BCSSS

International Encyclopedia of Systems and Cybernetics

2nd Edition, as published by Charles François 2004 Presented by the Bertalanffy Center for the Study of Systems Science Vienna for public access.

About

The International Encyclopedia of Systems and Cybernetics was first edited and published by the system scientist Charles François in 1997. The online version that is provided here was based on the 2nd edition in 2004. It was uploaded and gifted to the center by ASC president Michael Lissack in 2019; the BCSSS purchased the rights for the re-publication of this volume in 200?. In 2018, the original editor expressed his wish to pass on the stewardship over the maintenance and further development of the encyclopedia to the Bertalanffy Center. In the future, the BCSSS seeks to further develop the encyclopedia by open collaboration within the systems sciences. Until the center has found and been able to implement an adequate technical solution for this, the static website is made accessible for the benefit of public scholarship and education.

A B C D E F G H I J K L M N O P Q R S T U V W Y Z

SELF-SIMILARITY in the WEIERSTRAS FUNCTION 2)

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B. WEST and A. GOLDBERGER explain: "His function was a superposition of harmonic terms: a fundamental with frequency Ωo and unit amplitude, a second periodic term of frequency bΩo with amplitude 1/a, a third periodic term of frequency bΩo with amplitude 1/a, and so on. The resulting function is an infinite series of periodic terms, each term of which has a frequency that is a factor b larger than the preceeding and an amplitude that is a factor of 1/a smaller. Thus, in giving a functional form to CANTOR's ideas, WEIERSTRASS was the first scientist to construct a fractal function" (1987, p.360).

The WEIERSTRASS function offers some singular properties: "Because of the infinite layers of detail, one cannot draw a tangent to a fractal curve, which means that the function, although continuous, is not differentiable" (Ibid).

Its mode of construction implies that the curve is self-similar at any level. It also can be properly interpolated in a continuous way, however acquiring more and more closely defined values at microscopic levels.

At any chosen level the curve is a fluctuation of the more macroscopic level and the median of the immediately inferior level of fluctuations.

Moreover, any WEIERSTRASS function has its proper measure of self-similarity, in terms of frequency and amplitude, "which is precisely the fractal, or HAUSDORFF dimension: logna/lognb "(Ibid).

The WEIERSTRASS function is a "scaling relation, often called the renormalization group transformation" (Ibid).

Its analogy – unfortunately rather imperfect – with embedded complex cyclical processes is striking.

See: "equilibria (levels of)"; "cyclical or periodic".

→ "Resonance"

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  • 1) General information
  • 2) Methodology or model
  • 3) Epistemology, ontology and semantics
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Bertalanffy Center for the Study of Systems Science(2020).

To cite this page, please use the following information:

Bertalanffy Center for the Study of Systems Science (2020). Title of the entry. In Charles François (Ed.), International Encyclopedia of Systems and Cybernetics (2). Retrieved from www.systemspedia.org/[full/url]

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