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Introduction

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Dynamics with Chaos and Fractals

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 29))

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Abstract

In this chapter, the short outline of the conceptual content of the book is provided. Chaos and fractals as well as their connections and interrelations in terms of unpredictability, replication of chaos, fractals geometry, dynamics through fractal mappings, and abstract self-similarity are carefully described. Moreover, chaotic dynamics for fractals and applications on hybrid systems on a time scale, economic models, and weather and ocean dynamics are mentioned. Sources and consequences are discussed.

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References

  1. M.U. Akhmet, Hyperbolic sets of impact systems, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 15 (Suppl. S1), 1–2, in Proceedings of the 5th International Conference on Impulsive and Hybrid Dynamical Systems and Applications (Watan Press, Beijing, 2008)

    Google Scholar 

  2. M.U. Akhmet, Dynamical synthesis of quasi-minimal sets. Int. J. Bifurcat. Chaos 19, 2423–2427 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. M.U. Akhmet, Shadowing and dynamical synthesis. Int. J. Bifurcat. Chaos 19, 3339–3346 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. M.U. Akhmet, Devaney’s chaos of a relay system. Commun. Nonlinear Sci. Numer. Simulat. 14, 1486–1493 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. M.U. Akhmet, Li-Yorke chaos in the system with impacts. J. Math. Anal. Appl. 351, 804–810 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. M.U. Akhmet, Creating a chaos in a system with relay. Int. J. Qualit. Th. Diff. Eqs. Appl. 3, 3–7 (2009)

    MATH  Google Scholar 

  7. M.U. Akhmet, The complex dynamics of the cardiovascular system. Nonlinear Analysis 71, e1922–e1931 (2009)

    Article  MATH  Google Scholar 

  8. M.U. Akhmet, Homoclinical structure of the chaotic attractor. Commun. Nonlinear Sci. Numer. Simulat. 15, 819–822 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. M.U. Akhmet, Principles of Discontinuous Dynamical Systems (Springer, New York, 2010)

    Book  MATH  Google Scholar 

  10. M. Akhmet, E.M. Alejaily, Abstract Similarity, Fractals and Chaos. ArXiv e-prints, arXiv:1905.02198, 2019 (submitted)

    Google Scholar 

  11. M. Akhmet, E.M. Alejaily, Domain-structured chaos in a Hopfield neural network. Int. J. Bifurc. Chaos, 2019 (in press)

    Google Scholar 

  12. M. Akhmet, E.M. Alejaily, Chaos on the Multi-Dimensional Cube. ArXiv e-prints, arXiv:1908.11194, 2019 (submitted)

    Google Scholar 

  13. M.U. Akhmet, M.O. Fen, Chaotic period-doubling and OGY control for the forced Duffing equation. Commun. Nonlinear Sci. Numer. Simul. 17, 1929–1946 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. M.U. Akhmet, M.O. Fen, Replication of chaos. Commun. Nonlinear Sci. Numer. Simul. 18, 2626–2666 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. M.U. Akhmet, M.O. Fen, Chaos generation in hyperbolic systems. Discontinuity Nonlinearity Complexity 1, 353–365 (2012)

    Article  MATH  Google Scholar 

  16. M.U. Akhmet, M.O. Fen, Shunting inhibitory cellular neural networks with chaotic external inputs. Chaos 23, 023112 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Akhmet, M.O. Fen, Chaotification of impulsive systems by perturbations. Int. J. Bifurcat. Chaos 24, 1450078 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. M.U. Akhmet, M.O. Fen, Replication of discrete chaos. Chaotic Model. Simul. (CMSIM) 2, 129–140 (2014)

    Google Scholar 

  19. M. Akhmet, M.O. Fen, Attraction of Li-Yorke chaos by retarded SICNNs. Neurocomputing 147, 330–342 (2015)

    Article  Google Scholar 

  20. M. Akhmet, M.O. Fen, A. Kıvılcım, Li-Yorke chaos generation by SICNNs with chaotic/almost periodic postsynaptic currents. Neurocomputing 173, 580–594 (2016)

