Make your own free website on Tripod.com

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, 1993Reference ). 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 1950Reference 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) Reference had shown for river flows(Schepers et.al., 1992Reference) 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, 1991Reference).Long-range spatial correlations have been identified at the level of the DNA(Maddox,1992;Peng et.al.,1992Reference).Long-range correlations over time and space have also been investigated by Mandelbrot and Wallis(1969Reference) for geophyiscal records and more recently by Tang and Bak(1988Reference), 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, 1989Reference).Recent studies have identified turbulent cascades in foreign exchange markets(Ghashghaie et.al.,1996Reference) and power-laws governing epidemics have been reported(Rhodes and Anderson,1996Reference).

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, 1995Reference). 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, 1989Reference). 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., 1996Reference).

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, 1991Reference).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.(1993Reference)

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.(1987,1988Reference) 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.1992Reference) in the area of continuous phase transitions at critical points (Weinberg, 1993;Back et al., 1995Reference). 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., 1995Reference). 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, 1989,1991Reference).