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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Akhmet, M. O. Fen and E. M. Alejaily, Dynamics motivated by Sierpinski fractals, preprint, arXiv: 1811.07122. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8] M. Barnsley, Fractals Everywhere, Academic Press, Inc., Boston, MA, 1988.   Google Scholar
[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.  Google Scholar

[10]

E. Chen, Chaos for the Sierpinski carpet, J. Statist. Phys., 88 (1997), 979-984.  doi: 10.1023/B:JOSS.0000015182.90436.5b.  Google Scholar

[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.  Google Scholar

[12]

R. M. Crownover, Introduction to Fractals and Chaos, Jones and Bartlett, Boston, MA, 1995. Google Scholar

[13]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986.  Google Scholar

[14]

P. Diamond, Chaotic behavior of systems of difference equations, Internat. J. Systems Sci., 7 (1976), 953-956.  doi: 10.1080/00207727608941979.  Google Scholar

[15]

A. Dohtani, Occurrence of chaos in higher-dimensional discrete-time systems, SIAM J. Appl. Math., 52 (1992), 1707-1721.  doi: 10.1137/0152098.  Google Scholar

[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.  Google Scholar

[17] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.  doi: 10.1017/CBO9780511623738.  Google Scholar
[18]

K. J. Falconer, Sub-self-similar sets, Trans. Amer. Math. Soc., 347 (1995), 3121-3129.  doi: 10.1090/S0002-9947-1995-1264809-X.  Google Scholar

[19]

M. J. Feigenbaum, Universal behavior in nonlinear systems, Los Alamos Sci., 1 (1980), 4-27.   Google Scholar

[20]

M. Hata, On the structure of self-similar sets, Japan J. Appl. Math., 2 (1985), 381-414.  doi: 10.1007/BF03167083.  Google Scholar

[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.  Google Scholar

[22]

B. R. Hunt and V. Y. Kaloshin, Prevalence, in Handbook of Dynamical Systems, 3, Elsevier Science, Amsterdam, 2010, 43–87. Google Scholar

[23]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[27]

G. C. Layek, An Introduction to Dynamical Systems and Chaos, Springer, New Delhi, 2015. doi: 10.1007/978-81-322-2556-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., San Francisco, Calif., 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc., 42 (1946), 15-23.  doi: 10.1017/S0305004100022684.  Google Scholar

[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.  Google Scholar

[36]

H.-O. Peitgen, H. Jürgens and D. Saupe, Chaos and Fractals, Springer-Verlag, New York, 2004. doi: 10.1007/b97624.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[39]

E. Sander and J. A. Yorke, Period-doubling cascades galore, Ergodic Theory Dynam. Systems, 31 (2011), 1249-1267.  doi: 10.1017/S0143385710000994.  Google Scholar

[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.  Google Scholar

[41]

E. Schöll and H. G. Schuster, Handbook of Chaos Control, Wiley-VCH, Verlag GmbH & Co. KGaA, Weinheim, 2008. doi: 10.1002/9783527622313.  Google Scholar

[42]

D. W. Spear, Measure and self-similarity, Adv. Math., 91 (1992), 143-157.  doi: 10.1016/0001-8708(92)90014-C.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Akhmet, M. O. Fen and E. M. Alejaily, Dynamics motivated by Sierpinski fractals, preprint, arXiv: 1811.07122. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8] M. Barnsley, Fractals Everywhere, Academic Press, Inc., Boston, MA, 1988.   Google Scholar
[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.  Google Scholar

[10]

E. Chen, Chaos for the Sierpinski carpet, J. Statist. Phys., 88 (1997), 979-984.  doi: 10.1023/B:JOSS.0000015182.90436.5b.  Google Scholar

[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.  Google Scholar

[12]

R. M. Crownover, Introduction to Fractals and Chaos, Jones and Bartlett, Boston, MA, 1995. Google Scholar

[13]

R. L. Devaney, An Introduction to Chaotic Dynamical Systems, The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA, 1986.  Google Scholar

[14]

P. Diamond, Chaotic behavior of systems of difference equations, Internat. J. Systems Sci., 7 (1976), 953-956.  doi: 10.1080/00207727608941979.  Google Scholar

[15]

A. Dohtani, Occurrence of chaos in higher-dimensional discrete-time systems, SIAM J. Appl. Math., 52 (1992), 1707-1721.  doi: 10.1137/0152098.  Google Scholar

[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.  Google Scholar

[17] K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Mathematics, 85, Cambridge University Press, Cambridge, 1986.  doi: 10.1017/CBO9780511623738.  Google Scholar
[18]

K. J. Falconer, Sub-self-similar sets, Trans. Amer. Math. Soc., 347 (1995), 3121-3129.  doi: 10.1090/S0002-9947-1995-1264809-X.  Google Scholar

[19]

M. J. Feigenbaum, Universal behavior in nonlinear systems, Los Alamos Sci., 1 (1980), 4-27.   Google Scholar

[20]

M. Hata, On the structure of self-similar sets, Japan J. Appl. Math., 2 (1985), 381-414.  doi: 10.1007/BF03167083.  Google Scholar

[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.  Google Scholar

[22]

B. R. Hunt and V. Y. Kaloshin, Prevalence, in Handbook of Dynamical Systems, 3, Elsevier Science, Amsterdam, 2010, 43–87. Google Scholar

[23]

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[27]

G. C. Layek, An Introduction to Dynamical Systems and Chaos, Springer, New Delhi, 2015. doi: 10.1007/978-81-322-2556-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., San Francisco, Calif., 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

P. A. P. Moran, Additive functions of intervals and Hausdorff measure, Proc. Cambridge Philos. Soc., 42 (1946), 15-23.  doi: 10.1017/S0305004100022684.  Google Scholar

[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.  Google Scholar

[36]

H.-O. Peitgen, H. Jürgens and D. Saupe, Chaos and Fractals, Springer-Verlag, New York, 2004. doi: 10.1007/b97624.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar
[39]

E. Sander and J. A. Yorke, Period-doubling cascades galore, Ergodic Theory Dynam. Systems, 31 (2011), 1249-1267.  doi: 10.1017/S0143385710000994.  Google Scholar

[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.  Google Scholar

[41]

E. Schöll and H. G. Schuster, Handbook of Chaos Control, Wiley-VCH, Verlag GmbH & Co. KGaA, Weinheim, 2008. doi: 10.1002/9783527622313.  Google Scholar

[42]

D. W. Spear, Measure and self-similarity, Adv. Math., 91 (1992), 143-157.  doi: 10.1016/0001-8708(92)90014-C.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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