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Replication of chaos. (English) Zbl 1306.34067

Summary: We propose a rigorous method for replication of chaos from a prior one to systems with large dimensions. Extension of the formal properties and features of a complex motion can be observed such that ingredients of chaos united as known types of chaos, Devaney’s, Li-Yorke and obtained through period-doubling cascade. This is true for other appearances of chaos: intermittency, structure of the chaotic attractor, its fractal dimension, form of the bifurcation diagram, the spectra of Lyapunov exponents, etc. That is why we identify the extension of chaos through the replication as morphogenesis.To provide rigorous study of the subject, we introduce new definitions such as chaotic sets of functions, the generator and replicator of chaos, and precise description of ingredients for Devaney and Li-Yorke chaos in continuous dynamics. Appropriate simulations which illustrate the chaos replication phenomenon are provided. Moreover, in discussion form we consider inheritance of intermittency, replication of Shil’nikov orbits and quasiperiodical motions as a possible skeleton of a chaotic attractor. Chaos extension in an open chain of Chua circuits is also demonstrated.

MSC:

34C28 Complex behavior and chaotic systems of ordinary differential equations
34H10 Chaos control for problems involving ordinary differential equations
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[1] Fink, A. M., Almost periodic differential equations. Almost periodic differential equations, Lecture notes in mathematics (1974), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0325.34039
[2] Corduneanu, C., Almost periodic functions (1968), Interscience Publishers: Interscience Publishers New-York, London, Sydney · Zbl 0175.09101
[3] Thompson, J. M.T.; Stewart, H. B., Nonlinear dynamics and chaos (2002), John Wiley · Zbl 1174.37300
[4] Davies, J. A., Mechanisms of morphogenesis: the creation of biological form (2005), Elsevier Academic Press: Elsevier Academic Press United States of America
[5] Thom, R., Mathematical models of morphogenesis (1983), Ellis Horwood Limited: Ellis Horwood Limited Chichester, England
[6] Mandelbrot, B. M., The fractal geometry of nature (1982), Freeman: Freeman San Fransisco · Zbl 0504.28001
[7] Mandelbrot, B. M., Fractals: form, chance and dimension (1977), Freeman: Freeman San Fransisco · Zbl 0376.28020
[8] Milnor, J., Dynamics in one complex variable (2006), Princeton University Press: Princeton University Press United States of America · Zbl 1085.30002
[9] Branner, B.; Keen, L.; Douady, A.; Blanchard, P.; Hubbard, J. H.; Schleicher, D.; Devaney, R. L., Complex dynamical systems: the mathematics behind mandelbrot and julia sets, (Devaney, R. L. (1994), American Mathematical Society: American Mathematical Society United States of America)
[10] Peitgen, H. O.; Jürgens, H.; Saupe, D., Chaos and fractals: new frontiers of science (2004), Springer: Springer Verlag, New York · Zbl 1036.37001
[11] Gonzales-Miranda, J. M., Synchronization and control of chaos (2004), Imperial College Press: Imperial College Press London
[12] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Phys Rev Lett, 64, 821-825 (1990) · Zbl 0938.37019
[13] Kapitaniak, T., Synchronization of chaos using continuous control, Phys Rev E, 50, 1642-1644 (1994)
[14] Ding, M.; Ott, E., Enhancing synchronism of chaotic systems, Phys Rev E, 49, R945-R948 (1994)