    Article  Google Scholar 

  21. M. Akhmet, M.O. Fen, Replication of Chaos in Neural Networks, Economics and Physics (Higher Education Press, Beijing; Springer, Heidelberg, 2016)

    Book  MATH  Google Scholar 

  22. M. Akhmet, M.O. Fen, Input-output mechanism of the discrete chaos extension, in Complex Motions and Chaos in Nonlinear Systems, ed. by V. Afraimovich, J.A.T. Machado, J. Zhang (Springer, Switzerland, 2016), pp. 203–233

    Chapter  Google Scholar 

  23. M. Akhmet, M.O. Fen, E.M. Alejaily, Dynamics with fractals. Discontinuity Nonlinearity Complexity (in press)

    Google Scholar 

  24. M. Akhmet, M.O. Fen, E.M. Alejaily, Mapping Fatou-Julia Iterations. Proc. ICIME 2018, 64–67 (2018)

    Google Scholar 

  25. M. Akhmet, M.O. Fen, E.M. Alejaily, Extension of sea surface temperature unpredictability. Ocean Dynamics 69, 145–156 (2019)

    Article  Google Scholar 

  26. M. Akhmet, M.O. Fen, E.M. Alejaily, Generation of fractals as Duffing equation orbits. Chaos 29, 053113 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  27. E. Akin, S. Kolyada, Li-Yorke sensitivity. Nonlinearity 16, 1421–1433 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. K.G. Andersson, Poincaré’s discovery of homoclinic points. Arch. Hist. Exact Sci. 48, 133–147 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. B.-L. Hao, W.-M. Zheng, Applied Symbolic Dynamics and Chaos (World Scientific Publishing Company, 1998)

    Book  Google Scholar 

  30. C. Bandt, S. Graf, Self-similar sets 7. A characterization of self-similar fractals with positive Hausdorff measure. Proc. Am. Math. Soc. 114, 995–1001 (1992)

    MATH  Google Scholar 

  31. L. Barabasi, H.E. Stanley, Fractal Concepts in Surface Growth (Cambridge University Press, Cambridge, 1995)

    Book  MATH  Google Scholar 

  32. M.F. Barnsley, Fractals Everywhere (Academic Press, London, 1988)

    MATH  Google Scholar 

  33. M. Batty, P.A. Longley, Fractal Cities: A Geometry of Form and Function (Academic Press, London, 1994)

    MATH  Google Scholar 

  34. A.L. Bertozzi, Heteroclinic orbits and chaotic dynamics in planar fluid flows. SIAM J. Math. Anal. 19, 1271–1294 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  35. G.D. Birkhoff, Dynamical Systems, vol. 9 (Amer. Math. Soc., Colloquium Publications, Providence, 1927)

    MATH  Google Scholar 

  36. F. Blanchard, E. Glasner, S. Kolyada, A. Maass, On Li-Yorke pairs. J. Reine Angew. Math. 2002, 51–68 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Cartwright, J. Littlewood, On nonlinear differential equations of the second order I: The equation \(\ddot {y}- k(1 - y^2)'y + y = bk cos(\lambda t + a),\)k large. J. Lond. Math. Soc. 20, 180–189 (1945)

    Google Scholar 

  38. R. Chacon, J.D. Bejarano, Homoclinic and heteroclinic chaos in a triple-well oscillator. J. Sound Vib. 186, 269–278 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  39. G. Chen, Y. Huang, Chaotic Maps: Dynamics, Fractals and Rapid Fluctuations, Synthesis Lectures on Mathematics and Statistics (Morgan and Claypool Publishers, Texas, 2011)

    Google Scholar 

  40. C. Corduneanu, Almost Periodic Functions (Interscience Publishers, New York, London, Sydney, 1968)

    MATH  Google Scholar 

  41. R.M. Crownover, Introduction to Fractals and Chaos (Jones and Bartlett, Boston, MA, 1995)

    Google Scholar 

  42. R.L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison-Wesley, USA, 1989)

    MATH  Google Scholar 

  43. P. Diamond, Chaotic behavior of systems of difference equations. Int. J. Syst. Sci. 7, 953–956 (1976)

    Article  MATH  Google Scholar 

  44. A. Dohtani, Occurrence of chaos in higher dimensional discrete time systems. SIAM J. Appl. Math. 52, 1707–1721 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  45. G.A. Edgar, Measure, Topology, and Fractal Geometry (Springer, New York, 1990)

