Studies in growth patterns and fractals

Statistical Science, 1992

Physical Review E, 2013

We propose and investigate a simple model which describes the kinetics of aggregation of Brownian particles with stochastic self-replication. An exact solution and the scaling theory are presented alongside numerical simulation which fully support all theoretical findings. In particular, we show analytically that the particle size distribution function exhibits dynamic scaling and we verified it numerically using the idea of data-collapse. Besides, the conditions under which the resulting system emerges as a fractal are found, the fractal dimension of the system is given and the relationship between this fractal dimension and a conserved quantity is pointed out.

Bulletin of Mathematical Biology, 1989

In this paper we analyse time series data as the growth of organisms using markers such as treerings and otolith deposits (fish). The series studied belong to two tree species (Pinus uncinata, Fagus sylvatiea) and one fish species (Dicentrarchus labrax). Spectral analyses of the time series growth show that the main frequencies of fluctuation may be due to variations of the energy input. However, any causal explanation must consider the internal continuous readjustment in the system as reported by the corresponding chaotic properties of the asymptotic decay of the spectra time structure. Since the output of noisy and chaotic systems tend to show similar spectral densities, an attempt to differentiate them has been carried out. The chaotic behaviour has been characterized by the study of the attractors. The dimensions of these multiple topologies were 3.2 and 3.4 for the tree species and 2.3 for the fish species. Therefore, we are dealing with fractal attractors and the minimum number of variables that can be used to describe the systems are 4 and 3 respectively. It is suggested that some of the variables that most influence growth are those obtained by the response functions in the case of trees.

Alexandria Engineering Journal, 2014

A recent study has shown that everywhere real systems follow common ''fractals-general stacking behavior'' during their change pathways (or evolutionary life cycles). This fact leads to the emergence of the new discipline ''Fractals-General Science'' as a mother-discipline (and acting as upper umbrella) of existing natural and applied sciences to commonly handle their fractals-general change behavior. It is, therefore, the main targets of this short communication are to present the motives, objectives, relations with other existing sciences, and the development map of such new science. It is discussed that there are many foreseen illustrative applications in geology,

Self-similarity may originate from two origins, i.e., the process memory and the process' increments "infinite" variance. A distinction is attempted by employing the natural time χ. Concerning the first origin, we analyze recent data on Seismic Electric Signals, which support the view that they exhibit infinitely ranged temporal correlations. Concerning the second, slowly driven systems that emit bursts of various energies E obeying power-law distribution, i.e., P (E) ∼ E −γ ,are studied. An interrelation between the exponent γ and the variance κ1(≡ χ 2 − χ 2 ) is obtained for the shuffled (randomized) data. In the latter, the most probable value of κ1 is approximately equal to that of the original data. Finally, it is found that the differential entropy associated with the probability P (κ1) maximizes for γ around γ ≈ 1.6 to 1.7, which is comparable to the value determined experimentally in diverse phenomena, e.g., solar flares, icequakes, dislocation glide in stressed single crystals of ice e.t.c. It also agrees with the b-value in the Gutenberg-Richter law of earthquakes.

Applied Mathematical Modelling, 2014

Power law (PL) distributions have been largely reported in the modeling of distinct real phenomena and have been associated with fractal structures and self-similar systems. In this paper, we analyze real data that follows a PL and a double PL behavior and verify the relation between the PL coefficient and the capacity dimension of known fractals. It is to be proved a method that translates PLs coefficients into capacity dimension of fractals of any real data.

1997

In this lecture we present an overview of the physics of irreversible fractal growth process, with particular emphasis on a class of models characterized by quenched disorder. These models exhibit self-organization, with critical properties developing spontaneously, without the fine tuning of external parameters. This situation is different from the usual critical phenomena, and requires the introduction of new theoretical methods. Our approach to these problems is based on two concepts, the Fixed Scale Transformation, and the quenched-stochastic transformation, or Run Time Statistics (RTS), which maps a dynamics with quenched disorder into a stochastic process. These methods, combined together, allow us to understand the self-organized nature of models with quenched disorder and to compute analytically their critical exponents. In addition, it is also possible characterize mathematically the origin of the dynamics by avalanches and compare it with the continuous growth of other fractal models. A specific application to Invasion Percolation will be discussed. Some possible relations to glasses will also be mentioned.

Cornell University - arXiv, 2022

Numerous network models have been investigated to gain insights into the origins of fractality. In this work, we introduce two novel network models, to better understand the growing mechanism and structural characteristics of fractal networks. The Repulsion Based Fractal Model (RBFM) is built on the well-known Song-Havlin-Makse (SHM) model, but in RBFM repulsion is always present among a specific group of nodes. The model resolves the contradiction between the SHM model and the Hub Attraction Dynamical Growth model, by showing that repulsion is the characteristic that induces fractality. The Lattice Smallworld Transition Model (LSwTM) was motivated by the fact that repulsion directly influences the node distances. Through LSwTM we study the fractal-small-world transition. The model illustrates the transition on a fixed number of nodes and edges using a preferential-attachmentbased edge rewiring process. It shows that a small average distance works against fractal scaling, and also demonstrates that fractality is not a dichotomous property, continuous transition can be observed between the pure fractal and non-fractal characteristics.

Physical Review E, 1996

The fractal properties of models of randomly placed n-dimensional spheres (n=1,2,3) are studied using standard techniques for calculating fractal dimensions in empirical data (the box counting and Minkowski-sausage techniques). Using analytical and numerical calculations it is shown that in the regime of low volume fraction occupied by the spheres, apparent fractal behavior is observed for a range of scales between physically relevant cut-offs. The width of this range, typically spanning between one and two orders of magnitude, is in very good agreement with the typical range observed in experimental measurements of fractals. The dimensions are not universal and depend on density. These observations are applicable to spatial, temporal and spectral random structures. Polydispersivity in sphere radii and impenetrability of the spheres (resulting in short range correlations) are also introduced and are found to have little effect on the scaling properties. We thus propose that apparent fractal behavior observed experimentally over a limited range may often have its origin in underlying randomness.

Fractals in Physics, 1986

The fractal dimensions of various types of Jntersection sets of random fractals are discussed. ffris frÀipi a better understanding oÍ'issues concernlng-upper critical dimensionality' F'lory upp"oiimátions or universality oi critical behaviour w'lth respect to different excluded volume màttranisms. Diffusion on sel?-intersecting fractals, and on fractals with local bridges' is also iónjiaereA in the light of the above discussion, and accurate results for the relative exponents' obtained from ana'lysis of Monte-Carlo generated enumeratlons, are presented.