[15] Pecora, L. M.; Carroll, T. L., Driving systems with chaotic signals, Phys Rev A, 44, 2374-2383 (1991)
[16] Cuomo, K. M.; Oppenheim, A. V., Circuit implementation of synchronized chaos with applications to communications, Phys Rev Lett, 71, 65-68 (1993)
[17] Hadamard, J., Les surfaces à courbures opposées et leurs lignes géodésiques, J Math Pures Appl, 4, 27-74 (1898) · JFM 29.0522.01
[18] Morse, M.; Hedlund, G. A., Symbolic dynamics, Am J Math, 60, 815-866 (1938) · JFM 64.0798.04
[19] Wiggins, S., Global bifurcations and chaos: analytical methods (1988), Springer-Verlag: Springer-Verlag New York · Zbl 0661.58001
[20] Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1027.37002
[21] Devaney, R., An introduction to chaotic dynamical systems (1987), Addison-Wesley: Addison-Wesley United States of America
[22] Kennedy, J.; Yorke, J. A., Topological horseshoes, Trans Am Math Soc, 353, 2513-2530 (2001) · Zbl 0972.37011
[23] Grebogi, C.; Yorke, J. A., The impact of chaos on science and society (1997), United Nations University Press: United Nations University Press Tokyo
[24] Hale, J.; Koçak, H., Dynamics and bifurcations (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0745.58002
[25] Robinson, C., Dynamical systems: stability, symbolic dynamics, and chaos (1995), CRC Press: CRC Press Boca Raton/Ann Arbor/London/Tokyo · Zbl 0853.58001
[26] Akhmet, M. U., Devaney’s chaos of a relay system, Commun Nonlinear Sci Numer Simul, 14, 1486-1493 (2009) · Zbl 1221.37030
[27] Akhmet, M., Nonlinear hybrid continuous/discrete-time models (2011), Atlantis Press: Atlantis Press Amsterdam-Paris · Zbl 1328.93001
[28] Akhmet, M. U., Li-Yorke chaos in the impact system, J Math Anal Appl, 351, 804-810 (2009) · Zbl 1153.37017
[29] Smale, S., Differentiable dynamical systems, Bull Am Math Soc, 73, 747-817 (1967) · Zbl 0202.55202
[30] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1997), Springer-Verlag: Springer-Verlag New York
[31] Lorenz, E. N., Deterministic nonperiodic flow, J Atmos Sci, 20, 130-141 (1963) · Zbl 1417.37129
[32] Cartwright, M.; Littlewood, J., On nonlinear differential equations of the second order I: The equation \(\ddot{y} - k(1 - y^2)^\prime y + y = bkcos(\lambda t + a), k\) large, J London Math Soc, 20, 180-189 (1945) · Zbl 0061.18903
[33] Levinson, N., A second order differential equation with singular solutions, Ann Math, 50, 127-153 (1949) · Zbl 0041.42311
[34] Guckenheimer, J.; Williams, R. F., Structural stability of lorenz attractors, Publ Math, 50, 307-320 (1979) · Zbl 0436.58018
[35] Levi, M., Qualitative analysis of the periodically forced relaxation oscillations (1981), Memoirs of the American Mathematical Society: Memoirs of the American Mathematical Society United States of America · Zbl 0448.34032
[36] Akhmet, M., Principles of discontinuous dynamical systems (2010), Springer: Springer New York · Zbl 1204.37002
[37] Akhmet, M. U., Shadowing and dynamical synthesis, Int J Bifurcation Chaos, 19, 3339-3346 (2009) · Zbl 1182.34009
[38] Akhmet, M. U., Dynamical synthesis of quasi-minimal sets, Int J Bifurcation Chaos, 19, 2423-2427 (2009) · Zbl 1176.34009
[39] Akhmet, M. U., Creating a chaos in a system with relay, Int J Qual Theory Differ Equat Appl, 3, 3-7 (2009) · Zbl 1263.34060
[40] Akhmet, M. U.; Fen, M. O., Chaotic period-doubling and OGY control for the forced Duffing equation, Commun Nonlinear Sci Numer Simul, 17, 1929-1946 (2012) · Zbl 1253.34048
