Fractals

We reveal the existence of clustering properties in the temporal distribution of the earthquakes observed in a seismic active area of Southern Apennine Chain (Italy) by means of quantitatively fractal tools (Fano Factor and Allan Factor). Data consist in a sequence of seismic events instrumentally recorded during the period 1983–1995 in the Irpinia-Basilicata Region (Southern Italy), in which in past and recent years, many destructive events occurred. The analysis of the Fano and Allan Factors shows that the sequence of the occurrence times of events with magnitude Mth≥2.5 is characterized by a scale-invariant behavior from the time scale τ ~ 5 · 10³ s with a scaling coefficient α ~ 0.3. By gradually increasing the threshold magnitude up to Mth=3.1, the value of the scaling coefficient monotonically decreases, pointing out a falling-off in the correlation strength. Although the increasing of the threshold magnitude seems to act as a randomizing filter which removes clustered structures, no firm sign of Poissonian, memoryless behavior is detectable in our analysis.

In this paper, we study the effect of discreteness on various models for patterning in bacterial colonies (finite-size effect) and present two types of models to describe the growth of the bacterial colonies. The first model presented is the Communicating Walkers model (CWm), a hybrid model composed of both continuous fields and discrete entities — walkers, which are coarse-graining of the bacteria; coarse-graining may amplify the discreteness inherent to the biological system. Models of the second type are systems of reaction diffusion equations, where the branching of the pattern is due to non-constant diffusion coefficient of the bacterial field. The diffusion coefficient represents the effect of self-generated lubrication fluid on the bacterial movement. The representation of bacteria by a density field neglects their discreteness altogether. We implement the discreteness of the bacteria by introducing a cutoff in the growth term at low bacterial densities. We demonstrate that the cutoff does not improve the models in any way. The cutoff affects the dynamics by decreasing the effective surface tension of the front, making it more sensitive to anisotropy and decreasing the fractal dimension of the evolving patterns. We compare the continuous and semi-discrete models by introducing food chemotaxis and repulsive chemotactic signaling into the models. We find that the growth dynamics of the CWm and the growth dynamics of the Non-Linear Diffusion model (one of the continuous models) are affected in the same manner. From such similarities and from the insensitivity of the CWm to implicit anisotropy, we conclude that even the increased discreteness, introduced by the coarse-graining of the walkers, is small enough to be neglected. There are advantages and disadvantages to the two types of models. Employing both of them in parallel enables us to conclude that the discreteness of the bacteria does not significantly affect the growth dynamics (no finite-size effect).

We study the vibrational dynamics of cluster-cluster aggregations(CCA) by computer simulation with emphasis on the localization nature of vibrational excitations on the CCA. The frequency dependence of the inverse participation ratio P^-1(ω) obeys a power law such as P^-1(ω)∝ω^1.07±0.06 in two-dimensional system. In addition, we show that the time development of the average cluster size can be scaled by a characteristic time t_c. After a sufficiently long time, the cluster size obeys the power law behavior, depending on the sticking probability between particles.

Imposing restrictions on the allowed sequences of transformations of a standard IFS can give rise to attractors that are attractors of standard IFS consisting of a (larger) finite set of transformations, or a countably infinite set of transformations, or are not the attractor of any standard IFS. Whichever the case it can be read off from the transition graph of the restrictions.

In this paper, we study the spectrum of order 2m (m≥1) elliptic operator A in a bounded open set Ω∈R¹, with fractal boundary Γ=∂Ω and Minkowski dimension D∈(0, 1), thus proving the corresponding modified Weyl-Berry conjecture to be true, namely

In this letter, we present a general method for solving a wide range of nonlinear functional and finite difference equations, as well as iterated equations such as the Hénon and Mandelbrot equations. The method extends to differential equations using an Euler approximation to obtain a finite difference equation.

The box-count method is used to obtain fractal dimensions, limiting probabilities and relative entropies for fractal and fractal-like objects generated by matrix and summation substitution systems. Results are based on the fact that a transition matrix formalism can be developed for these systems, yielding box counts at each iteration. The transition matrix is non-negative, and many results on non-negative matrices can be applied. The formalism developed is, up to a multiplicative constant, equivalent to a Markov process.

An elastic block lattice model is proposed to simulate the spatio-temporal seismicity and stress patterns in the Earth's brittle crust. The famous Gutenberg-Richter magnitude-frequency law in seismology is reproduced. The synthetic catalogs generated by this model are analyzed by using a linked stress release model, which incorporates the stress transfer and spatial interactions. The results highlight the triggering mechanism of earthquake occurrence and the evidence that the crust may lie in a near-critical or self-organized critical state due to the long-range spatial interaction of elastic stress. The spatio-temporal complexity of seismicity is closely related to both nonlinear dynamics of faults and heterogeneities in a seismic region.

A Sierpinski gasket with continuous trajectories is presented as an example of the fractal transition that characterizes the behavior of dissipative dynamical systems excited by external temporal inputs. Using this example, we investigate the fractal transition from two points of views, i.e. a hierarchical structure and a noise effect. Depending on internal and external parameters, the structure can be geometrically classified as one of three types, i.e. totally disconnected, just-touching, and overlapping. For the totally disconnected structure, continuous trajectories and their starting points can be characterized by a definite hierarchical tree structure. Even for the just-touching and overlapping structure, a similar hierarchy exists. White noise contaminating the external inputs breaks the hierarchy. In particular, small clustered structures are sensitive to the noise. In such a case, the difference between trajectories and starting points is remarkable in the hierarchy.

König iteration functions are a generalization of Newton–Raphson method to determine roots of equations. These higher-degree rational functions possess additional fixed points, which are generally different from the desired roots. We first prove two new results: firstly, about these extraneous fixed points and, secondly, about the Julia sets of the König functions associated with the one-parameter family of quadratic polynomials. Then, after finding all the critical points of the König functions as applied to a one-parameter family of cubic polynomials, we examine the orbits of the ones available for convergence to an attracting periodic cycle, should such a cycle exist.

In the box-counting method, positioning of images do not significantly affect the estimation of the fractal dimension of river pattern on the brittle fracture surface, and that of dimple pattern on the ductile fracture surface of materials. A reasonable estimation of the fractal dimension can be made using the box-counting method by a single measurement on the fracture surface pattern. The fractal dimension of dimple pattern in pure Zn polycrystals (about 1.50) is larger than that of river pattern in soda-lime glass (about 1.30). Personal difference in image processing does not have a large influence on the estimation of the fractal dimension of grain-boundary fracture surface profile, compared with the effects of local variation in fracture pattern concerning image size.