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On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit
Abstract similarity, fractals and chaos
1. | Department of Mathematics, Middle East Technical University, Ankara, Turkey |
2. | Department of Fundamental Sciences, College of Engineering Technology, Houn, Lybya |
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.
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. |







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