Geoscience Reference
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12.3.1 The Lorentz Attractor
It can be argued that non-linear process modeling originated in 1963 when the
meteorologist Edward Lorenz ( 1963 ) published the paper entitled “Deterministic
nonperiodic flow” ( cf . Motter and Campbell 2013 ). Earlier, he had accidentally
discovered how miniscule changes in the initial state of a time series computer
simulation experiment grew exponentially with time to yield results that did not
resemble one another although the equations controlling the process were the same
in all experiments (Lorenz, 1963 ). It turned out that the notion of absolute predict-
ability, although it had been assumed intuitively to hold true in classical physics, is
in practice false for many systems. In the title of a talk Lorenz gave in 1972, he
aptly referred to this chaotic behaviour as the “butterfly effect” asking if the flap of a
butterfly in Brazil could set off a tornado in Texas (Fig. 12.8 ).
Box 12.2: Chaos Theory
Lorenz ( 1963 ) presented chaos theory by using a three-variable system of
nonlinear ordinary differential equations now known as the Lorenz equations.
The model derives from a truncated Fourier expansion of the partial differential
equations describing a thin, horizontal layer of fluid that is heated from below
and cooled from above. The equations can be written as: dx / dt
¼
a (
x + y ),
dy / dt
bz + xy ,where x represents intensity of con-
vective motion, y is proportional to the temperature difference between the
ascending and descending convective currents, and z indicates the deviation of
the vertical temperature profile from linearity. The parameters b and c represent
particulars of the flow geometry and rheology. Lorenz set them equal to b
¼
cx
y
xz ,and dz / dt
¼
¼
8/3
and c
10, respectively; leaving only the Rayleigh number a to vary. For small
c , the system has a stable fixed point at x
¼
¼
y
¼
z
¼
0, corresponding to no
convection. If 24.74
1, the system has two symmetrical fixed points,
representing two steady convective states. At c
>
c
¼
24.74, these two convective
states lose stability; at c
¼
28, the system shows nonperiodic trajectories. Such
trajectories orbit along a bounded region of 3-D space known as a chaotic
attractor, never intersecting themselves (Fig. 12.9 ). For larger values of c ,the
Lorenz equations exhibit different behaviors that have been catalogued by
Sparrow ( 1982 ).
It is beyond the scope of this topic to discuss the theory of chaos in more detail.
However, it is good to point out that chaos is closely connected to fractals and
multifractals. The attractor's geometry can be related to its fractal properties. For
example, although Lorenz could not resolve this from the numeric he applied, his
attractor has fractal dimension equal to approximately 2.06 (Motter and Campbell
2013 , p. 30). An excellent introduction to chaos theory and its relation to fractals
can be found in Turcotte ( 1997 ). The simplest nonlinear single differential equation
 
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