[41] Akhmet, M. U.; Fen, M. O., The period-doubling route to chaos in the relay system, Proc Dyn Syst Appl, 6, 22-26 (2012) · Zbl 1332.37017
[42] Kaneko, K.; Tsuda, I., Complex systems: chaos and beyond, a constructive approach with applications in life sciences (2000), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York
[43] Kaneko, K.; Tsuda, I., Chaotic itinerancy, Chaos, 13, 926-936 (2003) · Zbl 1080.37531
[44] Ikeda, K.; Matsumoto, K.; Otsuka, K., Maxwell-Bloch turbulence, Prog Theor Phys Suppl, 99, 295 (1989)
[45] Tsuda, I., Chaotic itinerancy as a dynamical basis of hermeneutics in brain and mind, World Futures, 32, 167-184 (1991)
[46] Tsuda, I., Dynamic link of memory-chaotic memory map in nonequilibrium neural networks, Neural Networks, 5, 313-326 (1992)
[47] Kaneko, K., Clustering, coding, switching, hierarchical ordering, and control in network of chaotic elements, Phys D, 41, 137-172 (1990) · Zbl 0709.58520
[48] Kaneko, K., Globally coupled circle maps, Phys D, 54, 5-19 (1991) · Zbl 0729.58503
[49] Sauer, T., Chaotic itinerancy based on attractors of one-dimensional maps, Chaos, 13, 947-952 (2003) · Zbl 1080.37555
[50] Freeman, W. J.; Barrie, J. M., Chaotic oscillations and the genesis of meaning in cerebral cortex, (Buzsaki, G.; Christen, Y., Temporal coding in the brain (1994), Springer-Verlag: Springer-Verlag Berlin), 13-37
[51] Kim, P.; Ko, T.; Jeong, H.; Lee, K. J.; Han, S. K., Emergence of chaotic itinerancy in simple ecological systems, Phys Rev E, 76, 065201(R), 1-4 (2007)
[52] Pomeau, Y.; Manneville, P., Intermittent transition to turbulence in dissipative dynamical systems, Commun Math Phys, 74, 189-197 (1980)
[53] Moon, F. C., Chaotic vibrations: an introduction for applied scientists and engineers (2004), John Wiley & Sons: John Wiley & Sons Hoboken, NJ
[54] Chua, L. O., Chua’s circuit: ten years later, IEICE Trans Fund Electron Commun Comput Sci, E77-A, 1811-1822 (1994)
[55] Kapitaniak, T.; Chua, L. O., Hyperchaotic attractors of unidirectionally-coupled Chua’s circuits, Int J Bifurcation Chaos Appl Sci Eng, 4, 477-482 (1994) · Zbl 0813.58037
[56] Lorenz, H. W., Nonlinear dynamical economics and chaotic motion (1989), Springer-Verlag: Springer-Verlag Berlin, New York · Zbl 0717.90001
[57] Sprott, J. C., Elegant chaos: algebraically simple chaotic flows (2010), World Scientific Publishing: World Scientific Publishing Singapore · Zbl 1222.37005
[58] Rössler, O. E., An equation for hyperchaos, Phys Lett A, 71, 155-157 (1979) · Zbl 0996.37502
[59] Kapitaniak, T.; Chua, L. O., Hyperchaotic attractors of unidirectionally-coupled Chua’s circuits, Int J Bifurcation Chaos Appl Sci Eng, 4, 477-482 (1994) · Zbl 0813.58037
[60] Kapitaniak, T., Transition to hyperchaos in chaotically forced coupled oscillators, Phys Rev E, 47, R2975-R2978 (1993)
[61] Kapitaniak, T.; Chua, L. O.; Zhong, G., Experimental hyperchaos in coupled Chua’s circuits, IEEE Trans Circuits Syst I Fund Theory Appl, 41, 499-503 (1994)
[62] Turing, A. M., The chemical basis of morphogenesis, Philos Trans R Soc London Ser B Biol Sci, 237, 37-72 (1952) · Zbl 1403.92034
[63] Schiff, J. L., Cellular automata: a discrete view of the world (2008), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. Hoboken, NJ · Zbl 1142.68052
[64] Smale, S., A mathematical model of two cells via Turing’s equation, (Some mathematical questions in biology, V, proc. seventh sympos., mathematical biology, Mexico city, 1973. Some mathematical questions in biology, V, proc. seventh sympos., mathematical biology, Mexico city, 1973, Lectures on mathematics in the life sciences, vol. 6 (1974), American Mathematical Society: American Mathematical Society Providence, RI), 15-26
