Vol. 6, No. 1-4, 2011

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Dynamical characterization of mixed fractal structures

Luiz Bevilacqua and Marcelo M. Barros

Vol. 6 (2011), No. 1-4, 51–69

We present a new technique to determine the fractal or self-similarity dimension of a sequence of curves. The geometric characterization of the sequence is obtained from the mechanical properties of harmonic oscillators with the same shape of the terms composing the given sequence of curves. The definition of “dynamical dimension” is briefly introduced with the help of simple examples. The theory is proved to be valid for a particular type of curves as those of the Koch family. The method is applied to more complex plane curves obtained by superposing two generators of the Koch family with different fractal dimensions. It is shown that this structure is composed by two series of objects one of which is fractal and the other which is not rigorously a fractal sequence but approaches asymptotically a fractal object. The notion of quasifractal structures is introduced. The results are shown to provide good information about the structure formation. It is shown that the dynamical dimension can identify randomness for certain fractal curves.

fractals, mixed fractals, dynamical dimension, random fractals
Received: 29 May 2010
Revised: 15 August 2010
Accepted: 3 October 2010
Published: 28 June 2011
Luiz Bevilacqua
Universidade Federal do Rio de Janeiro
Centro de Tecnologia, Bloco B, s/ 101
Cidade Universitária
21945-970 Rio de Janeiro
Marcelo M. Barros
Laboratório Nacional de Computação Científica
Rua Getúlio Vargas, 333
25651-075 Petropolis