Fractals dynamic models

Continentino M, Malozemoff AP (1986) Dynamical snsceptibility of spin glasses in the fractal cluster model. Phys Rev B 34 471-474... 

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]. 

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)] ... 

Chaos, intermittency and hysteresis in the dynamic model of a polymerization reactor. Chaos, Soliton Fractals, 1, 295-315. 

Interaction dynamics interpretation in the system is realized on the base of fractal structures. These structures ensure modeling of the situations development process and mechanisms of managing changes of fractal displays in the frame of the catastrophes dynamic model [12]. 

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... 

In the late 1980 s and the early 1990 s, two modelling theories have been successfully used in the modelling of irregular surfaces fractal analysis and molecular dynamics theories. 

From the discussion of various simulation methods, it is clear that they will continue to play an important role in further development of aggregation theories as they have advanced the state of knowledge over the last 20 years. The major limitation of the precise methods of Molecular and Brownian Dynamics continues to be difficulty associated with treatment of aggregates with complex geometry the same topic that limits the ability to model these systems using von Smoluchowski s approach. Research needs to be conducted on the hydrodynamics of interactions between fractal aggregates of increasing complexity in order to advance the current ability to describe these types of systems. 

Although the detailed features of the interactions involved in cortisol secretion are still unknown, some observations indicate that the irregular behavior of cortisol levels originates from the underlying dynamics of the hypothalamic-pituitary-adrenal process. Indeed, Ilias et al. [514], using time series analysis, have shown that the reconstructed phase space of cortisol concentrations of healthy individuals has an attractor of fractal dimension dj = 2.65 0.03. This value indicates that at least three state variables control cortisol secretion [515]. A nonlinear model of cortisol secretion with three state variables that takes into account the simultaneous changes of adrenocorticotropic hormone and corticotropin-releasing hormone has been proposed [516]. 

It was shown recently that disordered porous media can been adequately described by the fractal concept, where the self-similar fractal geometry of the porous matrix and the corresponding paths of electric excitation govern the scaling properties of the DCF P(t) (see relationship (22)) [154,209]. In this regard we will use the model of electronic energy transfer dynamics developed by Klafter, Blumen, and Shlesinger [210,211], where a transfer of the excitation... 

In order to establish the relationship between the static and dynamic fractal dimensions, the initial conditions of the classical static percolation model must be considered for the solution of differential equation (89) which can be written as 0 = Qs = 1 for D = Ds. Here the notation s corresponds to the static percolation model, and the condition s = 1 is fulfilled for an isotropic cubic hyperlattice. The solution of (89) with the above-mentioned initial conditions may be written as... 

---