Fractal dynamics

By looking at Eqs. (27) and (28), Eq. (32) confirms the customary way of relating P of the stretched exponential function, Eq. (19), to the relaxation time spectrum. The glassy state relaxation is dominated by the part of the spectrum having longer relaxation times. The fractal dynamics of holes are diffusive, and the diffusivity depends strongly on the tenuous structure in fractal lattices, v is the exponent in the power-law relationship between local diffusivity and diffusion length ... 

The fractal dynamics of holes are diffusive, and the diffusivity depends strongly on the tenuous structure in fractal lattices. The fractal dimension defines the self-similar connectivity of hole motions, the relaxation spectrum, and stretched exponential... 

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]. 

J. M. Hausdorff, S. L. Mitchell, R. Firtion, C. K. Peng, M. E. Cudkowicz, J. Y. Wei, and A. L. Goldberger, Altered fractal dynamics of gait Reduced stride-interval correlations with aging and Huntington s disease. J. Appl. Physiol. 82, (1997). 

J. W. Blaszczyk and W. Klonowski, Postural stability and fractal dynamics. Acta Neurobiol. Exp. 61, 105-112 (2001). 

Rouse model of randomly branched polymers exhibits fractal dynamics the relaxation time r(g) of a polymer section of g monomers has the same dependence on the number of monomers g as the whole chain [Eq. (8.144)] ... 

The full-time dependence of the stress relaxation modulus ot randomly branched unentangled polymers is best derived from the fractal dynamics of Section 8.8 using the relaxation rate spectrum P( ) ... 

Moon, F. C. (1992) Chaotic and Fractal Dynamics An Introduction for Applied Scientists and Engineers (Wiley, New York). 

It is also able to draw complex mathematical figures, including many fractals. Dynamics Solver is a powerful tool for studying differential equations, (eontinuous and diserete) nonlinear dynamieal systems, deterministic chaos, mechanics, and so forth. For instance, you can draw phase space portraits (including an optional direction field). Poincare maps, Liapunov exponents, histograms, bifurcation diagrams, attraction basis, and so forth. The results can be watehed (in perspective or not) from any direction and particular subspaces ean be analyzed. 

Dotsenko, V. S. (1985). Fractal Dynamics of Spin Glasses. J. Phys. C Solid State Phys., 18(15), 6023-6031. 

Balankin A. S. Fractal Dynamics of Deformable Sohd. Metally, 1992, JT22, p. 41-51. 

Alexander S., Courtens E., Vacher R. Vibrations of fractals dynamic scaling, couelation functions and inelastic light scattering. Physica A 1993 195 286-318... 

As on previous occasions, the reader is reminded that no very extensive coverage of the literature is possible in a textbook such as this one and that the emphasis is primarily on principles and their illustration. Several monographs are available for more detailed information (see General References). Useful reviews are on future directions and anunonia synthesis [2], surface analysis [3], surface mechanisms [4], dynamics of surface reactions [5], single-crystal versus actual catalysts [6], oscillatory kinetics [7], fractals [8], surface electrochemistry [9], particle size effects [10], and supported metals [11, 12]. 

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the chaos associated with motion on the attractor and some geometrical measure of the stmctural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the discussion we consider tliree-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

Wool, R.P, Dynamics and fractal structure of polymer interfaces. In Lee, L.-H. (Ed.), New Trends in Physics and Physical Chemistry of Polymers. Plenum Press, New York, 1989, p. 129. 

T. Matsuyama, K. Honda. Dynamics of rough surfaces growing on fractal substrates. J Phys Soc Jpn 66 2533, 1997. 

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

Since the phase space of a dissipative dynamical system contracts with time, we know that, in the long time limit, t oo, the motion will be confined to some fixed attractor, A. Moreover, becaust of the contraction, the dimension, D, of A, must be lower than that of the actual phase space. While D adds little information in the case of a noiichaotic attractor (we know immediately, and trivially, for example, that all fixed-points have D = 0, limit cycles have D = 1, 2-tori have D = 2, etc.), it is of significant interest for strange attractors, whose dimension is typically non-integer valued. Three of the most common measures of D are the fractal dimension, information dimension and correlation dimension. 

Lyapunov Dimension An interesting attempt to link a purely static property of an attractor, - as embodied by its fractal dimension, Dy - to a dynamic property, as expressed by its set of Lyapunov characteristic exponents, Xi, was, first made by Kaplan and Yorke in 1979 [kaplan79]. Defining the Lyapunov dimension, Dp, to be... 

Figure21. The division number A(e) for the trajectories of Ag and I" at 900 K. (Reprinted from M. Kubayashi andF. Shimojo, Molecular Dynamics Studies of Molten Agl.II. Fractal Behavior of Diffusion Trajectory, J. Phys. Soc. Jpn. 60 4076-4080, 1991, Fig. 6, with permission of the Physical Society of Japan.

Muthukumar and Winter [42] investigated the behavior of monodisperse polymeric fractals following Rouse chain dynamics, i.e. Gaussian chains (excluded volume fully screened) with fully screened hydrodynamic interactions. They predicted that n and d (the fractal dimension of the polymer if the excluded volume effect is fully screened) are related by... 

The power-law variation of the dynamic moduli at the gel point has led to theories suggesting that the cross-linking clusters at the gel point are self-similar or fractal in nature (22). Percolation models have predicted that at the percolation threshold, where a cluster expands through the whole sample (i.e. gel point), this infinite cluster is self-similar (22). The cluster is characterized by a fractal dimension, df, which relates the molecular weight of the polymer to its spatial size R, such that... 

---