    Book  MATH  Google Scholar 

  46. C. Ercai, Chaos for the Sierpinski carpet. J. Stat. Phys. 88, 979–984 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  47. K.J. Falconer, Sub-self-similar sets. Trans. Amer. Math. Soc. 347, 3121–3129 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  48. K. J. Falconer, The Geometry of Fractal Sets (Cambridge Univ. Press, Cambridge, 1985)

    Book  MATH  Google Scholar 

  49. P. Fatou, Sur les équations fonctionnelles, I, II, III. Bull. Soc. Math. France 47, 161–271 (1919); 48, 33–94 (1920); 48, 208–314 (1920)

    Google Scholar 

  50. M.O. Fen, Persistence of chaos in coupled Lorenz systems. Chaos Solitons Fractals 95, 200–205 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  51. M.O. Fen, M. Akhmet, Impulsive SICNNs with chaotic postsynaptic currents. Discret. Contin. Dyn. Syst. Ser. B 21, 1119–1148 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  52. M.O. Fen, F. Tokmak Fen, SICNNs with Li-Yorke chaotic outputs on a time scale. Neurocomputing 237, 158–165 (2017)

    Article  Google Scholar 

  53. M.O. Fen, F. Tokmak Fen, Replication of period-doubling route to chaos in impulsive systems. Electron. J. Qual. Theory Differ. Equ. 2019(58), 1–20 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  54. A.M. Fink, Almost Periodic Differential Equations (Springer, New York, 1974)

    Book  MATH  Google Scholar 

  55. G. Franceschetti, D. Riccio, Scattering, Natural Surfaces and Fractals (Academic Press, Burlington, 2007)

    MATH  Google Scholar 

  56. S.V. Gonchenko, L.P. Shil’nikov, D.V. Turaev, Dynamical phenomena in systems with structurally unstable Poincaré homoclinic orbits. Chaos 6, 15–31 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  57. J.M. Gonzáles-Miranda, Synchronization and Control of Chaos (Imperial College Press, London, 2004)

    Book  Google Scholar 

  58. C. Grebogi, J.A. Yorke, The Impact of Chaos on Science and Society (United Nations University Press, Tokyo, 1997)

    Google Scholar 

  59. J. Guckenheimer, P.J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, Heidelberg, Berlin, 1983)

    Book  MATH  Google Scholar 

  60. J.L.G. Guirao, M. Lampart, Li and Yorke chaos with respect to the cardinality of the scrambled sets. Chaos Solitons Fractals 24, 1203–1206 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  61. J. Hadamard, Les surfaces courbures opposes et leurs lignes godsiques. J. Math. Pures et Appl. 4, 27–74 (1898)

    MATH  Google Scholar 

  62. J. Hale, H. Koçak, Dynamics and Bifurcations (Springer, New York, 1991)

    Book  MATH  Google Scholar 

  63. M. Hata, On the structure of self-similar sets. Jpn. J. Appl. Math. 2, 381–414 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  64. J. Hutchinson, Fractals and self-similarity. Indiana Univ. J. Math. 30, 713–747 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  65. A.K. Janahmadov, M. Javadov, Fractal Approach to Tribology of Elastomers (Springer, Switzerland, 2018)

    Google Scholar 

  66. J.A. Kaandorp, Fractal Modelling: Growth and Form in Biology (Springer, New York, 2012)

    Google Scholar 

  67. J. Kennedy, J.A. Yorke, Topological horseshoes. Trans. Am. Math. Soc. 353, 2513–2530 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  68. J. Kigami, Analysis on Fractals (Cambridge Univ. Press, Cambridge, 2001)