[65] Drubi, F.; Ibáñez, S.; Rodriguez, J. A., Coupling leads to chaos, J Differ Equat, 239, 371-385 (2007) · Zbl 1133.34027
[66] Drubi, F.; Ibáñez, S.; Rodriguez, J. A., Singularities and chaos in coupled systems, Bull Belg Math Soc Simon Stevin, 15, 797-808 (2008) · Zbl 1153.37018
[67] Drubi, F.; Ibáñez, S.; Rodriguez, J. A., Hopf-Pitchfork singularities in coupled systems, Phys D, 240, 825-840 (2011) · Zbl 1220.37042
[68] Yuan, W-J.; Luo, X-S.; Jiang, P-Q.; Wang, B-H.; Fang, J-Q., Transition to chaos in small-world dynamical network chaos, Chaos Solitons Fract, 37, 799-806 (2008)
[69] Hale, J. K., Ordinary differential equations (1980), Krieger Publishing Company: Krieger Publishing Company Malabar, FL · Zbl 0433.34003
[70] Corduneanu, C., Principles of differential and integral equations (1977), Chelsea Publishing Company: Chelsea Publishing Company The Bronx, NY · Zbl 0208.10701
[71] Corduneanu, C., Integral equations and applications (2008), Cambridge University Press: Cambridge University Press New York · Zbl 1156.45001
[72] Li, T. Y.; Yorke, J. A., Period three implies chaos, Am Math Mon, 82, 985-992 (1975) · Zbl 0351.92021
[73] Čiklová, M., Li-Yorke sensitive minimal maps, Nonlinearity, 19, 517-529 (2006) · Zbl 1134.37307
[74] Kloeden, P.; Li, Z., Li-Yorke chaos in higher dimensions: a review, J Differ Equat Appl, 12, 247-269 (2006) · Zbl 1096.39019
[75] Akin, E.; Kolyada, S., Li-Yorke sensitivity, Nonlinearity, 16, 1421-1433 (2003) · Zbl 1045.37004
[76] Palmer, K., Shadowing in dynamical systems (2000), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, The Netherlands · Zbl 0997.37001
[77] Lerman, L. M.; Shil’nikov, L. P., Homoclinical structures in nonautonomous systems: nonautonomous chaos, Chaos, 2, 447-454 (1992) · Zbl 1055.37552
[78] Scheurle, J., Chaotic solutions of systems with almost periodic forcing, J Appl Math Phys (ZAMP), 37, 12-26 (1986) · Zbl 0616.58030
[79] Meyer, K. R.; Sell, G. R., Homoclinic orbits and Bernoulli bundles in almost periodic systems, Can Math Soc Conf Proc, 8, 527 (1987)
[80] Palmer, K. R.; Stoffer, D., Chaos in almost periodic systems, J Appl Math Phys (ZAMP), 40, 592-602 (1989) · Zbl 0698.58043
[81] Zadiraka, K. V., Investigation of irregularly perturbed differential equations, (Questions of the theory and history of differential equations (1968), Nauk. Dumka: Nauk. Dumka Kiev), 81-108, in Russian · Zbl 0248.34059
[82] Barbashin, E. A., Introduction to the theory of stability (1970), Wolters-Noordhoff Publishing: Wolters-Noordhoff Publishing Groningen · Zbl 0198.19703
[83] Alligood, K. T.; Sauer, T. D.; Yorke, J. A., Chaos: an introduction to dynamical systems (1996), Springer-Verlag: Springer-Verlag New York · Zbl 0867.58043
[84] Sander, E.; Yorke, J. A., Period-doubling cascades galore, Ergod Theory Dyn Syst, 31, 1249-1267 (2011) · Zbl 1223.37065
[85] Feigenbaum, M. J., Universal behavior in nonlinear systems, Los Alamos Sci/Summer, 4-27 (1980)
[86] Schuster, H. G.; Just, W., Deterministic chaos, an introduction (2005), Wiley-Vch: Wiley-Vch Federal Republic of Germany · Zbl 1094.37001
[87] (Zelinka, I.; Celikovsky, S.; Richter, H.; Chen, G., Evolutionary algorithms and chaotic systems (2010), Springer-Verlag: Springer-Verlag Berlin, Heidelberg) · Zbl 1186.68010
