2.1.2
Fractal Time Signals, and Power Laws
2.1.3
Self-Organized Criticality: Space-Time Fractals
2.1.2
Fractal Time Signals, and Power Laws
There are numerous power law relations in
science that have the selfsimilarity property. For example, the inverse
square law force, which is fundamental in gravitation and in electricity
and magnetism, has no intrinsic scale, it has the same form at all scales
under a linear scaling transformation (Deering and West, 1992; Wienberg,
1993 References ). The concept of fractals
may be used for modelling certain aspects of dynamics, i.e., temporal evolution
of spatially extended dynamical systems. Spatially extended dynamical systems
in nature exhibit fractal geometry to the spatial pattern and support
dynamical processes on all time scales,for example, the fractal
geometry to the global cloud cover pattern is associated with fluctuations
of meteorological parameters on all time scales from seconds to years.
The temporal fluctuations exhibit structure over multiple orders of temporal
magnitude in the same way that fractal forms exhibit details over
several orders of spatial magnitude. The frequency spectrum is broadband.
Selfsimilar variations on different time scales will produce a frequency
spectrum having an inverse power law distribution or 1/f - like distribution
and imply long-range temporal correlations signifying persistence or "memory".The
phenomenon of 1/f - noise spectrum first introduced by Van
Der Ziel in 1950 References is ubiquitous
to dynamical systems in nature and has a long history of more than 40 years
of observational documentation in all fields of science and other areas.
Power-law behaviour has been documented in the functioning of physiological
systems (Sun and Charef,1990; Suki et. al.,1994). Hurst (1951)
and Hurst et al. (1965) References
had shown for river flows (Schepers et.al., 1992 References
) that for a wide variety of data sets the degree of "memory" over time
spans of up to a millennium could be characterized by a power law relationship
(Bassingthwaighte and Beyer, 1991 References
). Long-range spatial correlations have been identified at the level of
the DNA (Maddox,1992; Peng et.al.,1992 References
). Long-range correlations over time and space have also been investigated
by Mandelbrot and Wallis (1969 References
) for geophyiscal records and more recently by Tang and Bak (1988 References
), and Bak et al.(1987,1988), for 1/f noise
in dynamical systems.The 1/f power law would seem to be natural
and white noise (flat distribution) would be the subject of involved investigation
(West and Shlesinger, 1989 References
). Recent studies have identified turbulent cascades in foreign exchange
markets (Ghashghaie et.al.,1996 References
) and power-laws governing epidemics have been reported (Rhodes and Anderson,1996
References
).
A major feature of
this correlation is that the amplitude of short-term and long-term fluctuations
are related to each other by the scale factor alone independent of details
of growth mechanisms from smaller to larger scale. The macroscopic pattern,
comprised of a multitude of subunits, functions as a unified whole independent
of details of dynamical processes governing its individual subunits (Mantegna
and Stanley, 1995 References ). Such a
concept that physical systems which consist of a large number of interacting
subunits obey universal laws that are independent of the microscopic details
is now acknowledged as a breakthrough in statistical physics. The variability
of individual elements in a system act cooperatively to establish regularity
and stability in the system as a whole (West and Shlesinger, 1989 References
). Scale invariance implies, knowledge of the properties of a model system
at short times or short length scales can be used to predict the behaviour
of a real system at large times and large length scales (Stanley, Amaral,
Buldyrev, Havlin et al., 1996 References
).
The fractal
dimension D of a temporal fractal can be computed
using recently developed algorithms. Since time series of a single variable
such as temperature in atmospheric flows may reflect the cumulative effect
of the multitude of factors governing flow dynamics, the fractal
dimension may indicate the number of parameters controlling the evolution
dynamics. However limitations in data length and computational algorithms
preclude exact determination of D (Lorenz, 1991 References
). It has not been possible to formulate governing equations based on a
knowledge of D for prediction purposes. A complete review
of applications of concepts in nonlinear dynamics and chaos in atmospheric
sciences has been given by Zeng et. al.(1993 References
)
The spatiotemporal
evolution of dynamical systems was not investigated as a unified whole
and fractal geometry to spatial pattern and fractal fluctuations
in time of dynamical processes were investigated as two separate multidisciplinary
areas of research till as late as 1987.
2.1.3
Self-Organized Criticality: Space-Time Fractals
Bak et.al.(1987,1988 References
) postulated in 1987 that fractal geometry to spatial pattern and
associated fractal fluctuations of dynamical processes in time are
signatures of self-organized criticality in the spatiotemporal evolution
of dynamical system. The relation between spatial and temporal power-law
behaviour was recognized much earlier in condensed matter physics where
long-range spatiotemporal correlations appear spontaneously at the critical
point for continuous phase transitions. The amplitude of large and small
scale fluctuation are obtained from the same mathematical function using
appropriate scale factor, i.e., ratio of the scale lengths. This property
of self similarity is often called a renormalization group relation in
physics (Wilson, 1979; West, 1990; Peitgen et. al.,1992 References
) in the area of continuous phase transitions at critical points (Weinberg,
1993; Back et. al., 1995 References
). When a system is poised at a critical point between two macroscopic
phases, e.g., vapour to liquid, it exhibits dynamical structures on all
available spatial scales, even though the underlying microscopic interactions
tend to have a characteristic length scale (Back et. al.,
1995
References ). But, in order to arrive
at the critical point, one has to fine-tune an external control parameter,
such as temperature, pressure or magnetic field, in contrast to the phenomena
described above which occur universally without any fine tuning.The explanation
is that open extended dissipative dynamical systems, i.e., systems not
in thermodynamic equilibrium may go automatically to the critical state
as long as they are driven slowly : the critical state is self-organized
(Tang and Bak,1988; Bak and Chen, 1989,1991 References
).