Abstract Fractals
DOI:
https://doi.org/10.5890/DNC.2021.03.009Abstract
We develop a new definition of fractals which can be considered as an abstraction of the fractals determined through self-similarity. The definition is formulated through imposing conditions which govern a relation between subsets of a metric space to build a porous self-similar structure. Examples are provided to confirm that the definition satisfies a large class of self-similar fractals. The new concepts create new frontiers for fractals and chaos investigations.References
[1] Addison, P.S. (1997), Fractals and Chaos: An Illustrated Course, Institute of Physics Publishing: Bristol, UK.
[2] Mandelbrot, B.B. (1983), The Fractal Geometry of Nature, Freeman: New York.
[3] Zmeskal, O., Dzik, P., and Vesely M. (2013), Entropy of fractal systems, Comput. Math. Appl., 66, 135-146.
[4] Anovitz, L.M. and Cole, D.R. (2015), Characterization and analysis of porosity and pore structures, Rev. Mineral. Geochem., 80, 61-164.
[5] Ganji, D.D. and Kachapi S.H.H. (2015), Application of Nonlinear Systems in Nanomechanics and Nanofluids: Analytical Methods and Applications, Elsevier Inc: New York.
[6] Davis, H.T. (1989), On the fractal character of the porosity of natural sandstone, Europhys. Lett. 8, 629-632.
[7] Yu, B. and Liu, W. (2004), Fractal analysis of permeabilities for porous media, AIChE J 50(1), 46-57.
[8] Huang, H. (1993), Porosity-size relationship of drilling mud flocs: Fractal structure, Clay Clay Minerals, 41, 373-379.
[9] Guyon, E., Mitescu, C.D., Hulin, J.P., and Roux, S. (1989), Fractals and percolation in porous media and flows?, Physica D, 38, 172-178.
[10] Cai, J.C., Yu, B.M., Zuo, M.Q., and Mei, M.F. (2010), Fractal analysis of surface roughness of particles in porous media, Chin. Phys. Lett., 27(2), 024705.
[11] Puzenko, A., Kozlovich, N., Gutina, A., and Feldman, Y. (1999), Determination of pore fractal dimensions and porosity of silica glasses from the dielectric response at percolation, Phys. Rev. B, 60, 14348.
[12] Tang, H.P., Wang, J.Z., Zhu, J.L., Ao, Q.B., Wang, J.Y., Yang, B.J., and Li, Y.N. (2012), Fractal dimension of pore-structure of porous metal materials made by stainless steel powder, Powder Technol, 217, 383-387.
[13] Xia, Y., Cai, J., Wei, W., Hu, X., Wang, X.I.N., and Ge, X. (2018), A new method for calculating fractal dimensions of porous media based on pore size distribution, Fractals, 26, 1850006.
[14] Akhmet, M. and Alejaily E.M. (2018), Abstract Similarity, Chaos and Fractals, arXiv:1905.02198
[15] Akhmet, M. and Fen, M.O. (2016), Unpredictable points and chaos, Commun. Nonlinear Sci. Numer. Simulat., 40, 1-5.
[16] Akhmet, M., and Fen, M.O. (2016), Poincar{e} chaos and unpredictable functions, Commun. Nonlinear Sci. Numer. Simulat., 48, 85-94.
[17] Hutchinson, J. (1981), Fractals and self-similarity, Indiana Univ. J. Math., 30, 713-47.
[18] Barnsley, M.F. and Demko, S. (1985), Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A, 399, 243-275.
[19] Barnsley, M.F. (1998), Fractals Everywhere, Academic Press: New York.
[20] Akhmet, M. and Alejaily E.M. (2019), Chaos in the multidimensional cube, arXiv:1908.11194.
[21] Akhmet, M. and Alejaily E.M. (2019), Domain structured chaos in Hopfield neural networks, Int. J. Bifurc. Chaos, (in press).
[22] Akhmet, M. and Alejaily, E.M. (2019), Abstract Similarity, Fractals and Chaos, Discrete and Continuous Dynamical Systems, Ser. B., doi:10.3934/dcdsb.2020191 (online first).
Article Metrics
Usage tracking begins September 1, 2026.