Dynamical Characterization of Fractal Objects: Determination of the Fine Fractal Topology Using the Energy Cover

TL;DR

This paper extends a dynamical method for analyzing fractal objects to complex curves, using harmonic oscillators to determine fine fractal topology and classify fractal structures based on their energy response.

Contribution

It introduces a novel dynamical approach to characterize and distinguish fine fractal structures, including randomly generated and Weierstrass-Mandelbrot curves, based on their energy cover responses.

Findings

The method accurately estimates fractal dimensions from dynamical responses.

It can distinguish between perfectly and partially fractal structures.

The technique detects affine similarity in fractal sets.

Abstract

The foundation of the theory presented here has already been proved to be effective for the case of curves belonging to the Koch family. The present paper extends the investigation to more complex curves, namely randomly generated curves and the Weierstrass-Mandelbrot curve. The analysis is focused on numerical experiments. The results obtained with the numerical analysis allow advancing some interesting proposition concerning the fine fractal structure of plane curves. The method uses the dynamical response of appropriate harmonic oscillators built up according to the geometry of the fractal set. For the plane motion three fundamental periods are determined. Each period plotted against the characteristic length of the corresponding curve in a log-log scale estimates the fractal dimension. It is shown that each degree of freedom corresponds to a distinct energy covering associate to a certain topological property. Using this technique it is possible to distinguish the fine fractal structure of plane curves. The dynamic technique is proved to work also for the identification problem, that is, to determine the fractal characteristics embedded in some sample of a given fractal set. It is proposed the classification of fractal structures into at least two categories, perfectly fractal and partially fractal. The affine similarity can be precisely detected using this method.