Fractals functions

The lifetimes T 9q) displayed in Fig. 9.5 were calculated for fixed Iq. In order to obtain a better impression where the firactal features of T are located in the 9o, lo plane, T 9o,lo) was calculated as a function of the two initial conditions 9o and Iq for e — 0.5 and p = 1. The result is shown in Fig. 9.6 in the form of a grey-scale plot. The darker the shades in Fig. 9.6 the longer lived the molecule. Again there are apparently unresolved regions in Fig. 9.6(a). A magnification of the framed detail of Fig. 9.6(a) is shown in Fig. 9.6(b). Again there are apparently unresolved structures in Fig. 9.6(b). As before in the one-dimensional case one will never be able to resolve the two-dimensional features in T 9q, Iq) as more and more structme appears on smaller and smaller scales. Thus, T 9q, Iq) is a fractal function embedded in the two-dimensional 9q — Iq) space. 

In this section we describe some of the essential features of fractal functions starting from the simple dynamical processes described by functions that are fractal (such as the Weierstrass function) and that are continuous everywhere but are nowhere differentiable. This idea of nondifferentiability leads to the introduction of the elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. We find that the relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, the changes in time of phenomena that are best described by fractal functions are probably best described by fractional equations of motion, as well. In any event, this latter perspective is the one we developed elsewhere [52] and discuss herein. Others have also made inquiries along these lines [70] ... 

The fractal dynamics of complex physiologic systems can be modeled using the fractional rather than the ordinary calculus because the changes in the fractal functions necessary to describe physiologic complexity remain finite in the former formalism but diverge in the latter [53]. 

Appendix. Derivative of Fractal Functions. In general, functions for which the total increment,... 

Wiener s process (i.e., Brownian motion) and Kolmogorov s turbulence (i.e., a nonsmooth vector field) may be cited as examples of phenomena which can be described by continuous, nowhere differentiable functions (fractal functions). 

The coefficients of the series (606) depend both on the fractional derivative of gth order of the fractal function/(x) at the point x = x0 and on the branching index j of the fractal ensemble for which the function/(x) is specified. 

If one chooses radius to be the smallest, the density of spheres with radius is a fractal function of the radius r of the sphere in which it is contained. That such a structure indeed gives a power-law dependence on K is shown for the theoretically computed spectrum in Fig.(5.24b)I J. 

Continued refinement of this proposed wave structure in space-time must result in a fractal function, or self-similarity, wherein the curve, at any scale, contains a smaller copy of itself. 

Berry, M.V. and Lewis, Z.V. (1980) On the Weierstrass-Mandelbrot fractal function. Proceedings of Royal Society, London, A 370 459-84. 

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