Abstract similarity, fractals and chaos

Marat Akhmet; Ejaily Milad Alejaily

doi:10.3934/dcdsb.2020191

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May 2021, 26(5): 2479-2497. doi: 10.3934/dcdsb.2020191

Abstract similarity, fractals and chaos

Marat Akhmet ^1,, and Ejaily Milad Alejaily ^2,

1.	Department of Mathematics, Middle East Technical University, Ankara, Turkey
2.	Department of Fundamental Sciences, College of Engineering Technology, Houn, Lybya

^* Corresponding author: Tel.: +90 312 210 5355, Fax: +90 312 210 2972

The art of doing mathematics consists in finding that special case which contains all the germs of generalit David Hilbert

Received August 2019 Revised March 2020 Published May 2021 Early access June 2020

Fund Project: The first author has been supported by a grant (118F161) from TÜBİTAK, the Scientific and Technological Research Council of Turkey

A new mathematical concept of abstract similarity is introduced and is illustrated in the space of infinite strings on a finite number of symbols. The problem of chaos presence for the Sierpinski fractals, Koch curve, as well as Cantor set is solved by considering a natural similarity map. This is accomplished for Poincaré, Li-Yorke and Devaney chaos, including multi-dimensional cases. Original numerical simulations illustrating the results are presented.

Keywords: Abstract self-similarity, self-similar space, similarity map, fractals, chaos, multi-dimensional chaotic maps.

Mathematics Subject Classification: Primary: 28A80, 65P20; Secondary: 37B10.

Citation: Marat Akhmet, Ejaily Milad Alejaily. Abstract similarity, fractals and chaos. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2479-2497. doi: 10.3934/dcdsb.2020191

References:

