Abstract

For the first time in the world, a universal generalized method for mono- and multifractals synthesis is proposed, which allows combining and reproducing almost all known artificial and semi-artificial methods of fractal synthesis on a single natural platform - Brownian point dynamics (BPD) in the field of N gravitational forces.

On the basis of the BPD method, a set of algorithms for the synthesis of fractal sets of a wide range of characteristics and properties, such as monofractal, multifractal, isotropic, anisotropic, rectangular, hexagonal, ring, single- and multi-loop, etc., has been developed.

The fractals synthesized by the BPD method enable the production of solid samples of these fractals with predetermined characteristics using modern 3D printing technologies and experimentally study their porosity, fractal dimensions, residual strength, aerodynamic and hydrodynamic drag, etc.

It is shown that the Brownian dynamics of a point in a field of N gravitational forces clearly demonstrates the generation of a fractal order by Brownian chaos with a cascade of discrete levels ordered by scaling, i.e., an important philosophical result of the demonstration of order from chaos. The influence of minimal values of N on the synthesis of linear, planar or volumetric multifractal sets is demonstrated.

Keywords: chaos; multifractal; synthesis of multifractals; tensor structure of multifractals; inverse problem of fractal analysis

References

1. Ihor Sliusar., et al. “Synthesis of Quasi-Fractal Ring Antennas”. Problems of Infocommunications. Science and Technology 2019 International Scientific-Practical Conference (2019).
2. Martin Langhammer., et al. “Fractal Synthesis: Invited Tutorial”. FPGA '19: Proceedings of the 2019 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays (2019): 202-211.
3. G Ruello., et al. “Synthesis, construction, and validation of a fractal surface”. IEEE Transactions on Geoscience and Remote Sensing 44.6 (2006): 1403-1412.
4. Daniele Riccio and Giuseppe Ruello. “Synthesis of Fractal Surfaces for Remote-Sensing Applications”. IEEE Transactions on Geoscience and Remote Sensing 53.7 (2015).
5. DH Werner and PL Werner. “On the synthesis of fractal radiation patterns”. Radio Science (1995).
6. Grabar IG and Kubrak YO. Synthesis of Multifractals by Brownian Dynamics of a Point in a Field of N Central Forces. 16th Chaotic Modeling and Simulation International Conference (2024): 157-174.
7. G Grabar. Synthesis of multifractals, Zhytomyr, Polissia National University (2023): 200 (in Ukrainian).
8. G Grabar and OI Grabar. “Influence of the stationary and perturbed state of the central force field on the fractal characteristics of the attractor”. Scientific Horizons 23.11 (2020): 39-52.
9. G Grabar and OI Grabar. “Fractal ornaments: design, properties, 3D production, Zhytomyr”. Polissia National University (2021): 308 (in Ukrainian).
10. G Grabar., et al. “Chaos and a quantitative modeling of the kinetics of phase transitions on the final measure areas”. 10th International Conference on Chaotic Modeling and Simulation, CHAOS 2017; Casa Convalescencia of the Universitat Autonoma de Barcelona; Barcelona; Spain; 30 May 2017 through 2 June 2017; Code 151090, 277{288, 2017.
11. Alexander S Balankin, Julián Patiño Ortiz and Miguel PATIÑO Ortiz. “Inherent Features of Fractal Sets and Key Attributes of Fractal Models”. Fractals 30.4 (2022): 2250082.
12. Alexander S Balankin., et al. “Formation factors for a class of deterministic models of pre-fractal pore-fracture networks”. Chaos Solitons & Fractals 162.9 (2022): 112452.
13. Michael F. Barnsley and Hawley Rising. “Fractals Everywhere”. Morgan Kaufmann (1993).
14. ?? Mandelbrot. “The Fractal Geometry of Nature”. New York: W.H. Freeman (1982): 656.

Citation

Ivan G Grabar., et al. “A New Universal General ized Method for Multifractals Synthesis". Clareus Scientific Science and Engineering 2.3 (2025): 11-19.

Copyright

© 2025 Ivan G Grabar., et al. Licensee Clareus Scientific Publications. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license.