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FRACTAL RELAXATION SYSTEMS. Part I: Singularity Structure Analysis
Journal article   Peer reviewed

FRACTAL RELAXATION SYSTEMS. Part I: Singularity Structure Analysis

YUAN-YING Tsao and BANU Onaral
International journal of general systems, v 17(4), pp 359-377
01 Nov 1990
DOI: https://doi.org/10.1080/03081079008935118
Featured in Collection :   UN Sustainable Development Goals @ Drexel

Abstract

1/f-type attenuation Fractal relaxation systems fractional power attentualion location ratio power-law behavior self-similarity singularity structure structure scale view scale
The magnitude spectral density of many physical phenomena such as electrical noise, the relaxation of polarized dielectrics, viscous and magnetic materials, and the interface between two dissimilar conducting materials attenuate following a fractional power function dependence on frequency. Such systems are recognized as fractal systems distinguished by the 1/f-type attenuation in the magnitude spectrum and characterized by global parameters such as fractal dimension and global corners. Fractal systems which relax to steady state by a distribution of purely real exponentials are recognized as fractal relaxation systems. This is the first part of a series of planned articles, each focusing on a particular aspect of fractal relaxation systems. In this part, the singularity structure model is proposed to represent the steady state frequency response of fractal relaxation systems in the linear range. The singularity structure model of fractal relaxation systems can be mathematically represented by a rational system function with simple poles and zeros arranged in a self-similar pattern in the complex frequency plane. We show that the local singularity structure, i.e. the placement of poles and zeros, can be generated by a recursive procedure following a simple rule which relates the global parameters to local ones. To illustrate the approach and test the model, we first synthesize a fractal R-C circuit; then, we demonstrate the strength of the method by analyzing experimental impedance data collected from a polarized metal electrode-electrolyte interface over six decades of frequency. The standard singularity structure is also introduced to generalize the concept of singularity structure. Forthcoming articles in the series will focus on the analysis of fractal relaxation systems using the discrete distribution of relaxation times and on the concepts of 'structure'and 'view' scales. We promote the distribution of relaxation times as a powerful method to analyze fractal systems and note that it carries information identical to the singularity structure. In fractal systems, the observer becomes part of the modelling process. This is due to the self-similar property of fractal systems: the structure scale of the singularity function is determined according to the view scale chosen by the observer. The implication of this property on the conventional model order concept is elaborated.

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5 citations in Web of Science
7 citations in Scopus

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Web of Science research areas
Computer Science, Theory & Methods
Ergonomics
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