[1]	T. Addabbo, A. Fort, S. Rocchi and V. Vignoli, Digitized chaos for pseudo-random number generation in cryptography, in Chaos-Based Cryptography, Studies in Computational Intelligence, 354, Springer, Berlin, Heidelberg, 2011, 67–97. doi: 10.1007/978-3-642-20542-2.
[2]	M. Akhmet and M. O. Fen, Unpredictable points and chaos, Commun. Nonlinear Sci. Numer. Simul., 40 (2016), 1-5. doi: 10.1016/j.cnsns.2016.04.007.
[3]	M. Akhmet and M. O. Fen, Poincaré chaos and unpredictable functions, Commun. Nonlinear Sci. Numer. Simul., 48 (2017), 85-94. doi: 10.1016/j.cnsns.2016.12.015.
[4]	M. Akhmet and M. O. Fen, Replication of Chaos in Neural Networks, Economics and Physics, Nonlinear Physical Science, Springer, Heidelberg, 2016. doi: 10.1007/978-3-662-47500-3.
[5]	M. Akhmet, M. O. Fen and E. M. Alejaily, Dynamics motivated by Sierpinski fractals, preprint, arXiv: 1811.07122.
[6]	A. Avila and C. Gustavo Moreira, Bifurcations of unimodal maps, in Dynamical Systems. Part II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003, 1–22.
[7]	C. Bandt and S. Graf, Self-similar sets. Ⅶ. A characterization of self-similar fractals with positive Hausdorff measure, Proc. Amer. Math. Soc., 114 (1992), 995-1001. doi: 10.2307/2159618.
[8]	M. Barnsley, Fractals Everywhere, Academic Press, Inc., Boston, MA, 1988.
[9]	G. Boeing, Visual analysis of nonlinear dynamical systems: Chaos, fractals, self-similarity and the limits of prediction, Systems, 4 (2016), 1-18. doi: 10.3390/systems4040037.
[10]	E. Chen, Chaos for the Sierpinski carpet, J. Statist. Phys., 88 (1997), 979-984. doi: 10.1023/B:JOSS.0000015182.90436.5b.
[11]	G. Chen and Y. Huang, Chaotic Maps. Dynamics, Fractals, and Rapid Fluctuations, Synthesis Lectures on Mathematics and Statistics, 11, Morgan & Claypool Publishers, Williston, VT, 2011. doi: 10.2200/S00373ED1V01Y201107MAS011.
[12]	R. M. Crownover, Introduction to Fractals and Chaos, Jones and Bartlett, Boston, MA, 1995.
[13]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986.
[14]	P. Diamond, Chaotic behavior of systems of difference equations, Internat. J. Systems Sci., 7 (1976), 953-956. doi: 10.1080/00207727608941979.
[15]	A. Dohtani, Occurrence of chaos in higher-dimensional discrete-time systems, SIAM J. Appl. Math., 52 (1992), 1707-1721. doi: 10.1137/0152098.
[16]	G. A. Edgar, Measure, Topology, and Fractal Geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1990. doi: 10.1007/978-0-387-74749-1.
[17]	K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986. doi: 10.1017/CBO9780511623738.
[18]	K. J. Falconer, Sub-self-similar sets, Trans. Amer. Math. Soc., 347 (1995), 3121-3129. doi: 10.1090/S0002-9947-1995-1264809-X.
[19]	M. J. Feigenbaum, Universal behavior in nonlinear systems, Los Alamos Sci., 1 (1980), 4-27.
[20]	M. Hata, On the structure of self-similar sets, Japan J. Appl. Math., 2 (1985), 381-414. doi: 10.1007/BF03167083.
[21]	M. Hata, Topological aspects of self-similar sets and singular functions, in Fractal Geometry and Analysis, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 346, Kluwer Acad. Publ., Dordrecht, 1991,255–276. doi: 10.1007/978-94-015-7931-5_6.
[22]	B. R. Hunt and V. Y. Kaloshin, Prevalence, in Handbook of Dynamical Systems, 3, Elsevier Science, Amsterdam, 2010, 43–87.
[23]	J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.
[24]	P. E. T. Jorgensen, Analysis and Probability: Wavelets, Signals, Fractals, Graduate Texts in Mathematics, 234, Springer, New York, 2006. doi: 10.1007/978-0-387-33082-2.
[25]	S. H. Kellert, In the Wake of Chaos. Unpredictable Order in Dynamical Systems, Science and its Conceptual Foundations, University of Chicago Press, Chicago, IL, 1993. doi: 10.7208/chicago/9780226429823.001.0001.
[26]	K.-S. Lau, S.-M. Ngai and H. Rao, Iterated function systems with overlaps and the self-similar measures, J. London Math. Soc. (2), 63 (2001), 99-116. doi: 10.1112/S0024610700001654.
[27]	G. C. Layek, An Introduction to Dynamical Systems and Chaos, Springer, New Delhi, 2015. doi: 10.1007/978-81-322-2556-0.
[28]	T. Y. Li and and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. doi: 10.1080/00029890.1975.11994008.
[29]	R. Li and X. Zhou, A note on chaos in product maps, Turkish J. Math., 37 (2013), 665-675. doi: 10.3906/mat-1101-71.
[30]	B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., San Francisco, Calif., 1982.
[31]	F. R. Marotto, Snap-back repellers imply chaos in ${\bf R}^n$, J. Math. Anal. Appl., 63 (1978), 199-223. doi: 10.1016/0022-247X(78)90115-4.
[32]	C. Masoller, A. C. Sicardi Schifino and L. Romanelli, Characterization of strange attractors of Lorenz model of general circulation of the atmosphere, Chaos Solit. Fract., 6 (1995), 357-366. doi: 10.1016/0960-0779(95)80041-E.
[33]	F. C. Moon, Chaotic and Fractal Dynamics. An Introduction for Applied Scientists and Engineers, John Wiley & Sons, Inc., New York, 1992. doi: 10.1002/9783527617500.
[34]	P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc., 42 (1946), 15-23. doi: 10.1017/S0305004100022684.
[35]	S.-M. Ngai and Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2), 63 (2001), 655-672. doi: 10.1017/S0024610701001946.
[36]	H.-O. Peitgen, H. Jürgens and D. Saupe, Chaos and Fractals, Springer-Verlag, New York, 2004. doi: 10.1007/b97624.
[37]	Y. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture, Comm. Math. Phys., 182 (1996), 105-153. doi: 10.1007/BF02506387.
[38]	Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. doi: 10.7208/chicago/9780226662237.001.0001.
[39]	E. Sander and J. A. Yorke, Period-doubling cascades galore, Ergodic Theory Dynam. Systems, 31 (2011), 1249-1267. doi: 10.1017/S0143385710000994.
[40]	S. Sato and K. Gohara, Fractal transition in continuous recurrent neural networks, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 421-434. doi: 10.1142/S0218127401002158.
[41]	E. Schöll and H. G. Schuster, Handbook of Chaos Control, Wiley-VCH, Verlag GmbH & Co. KGaA, Weinheim, 2008. doi: 10.1002/9783527622313.
[42]	D. W. Spear, Measure and self-similarity, Adv. Math., 91 (1992), 143-157. doi: 10.1016/0001-8708(92)90014-C.
[43]	S. Stella, On Hausdorff dimension of recurrent net fractals, Proc. Amer. Math. Soc., 116 (1992), 389-400. doi: 10.1090/S0002-9939-1992-1094507-X.
[44]	S. Wiggins, Global Bifurcation and Chaos. Analytical Methods, Applied Mathematical Sciences, 73, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1042-9.
[45]	G. M. Zaslavsky, M. Edelman and B. A. Niyazov, Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics, Chaos, 7 (1997), 159-181. doi: 10.1063/1.166252.
[46]	E. Zeraoulia and J. C. Sprott, Robust Chaos and Its Applications, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 79, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/8296.
[47]	O. Zmeskal, P. Dzik and M. Vesely, Entropy of fractal systems, Comput. Math. Appl., 66 (2013), 135-146. doi: 10.1016/j.camwa.2013.01.017.

