Chaotic fractal structures models

The hierarchical structure model is generalized and applied to study the viscoelastic properties of a two-component inhomogeneous medium with chaotic, fractal structure. It is shown that just as the results obtained recently using the Hashin-Strikman model, the present model predicts the possibility of obtaining composites with an effective shear and dumping coefficient much higher than those characterizing the individual component phases. The viscoelastic properties of the fractal medium, however, differ qualitatively from the properties of the Hashin-Strikman medium. 

We focus on aggregation in model, regular and chaotic, flows. Two aggregation scenarios are considered In (i) the clusters retain a compact geometry—forming disks and spheres—whereas in (ii) fractal structures are formed. The primary focus of (i) is kinetics and self-similarity of size distributions, while the main focus of (ii) is the fractal structure of the clusters and its dependence with the flow. 

In Section 2.3 we studied the tent map, a schematic model for ionization that was able to produce fractal structures as a result of ionization. An important question is therefore whether the results presented in Section 2.3 are only of academic interest, or whether fractal structures can appear as a result of ionization in physical systems. In order to answer this question we return to the microwave-driven one-dimensional hydrogen atom. As we know from the previous chapter, this model is ionizing and realistic enough to qualitatively reproduce measured ionization data. Therefore this model is expected to be a fair representative for a large class of chaotic ionization processes. 

One-electron atoms subjected to a time-dependent external field provide physically realistic examples of scattering systems with chaotic classical dynamics. Recent work on atoms subjected to a sinusoidal external field or to a periodic sequence of instantaneous kicks is reviewed with the aim of exposing similarities and differences to frequently studied abstract model systems. Particular attention is paid to the fractal structure of the set of trapped unstable trajectories and to the long time behavior of survival probabilities which determine the ionization rates of the atoms. Corresponding results for unperturbed two-electron atoms are discussed. 

Chaotic fractal sets on rectangular lattices have been used to the define the effective conductivity of the composite material. The effective conductivity of the composite material is defined using the fractal random structure model of a composite and the iteration method of averaging. Comparison of the calculation with experimental data is also given. 

We have also studied the elastic properties of a nonuniform medium with chaotic structure in which one phase has a negative shear modulus. The analysis may be made using the fractal hierarchical structure model. 

Conclusion. Calculations of the dependence of the conductivity and the relative permittivity of chaotic hierarchical self-similar structures of composites were performed using a fractal model in the entire range of concentrations of inhomogeneities at various frequencies of an external field. The metal-insulator transition was shown to occur not only near the percolation threshold. It was also shown that the transition depends on the concentration of the metallic phase and the frequency of the external field. 

These phE fluctuations may shed light on the dynamics of possible chaotic processes responsible for the fractal nature of many fracture surfaces. Measurements of the fractal dimensions of various fracture surfaces have been reported by several authors.(65-6 ) since the underlying structure of many chaotic systems is fractal, this suggests that fracture is to some degree chaotic, that is, deterministic, yet aperiodic and unpredictable. Some simple models of the fracture process have been proposed which display chaos or result in fractal... 

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