Dynamical characterization of mixed fractal structures

Abstract

It is because of people like Marie-Louise and Charles that it is worth fighting for a better world.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.

The inverse problem for fractal curves solved with the dynamical approach method

“…With the dynamical approach it is possible to determine the fractal dimension of a given sample by exploring variables that are not accessible to classical methods. In [6][7][8][9] it was shown that the dynamical dimension of a sequence of subsamples obtained by successively cutting the previous one from the original sample led to its fractal dimension.…”

“…This non-homogeneous distribution of the bending moment leads to the inclusion of a disturbance term Ω k in the determination of the period of oscillation. It can be shown that for regular fractal curves the term Ω k → 1 for large values of k [6][7][8][9]. Therefore the angular coefficients of the straight lines corresponding to the functions log(T k /T0) × log(λ k /L0) for the three initial conditions M, H and V are related with the Hausdorff dimension as DM = DH = DV = (1 − F )/2 since all three angular coefficients coincide for the cases under consideration, FM = FH = FV = F .…”