show all references

The art of doing mathematics consists in finding that special case which contains all the germs of generalit

David Hilbert

References:

[1]	T. Addabbo, A. Fort, S. Rocchi and V. Vignoli, Digitized chaos for pseudo-random number generation in cryptography, in Chaos-Based Cryptography, Studies in Computational Intelligence, 354, Springer, Berlin, Heidelberg, 2011, 67–97. doi: 10.1007/978-3-642-20542-2.
[2]	M. Akhmet and M. O. Fen, Unpredictable points and chaos, Commun. Nonlinear Sci. Numer. Simul., 40 (2016), 1-5. doi: 10.1016/j.cnsns.2016.04.007.
[3]	M. Akhmet and M. O. Fen, Poincaré chaos and unpredictable functions, Commun. Nonlinear Sci. Numer. Simul., 48 (2017), 85-94. doi: 10.1016/j.cnsns.2016.12.015.
[4]	M. Akhmet and M. O. Fen, Replication of Chaos in Neural Networks, Economics and Physics, Nonlinear Physical Science, Springer, Heidelberg, 2016. doi: 10.1007/978-3-662-47500-3.
[5]	M. Akhmet, M. O. Fen and E. M. Alejaily, Dynamics motivated by Sierpinski fractals, preprint, arXiv: 1811.07122.
[6]	A. Avila and C. Gustavo Moreira, Bifurcations of unimodal maps, in Dynamical Systems. Part II, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2003, 1–22.
[7]	C. Bandt and S. Graf, Self-similar sets. Ⅶ. A characterization of self-similar fractals with positive Hausdorff measure, Proc. Amer. Math. Soc., 114 (1992), 995-1001. doi: 10.2307/2159618.
[8]	M. Barnsley, Fractals Everywhere, Academic Press, Inc., Boston, MA, 1988.
[9]	G. Boeing, Visual analysis of nonlinear dynamical systems: Chaos, fractals, self-similarity and the limits of prediction, Systems, 4 (2016), 1-18. doi: 10.3390/systems4040037.
[10]	E. Chen, Chaos for the Sierpinski carpet, J. Statist. Phys., 88 (1997), 979-984. doi: 10.1023/B:JOSS.0000015182.90436.5b.
[11]	G. Chen and Y. Huang, Chaotic Maps. Dynamics, Fractals, and Rapid Fluctuations, Synthesis Lectures on Mathematics and Statistics, 11, Morgan & Claypool Publishers, Williston, VT, 2011. doi: 10.2200/S00373ED1V01Y201107MAS011.
[12]	R. M. Crownover, Introduction to Fractals and Chaos, Jones and Bartlett, Boston, MA, 1995.
[13]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986.
[14]	P. Diamond, Chaotic behavior of systems of difference equations, Internat. J. Systems Sci., 7 (1976), 953-956. doi: 10.1080/00207727608941979.
[15]	A. Dohtani, Occurrence of chaos in higher-dimensional discrete-time systems, SIAM J. Appl. Math., 52 (1992), 1707-1721. doi: 10.1137/0152098.
[16]	G. A. Edgar, Measure, Topology, and Fractal Geometry, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1990. doi: 10.1007/978-0-387-74749-1.
[17]	K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986. doi: 10.1017/CBO9780511623738.
[18]	K. J. Falconer, Sub-self-similar sets, Trans. Amer. Math. Soc., 347 (1995), 3121-3129. doi: 10.1090/S0002-9947-1995-1264809-X.
[19]	M. J. Feigenbaum, Universal behavior in nonlinear systems, Los Alamos Sci., 1 (1980), 4-27.
[20]	M. Hata, On the structure of self-similar sets, Japan J. Appl. Math., 2 (1985), 381-414. doi: 10.1007/BF03167083.
[21]	M. Hata, Topological aspects of self-similar sets and singular functions, in Fractal Geometry and Analysis, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 346, Kluwer Acad. Publ., Dordrecht, 1991,255–276. doi: 10.1007/978-94-015-7931-5_6.
[22]	B. R. Hunt and V. Y. Kaloshin, Prevalence, in Handbook of Dynamical Systems, 3, Elsevier Science, Amsterdam, 2010, 43–87.
[23]	J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.
[24]	P. E. T. Jorgensen, Analysis and Probability: Wavelets, Signals, Fractals, Graduate Texts in Mathematics, 234, Springer, New York, 2006. doi: 10.1007/978-0-387-33082-2.
[25]	S. H. Kellert, In the Wake of Chaos. Unpredictable Order in Dynamical Systems, Science and its Conceptual Foundations, University of Chicago Press, Chicago, IL, 1993. doi: 10.7208/chicago/9780226429823.001.0001.
[26]	K.-S. Lau, S.-M. Ngai and H. Rao, Iterated function systems with overlaps and the self-similar measures, J. London Math. Soc. (2), 63 (2001), 99-116. doi: 10.1112/S0024610700001654.
[27]	G. C. Layek, An Introduction to Dynamical Systems and Chaos, Springer, New Delhi, 2015. doi: 10.1007/978-81-322-2556-0.
