Selected Topics for Further Research - Geomathematics: Theoretical Foundations, Applications and Future Developments - page 479

where F ¼ 4.669202 is the so-called Feigenbaum constant. In the limit,

1) 1 a 1 ¼ 4, windows

of chaos and multiple cycles occur. Logistic maps for different values of a including
some that show chaotic behavior are shown by Turcotte ( 1997 , Figs. 10.1, 10.2, 10.3,
10.4, 10.5, and 10.6). Sornette et al. ( 1991 ) and Dubois and Chemin ´ e( 1991 ) have
treated the return periods for eruptions of the Piton de la Fournaise on R ´ union Island
and Mauna Lao and Kilauea in Hawaii as return maps that resemble chaotic maps
resulting from the logistic model.
As pointed out in the chapters on fractals and multifractals, nonlinear process
modeling is providing new clues to answers of where the randomness in nature
comes from. From chaos theory it is known that otherwise deterministic Earth
process models can contain terms that generate purely random responses. The
solutions of such equations may contain unstable fixed points or bifurcations. In
the previous chapter it was shown that multifractals provide a novel way of
approach to problem-solving in situations where the attributes display strongly
positively skewed frequency distributions. The preceding examples of deterministic
processes result in chaotic results. However, one can ask the question of whether
there exist deterministic processes that can fully explain fractals and multifractals?
The power-law models related to fractals and multifractals can be partially
explained on the basis of the concept of self-similarity (Box 11.1 ). Various chaotic
patterns observed for element concentration values in rocks and orebodies could be
partially explained as the results of multiplicative cascade models such as the model
of de Wijs. These models invoke random elements such as increases of element
concentration that are not deterministic.
As already discussed in Sect. 10.1 , successful applications of non-linear model-
ing in geoscience include the following: Rundle et al. ( 2003 ) showed that the
Gutenberg-Richter frequency-magnitude relation is a combined effect of the geo-
metrical (fractal) structure of fault networks and the non-linear dynamics of seis-
micity. Most weather-related processes taking place in the atmosphere including
cloud formation and rainfall are multifractal (Lovejoy and Schertzer 2007 ; Sharma
1995 ). Other space-related non-linear processes include “current disruption” and
“magnetic reconnection” scenarios (Uritsky et al. 2008 ). Within the solid Earth's
crust, processes involving the release of large amounts of energy over very short
intervals of time including earthquakes (Turcotte 1997 ), landslides (Park and Chi
2008 ), flooding (Gupta et al. 2007 ) and forest fires (Malamud et al. 1998 ) are
non-linear and result in fractals or multifractals.

( F ( Fa i a i1 ) ¼ 3.569946. In the region 3.569946 < a <

12.4 Three-Parameter Model of de Wijs

This section is concerned with the lognormal, and its logbinomial approximation,
in connection with a three-parameter version of the model of de Wijs. The three
parameters are: overall average element concentration value ( ), dispersion index ( d ),

and apparent number of subdivisions of the environment ( N ). Multifractal theory

ξ