    Book  MATH  Google Scholar 

  69. P. Kloeden, Z. Li, Li-Yorke chaos in higher dimensions: a review. J. Differ. Equ. Appl. 12, 247–269 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  70. M. Kuchta, J. Smítal, Two Point Scrambled Set Implies Chaos. European Conference on Iteration Theory (ECIT 87) (World Sci. Publishing, Singapore, 1989), pp. 427–430

    Google Scholar 

  71. V. Lakshmikantham, D. Trigiante, Theory of Difference Equations: Numerical Methods and Applications (Marcel Dekker, USA, 2002)

    Book  MATH  Google Scholar 

  72. K.S. Lau, S.M. Ngai, H. Rao, Iterated function systems with overlaps and the self-similar measures. J. Lond. Math. Soc. 63, 99–115 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  73. N. Levinson, A second order differential equation with singular solutions. Ann. Math. 50, 127–153 (1949)

    Article  MathSciNet  MATH  Google Scholar 

  74. T.Y. Li, J.A. Yorke, Period three implies chaos. Am. Math. Monthly 82, 985–992 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  75. P. Li, Z. Li, W.A. Halang, G. Chen, Li-Yorke chaos in a spatiotemporal chaotic system. Chaos Solitons Fractals 33, 335–341 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  76. S. Libeskind, Euclidean and Transformational Geometry: A Deductive Inquiry (Jones and Bartlett Publishers, Sudbury, MA, 2008)

    MATH  Google Scholar 

  77. E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  78. B.B. Mandelbrot, Les Objets Fractals: Forme, Hasard, et Dimension (Flammarion, Paris, 1975)

    MATH  Google Scholar 

  79. B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, New York, 1983)

    Book  Google Scholar 

  80. B.B. Mandelbrot, Fractals and Chaos: The Mandelbrot Set and Beyond (Springer, New York, 2004)

    Book  MATH  Google Scholar 

  81. F.R. Marotto, Snap-back repellers imply chaos in \(\mathbb R^{n}\). J. Math. Anal. Appl. 63, 199–223 (1978)

    Google Scholar 

  82. E.W. Mitchell, S.R. Murray (eds.), Classification and Application of Fractals: New Research (Nova Science Publishers, New York, 2012)

    Google Scholar 

  83. F.C. Moon, Chaotic and Fractal Dynamics: An Introduction for Applied Scientists and Engineers (Wiley, New York, 1992)

    Book  Google Scholar 

  84. P.A.P. Moran, Additive functions of intervals and Hausdorff measure. Proc. Cambridge Philos. Soc. 42, 15–23 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  85. M. Morse, G.A. Hedlund, Symbolic dynamics. Am. J. Math. 60, 815–866 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  86. S.M. Ngai, Y. Wang, Hausdorff dimension of overlapping self-similar sets. J. Lond. Math. Soc. 63, 655–672 (2001)

    Article  MATH  Google Scholar 

  87. H-O. Peitgen, H. Jürgens, D. Saupe, Chaos and Fractals (Springer, New York, 2004)

    Book  MATH  Google Scholar 

  88. Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications (University of Chicago Press, Chicago, 1997)

    Book  MATH  Google Scholar 

  89. Y. Pesin, H. Weiss, On the dimension of deterministic and random cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture. Comm. Math. Phys. 182, 105–153 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  90. L. Pietronero, E. Tosatti, Fractals in Physics (North-Holland, Amsterdam, 2012)

    MATH  Google Scholar 

  91. H. Poincaré, Sur le probléme des trois corps et les équations de la dynamique. Acta Math. 13, 1–270 (1880)

    MathSciNet  MATH  Google Scholar 

  92. H. Poincaré, Les méthodes nouvelles de la mécanique céleste, Vol. 1, 2 (Gauthier-Villars, Paris, 1892)

    Google Scholar 

  93. H. Poincaré, Les methodes nouvelles de la mecanique celeste, Vol. III, Paris, 1899; reprint (Dover, New York, 1957)

    Google Scholar 

  94. C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (CRC Press, Boca Raton, 1995)

    MATH  Google Scholar 

  95. G.R. Sell, Topological Dynamics and Ordinary Differential Equations (Van Nostrand Reinhold Company, London, 1971)

    MATH  Google Scholar 

  96. Y. Shi, G. Chen, Chaos of discrete dynamical systems in complete metric spaces. Chaos Solitons Fractals 22, 555–571 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  97. Y. Shi, G. Chen, Discrete chaos in Banach spaces. Sci. China Ser. A Math. 48, 222–238 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  98. L.P. Shil’nikov, On a Poincaré-Birkhoff problem. Math. USSR-Sbornik 3, 353–371 (1967)

