the temperature difference between descending and ascending
currents, and 
to the difference in vertical temperature profile from linearity in his system of
equations. From these, he obtained the simplified equations
, 
, 
, and
is the Prandtl number, Ra is the Rayleigh number, 
is the critical Rayleigh number, and 
is a geometric factor (Tabor 1989, p. 206). Lorenz took
and 
.
and capacity dimension 
(Grassberger and Procaccia 1983). For more details,
see Lichtenberg and Lieberman (1983, p. 65) and Tabor (1989, p. 204). As
one of his list of challenging problems for mathematics (Smale's
problems), Smale (1998, 2000) posed the open question of whether the Lorenz attractor
is a strange attractor. This question was answered
in the affirmative by Tucker (2002), whose technical proof makes use of a combination
of normal form theory and validated interval arithmetic.
if 
.