[88] Sato, S.; Sano, M.; Sawada, Y., Universal scaling property in bifurcation structure of Duffing’s and of generalized Duffing’s equations, Phys Rev A, 28, 1654-1658 (1983)
[89] Horn, R. A.; Johnson, C. R., Matrix analysis (1992), Cambridge University Press: Cambridge University Press United States of America
[90] Pyragas, K., Continuous control of chaos by self-controlling feedback, Phys Rev A, 170, 421-428 (1992)
[91] Fradkov, A. L., Cybernetical physics (2007), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 1117.81002
[92] Arneodo, A.; Coullet, P.; Tresser, C., Oscillators with chaotic behavior an illustration of a theorem by Shil’nikov, J Statist Phys, 27, 171-182 (1982) · Zbl 0522.58033
[93] Shilnikov, L. P., A case of the existence of a denumerable set of periodic motions, Sov Math Dokl, 6, 163-166 (1965)
[94] Anishchenko, V. S.; Kapitaniak, T.; Safonova, M. A.; Sosnovzeva, O. V., Birth of double-double scroll attractor in coupled Chua circuits, Phys Lett A, 192, 207-214 (1994) · Zbl 0961.34501
[95] Chua, L. O.; Komuro, M.; Matsumoto, T., The double scroll family. Parts I and II, IEEE Trans Circuit Syst, CAS-33, 1072-1118 (1986) · Zbl 0634.58015
[96] Matsumoto, T.; Chua, L. O.; Komuro, M., The double scroll, IEEE Trans Circuit Syst, CAS-32, 797-818 (1985) · Zbl 0578.94023
[97] Chua, L. O.; Wu, C. W.; Huang, A.; Zhong, G., A universal circuit for studying and generating chaos. Part I: Routes to chaos, IEEE Trans Circuits Syst I Fund Theory Appl, 40, 732-744 (1993) · Zbl 0844.58052
[98] Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys Rev Lett, 64, 1196-1199 (1990) · Zbl 0964.37501
[99] Schuster, H. G., Handbook of chaos control (1999), Wiley-Vch: Wiley-Vch Weinheim · Zbl 0997.93500
[100] Dowell, E. H.; Pazeshki, C., On the understanding of chaos in Duffing’s equation including a comparison with experiment, J Appl Mech, 53, 5-9 (1986) · Zbl 0592.73053
[101] Shilnikov, L., Bifurcations and strange attractors, Proceedings of the international congress of mathematicians, vol. III (2002), Higher Ed. Press: Higher Ed. Press Beijing, pp. 349-372 · Zbl 1055.37026
[102] Rössler, O. E., An equation for continuous chaos, Phys Lett, 57A, 397-398 (1976) · Zbl 1371.37062
[103] (Von Neumann, J.; Burks, A. W., The theory of self-reproducing automata (1966), University of Illinois Press: University of Illinois Press Urbana and London)
[104] Thom, R., Stabilité structurelle et morphogén \(\overset{`}{e}\) se (1972), W.A. Benjamin: W.A. Benjamin New York
[105] Courtat, T.; Gloaguen, C.; Douady, S., Mathematics and morphogenesis of cities: a geometrical approach, Phys Rev E, 83, 036106, 1-12 (2011)
[106] Roudavski, S., Towards morphogenesis in architecture, Int J Archit Comput, 7, 345-374 (2009)
[107] Taber, L. A., Towards a unified theory for morphomechanics, Philos Trans R Soc A, 367, 3555-3583 (2009) · Zbl 1185.74067
[108] Bourgine, P.; Lesne, A., Morphogenesis: origins of patterns and shapes (2011), Springer-Verlag: Springer-Verlag Berlin, Heidelberg
[109] Hagége, C., The language builder: an essay on the human signature in linguistic morphogenesis (1993), John Benjamins Publishing Co.: John Benjamins Publishing Co. Amsterdam, The Netherlands
[110] Archer, M. S., Realistic social theory: the morphogenetic approach (1995), Cambridge University Press: Cambridge University Press Cambridge
[111] Buckley, W., Sociology and modern systems theory (1967), Prentice Hall: Prentice Hall New Jersey
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