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Chaos Extension in Hyperbolic Systems

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Replication of Chaos in Neural Networks, Economics and Physics

Part of the book series: Nonlinear Physical Science ((NPS))

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Abstract

This chapter is devoted to the investigation of chaos in the dynamics of chaotically perturbed hyperbolic systems. Extension of chaos in the sense of Devaney and Li–Yorke is taken into account for unidirectionally coupled systems. The rigorously proved results are supported by simulations. A method for controlling the extended chaos is also presented.

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References

  1. M.U. Akhmet, Creating a chaos in a system with relay. Int. J. Qual. Theory Differ. Equ. Appl. 3, 3–7 (2009)

    MATH  Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. 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  ADS  MATH  Google Scholar 

  5. M.U. Akhmet, M.O. Fen, The period-doubling route to chaos in the relay system, in Proceedings of Dynamic Systems and Applications, vol. 6, ed. by G.S. Ladde, N.G. Medhin, C. Peng, M. Sambandham (Dynamic Publisher Inc., Atlanta, 2012), pp. 22–26

    Google Scholar 

  6. M.U. Akhmet, M.O. Fen, Chaos generation in hyperbolic systems. Interdiscip. J. Discontin. Nonlinearity Complex. 1, 367–386 (2012)

    Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  10. M.U. Akhmet, The complex dynamics of the cardiovascular system. Nonlinear Anal.: Theory Methods Appl. 71, e1922–e1931 (2009)

    Article  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  12. O.E. Rössler, An equation for continuous chaos. Phys. Lett. A 57, 397–398 (1976)

    Article  ADS  Google Scholar 

  13. L.O. Chua, M. Komuro, T. Matsumoto, The double scroll family, parts I and II. IEEE Trans. Circuit Syst. CAS–33, 1072–1118 (1986)

    Article  MathSciNet  ADS  Google Scholar 

  14. L.O. Chua, C.W. Wu, A. Huang, G. Zhong, A universal circuit for studying and generating chaos-part I: routes to chaos. IEEE Trans. Circuits Syst.-I: Fundam. Theory Appl. 40, 732–744 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Cartwright, J. Littlewood, On nonlinear differential equations of the second order I: the equation \(\ddot{y}- k(1 - y^2)^{\prime }y + y = bk cos(\lambda t + a)\), \(k\) large. J. Lond. Math. Soc. 20, 180–189 (1945)

    Google Scholar 

  16. M. Levi, Qualitative Analysis of the Periodically Forced Relaxation Oscillations (Memoirs of the American Mathematical Society, Providence, 1981)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. A.C.J. Luo, Regularity and Complexity in Dynamical Systems (Springer, New York, 2012)

    Book  MATH  Google Scholar 

  19. F.C. Moon, Chaotic Vibrations: An Introduction for Applied Scientists and Engineers (Wiley, Hoboken, 2004)

    Book  Google Scholar 

  20. J.M.T. Thompson, H.B. Stewart, Nonlinear Dynamics and Chaos (Wiley, Chichester, 2002)

    Google Scholar 

  21. R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, New York, 1992)

    Google Scholar 

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

    Article  MathSciNet  ADS  MATH  Google Scholar 

  23. M. C̆iklová, Li–Yorke sensitive minimal maps. Nonlinearity 19, 517–529 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  24. R.L. Devaney, An Introduction to Chaotic Dynamical Systems (Addison, Menlo Park, 1989)

    MATH  Google Scholar 

  25. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. K. Palmer, Shadowing in Dynamical Systems: Theory and Applications (Kluwer Academic Publishers, Dordrecht, 2000)

    Book  Google Scholar 

  28. J.K. Hale, Ordinary Differential Equations (Krieger Publishing Company, Malabar, 1980)

    MATH  Google Scholar 

  29. K. Pyragas, Continuous control of chaos by self-controlling feedback. Phys. Rev. A 170, 421–428 (1992)

    Google Scholar 

  30. A.L. Fradkov, Cybernetical Physics: From Control of Chaos to Quantum Control (Springer, Berlin, 2007)

    Google Scholar 

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

    Book  Google Scholar 

  32. E. Schöll, H.G. Schuster, Handbook of Chaos Control (Wiley, Weinheim, 2008)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Middle East Technical University, 06800, Ankara, Turkey

    Marat Akhmet

  2. Neuroscience Institute, Georgia State University, 30303, Atlanta, GA, USA

    Mehmet Onur Fen

Authors
  1. Marat Akhmet
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  2. Mehmet Onur Fen
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Correspondence to Marat Akhmet .

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© 2016 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg

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Akhmet, M., Fen, M.O. (2016). Chaos Extension in Hyperbolic Systems. In: Replication of Chaos in Neural Networks, Economics and Physics. Nonlinear Physical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47500-3_3

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  • DOI: https://doi.org/10.1007/978-3-662-47500-3_3

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-47499-0

  • Online ISBN: 978-3-662-47500-3

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

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