[28]	T. Y. Li and and J. A. Yorke, Period three implies chaos, Amer. Math. Monthly, 82 (1975), 985-992. doi: 10.1080/00029890.1975.11994008.
[29]	R. Li and X. Zhou, A note on chaos in product maps, Turkish J. Math., 37 (2013), 665-675. doi: 10.3906/mat-1101-71.
[30]	B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., San Francisco, Calif., 1982.
[31]	F. R. Marotto, Snap-back repellers imply chaos in ${\bf R}^n$, J. Math. Anal. Appl., 63 (1978), 199-223. doi: 10.1016/0022-247X(78)90115-4.
[32]	C. Masoller, A. C. Sicardi Schifino and L. Romanelli, Characterization of strange attractors of Lorenz model of general circulation of the atmosphere, Chaos Solit. Fract., 6 (1995), 357-366. doi: 10.1016/0960-0779(95)80041-E.
[33]	F. C. Moon, Chaotic and Fractal Dynamics. An Introduction for Applied Scientists and Engineers, John Wiley & Sons, Inc., New York, 1992. doi: 10.1002/9783527617500.
[34]	P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc., 42 (1946), 15-23. doi: 10.1017/S0305004100022684.
[35]	S.-M. Ngai and Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2), 63 (2001), 655-672. doi: 10.1017/S0024610701001946.
[36]	H.-O. Peitgen, H. Jürgens and D. Saupe, Chaos and Fractals, Springer-Verlag, New York, 2004. doi: 10.1007/b97624.
[37]	Y. Pesin and H. Weiss, On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture, Comm. Math. Phys., 182 (1996), 105-153. doi: 10.1007/BF02506387.
[38]	Y. Pesin, Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997. doi: 10.7208/chicago/9780226662237.001.0001.
[39]	E. Sander and J. A. Yorke, Period-doubling cascades galore, Ergodic Theory Dynam. Systems, 31 (2011), 1249-1267. doi: 10.1017/S0143385710000994.
[40]	S. Sato and K. Gohara, Fractal transition in continuous recurrent neural networks, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), 421-434. doi: 10.1142/S0218127401002158.
[41]	E. Schöll and H. G. Schuster, Handbook of Chaos Control, Wiley-VCH, Verlag GmbH & Co. KGaA, Weinheim, 2008. doi: 10.1002/9783527622313.
[42]	D. W. Spear, Measure and self-similarity, Adv. Math., 91 (1992), 143-157. doi: 10.1016/0001-8708(92)90014-C.
[43]	S. Stella, On Hausdorff dimension of recurrent net fractals, Proc. Amer. Math. Soc., 116 (1992), 389-400. doi: 10.1090/S0002-9939-1992-1094507-X.
[44]	S. Wiggins, Global Bifurcation and Chaos. Analytical Methods, Applied Mathematical Sciences, 73, Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1042-9.
[45]	G. M. Zaslavsky, M. Edelman and B. A. Niyazov, Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics, Chaos, 7 (1997), 159-181. doi: 10.1063/1.166252.
[46]	E. Zeraoulia and J. C. Sprott, Robust Chaos and Its Applications, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 79, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/8296.
[47]	O. Zmeskal, P. Dzik and M. Vesely, Entropy of fractal systems, Comput. Math. Appl., 66 (2013), 135-146. doi: 10.1016/j.camwa.2013.01.017.

Figure 1. (a) The first step of abstract self-similar set construction. (b) The illustration of the boundary agreement

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Figure 2. Examples of the $ 2^{nd} $ and the $ 3^{rd} $ order subsets of the Sierpinski carpet

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Figure 3. A trajectory of the point under the similarity map

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Figure 4. The construction of abstract self-similar set corresponding to the Sierpinski gasket

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Figure 5. The construction of abstract self-similar set corresponding to the Koch curve

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Figure 6. The $ 1^{st} $ and the $ 2^{nd} $ order subsets for the Cantor set

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Figure 7. The construction of abstract self-similar set using the map (12)

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Figure 8. The first three iterations of DASS construction using the map (13)

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Figure 9. The two trajectories of the system (13)

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