Abstract
In many cases occurring in the real world and studied in science and engineering, non-homogeneous fractal forms often emerge with striking characteristics of cyclicity or periodicity. The authors, for example, have repeatedly traced these characteristics in hydrological basins, hydraulic networks, water demand, and various datasets. But, unfortunately, today we do not yet have well-developed and at the same time simple-to-use mathematical models that allow, above all scientists and engineers, to interpret these phenomena. An interesting idea was firstly proposed by Sergeyev in 2007 under the name of “blinking fractals.” In this paper we investigate from a pure geometric point of view the fractal properties, with their computational aspects, of two main examples generated by a system of multiple rules and which are enlightening for the theme. Strengthened by them, we then propose an address for an easy formalization of the concept of blinking fractal and we discuss some possible applications and future work.
Highlights
The word “fractal” was coined by B
The main characteristic of a fractal, as it is well known, is the property of self-similarity at different scales, and many abstract mathematical models have been created by focusing on this property
A fractal is mathematically described by a generating rule or an iterated mechanism, but in the real world it is not difficult to find examples in which it clearly emerges that a single simple rule is not enough to build the fractal
Summary
The word “fractal” was coined by B. Maiolo applied in contexts with different mass concentrations and in chaotic dynamics, for instance, to the Sun’s magnetic field, human heartbeat and brain activity, turbulent dynamics in fluids, meteorology, geophysics, and finance, internet traffic, and others (see Falconer 2014; Harte 2001; Ivanov et al 1999; Stanley and Meakin 1988; Veneziano and Essiam 2003 and their references) Another interesting example is given by the superfractal formalism introduced in 2002 by Barnsley, Hutchinson and Stenflo. If A and B are two nonempty subsets of RN we define their Hausdorff distance dH ( A, B) by dH ( A, B) := inf ε ≥ 0 : A ⊆ B{ε} and B ⊆ A{ε} . (1)
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