    Article  MATH  Google Scholar 

  99. L. Shilnikov, Homoclinic chaos, in Nonlinear Dynamics, Chaotic and Complex Systems, ed. by E. Infeld, R. Zelazny, A. Galkowski (Cambridge University Press, Cambridge, 1997), pp. 39–63

    Google Scholar 

  100. S. Smale, Diffeomorphisms with many periodic points, in Differential and Combinatorial Topology, ed. by S.S. Cairns (Princeton University Press, Princeton, 1965), pp. 63–80

    Chapter  Google Scholar 

  101. D.W. Spear, Measure and self-similarity. Adv. Math. 91, 143–157 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  102. S. Stella, On Hausdorff dimension of recurrent net fractals. Proc. Am. Math. Soc. 116, 389–400 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  103. B.Yu. Sternin, V.E. Shatalov, Differential Equations on Complex Manifolds (Kluwer Academic Publishers, Dordrecht, 1994)

    Book  MATH  Google Scholar 

  104. R.S. Strichartz, Differential Equations on Fractals: A Tutorial (Princeton University Press, Princeton, 2006)

    Book  MATH  Google Scholar 

  105. M. Takayasu, H. Takayasu, Fractals and economics, in Complex Systems in Finance and Econometrics, ed. by R.A. Meyers (Springer, New York, 2011), pp. 444–463

    Google Scholar 

  106. Y. Ueda, Random phenomena resulting from non-linearity in the system described by Duffing’s equation. Trans. Inst. Electr. Eng. Jpn. 98A, 167–173 (1978)

    Google Scholar 

  107. G.K. Vallis, El Niño: A chaotic dynamical system? Science 232, 243–245 (1986)

    Article  Google Scholar 

  108. G.K. Vallis, Conceptual models of El Niño and the southern oscillation. J. Geophys. 93, 13979–13991 (1988)

    Article  Google Scholar 

  109. J.L. Véhel, E. Lutton, C. Tricot (Eds.), Fractals in Engineering: From Theory to Industrial Applications (Springer, New York, 1997)

    Google Scholar 

  110. T. Vicsek, Fractal Growth Phenomena, 2nd edn. (World Scientific, Singapore, 1992)

    Book  MATH  Google Scholar 

  111. H.J. Vollrath, The understanding of similarity and shape in classifying tasks. Educ. Stud. Math. 8, 211–224 (1977)

    Article  Google Scholar 

  112. S. Wiggins, Global Bifurcation and Chaos: Analytical Methods (Springer, New York, Berlin, 1988)

    Book  MATH  Google Scholar 

  113. L. Zhao, W. Li, L. Geng, Y. Ma, Artificial neural networks based on fractal growth, in Advances in Automation and Robotics 2, Lecture Notes in Electrical Engineering, vol. 123, ed. by G. Lee (Springer, Berlin, 2011), pp. 323–330

    Google Scholar 

  114. O. Zmeskal, P. Dzik, M. Vesely, Entropy of fractal systems. Comput. Math. Appl. 66, 135–146 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Middle East Technical University, Çankaya, Ankara, Turkey

    Marat Akhmet

  2. Department of Mathematics, TED University, Çankaya, Ankara, Turkey

    Mehmet Onur Fen

  3. Department of Fundamental Sciences, College of Engineering Technology, Houn, Libya

    Ejaily Milad Alejaily

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  1. Marat Akhmet
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  2. Mehmet Onur Fen
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  3. Ejaily Milad Alejaily
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Akhmet, M., Fen, M.O., Alejaily, E.M. (2020). Introduction. In: Dynamics with Chaos and Fractals. Nonlinear Systems and Complexity, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-030-35854-9_1

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