Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com


Chaos Generation in Hyperbolic Systems

Discontinuity, Nonlinearity, and Complexity 1(4) (2012) 367--386 | DOI:10.5890/DNC.2012.10.001

M.U. Akhme; M.O. Fen

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

Download Full Text PDF

 

Abstract

In the present paper, we consider extension of chaos in hyperbolic systems with arbitrary large dimensions. Our investigations comprise chaos in the sense of both Devaney and Li-Yorke. We provide a mechanism for unidirectionally coupled systems through the insertion of chaos from one system to another, where the latter is initially nonchaotic. In our procedure for the chaos extension, we take advantage of chaotic sets of functions to provide mathematically approved results. The theoretical results are supported through the simulations for the extension of chaos generated by a Duffing’s oscillator. A control procedure for the extended chaos is demonstrated numerically in the paper.

References

  1. [1]  Akhmet, M.U. and Fen, M.O., Morphogenesis of chaos, arXiv:1205.1166v1[nlin.CD], (submitted).
  2. [2]  Akhmet, M.U. and Fen, M.O., Entrainment of chaos, arXiv:1209.1765v1[nlin.CD], (submitted).
  3. [3]  Akhmet, M.U. (2009), Devaney's chaos of a relay system, Commun. Nonlinear Sci. Numer. Simulat., 14, 1486- 1493.
  4. [4]  Akhmet, M.U. (2009), Li-Yorke chaos in the impact system, J. Math. Anal. Appl., 351, 804-810.
  5. [5]  Akhmet, M.U. (2009), Dynamical synthesis of quasi-minimal sets, Int. J. Bifur. Chaos, 19, 2423-2427.
  6. [6]  Akhmet, M.U. and Fen, M.O. (2012), Chaotic period-Doubling and OGY control for the forced Duffing equation, Commun. Nonlinear Sci. Numer. Simulat., 17, 1929-1946.
  7. [7]  Lorenz, E.N. (1963), Deterministic nonperiodic flow, J. Atmos. Sci, 20, 130-141.
  8. [8]  Rössler, O.E. (1976), An equation for continuous chaos, Phys. Lett., 57A, 397-398.
  9. [9]  Chua, L.O., Wu, C.W., Huang, A. and Zhong, G. (1993), A universal circuit for studying and generating chaos- Part I: Routes to chaos, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 40, 732-744.
  10. [10]  Chua, L.O., Komuro, M., and Matsumoto, T. (1986), The double scroll family, parts I and II, IEEE Trans. Circuit Syst., CAS-33, 1072-1118.
  11. [11]  Cartwright, M. and J. Littlewood (1945), On nonlinear differential equations of the second order I: The equation y-k(1-y2)'y+y = bkcos(λt+a), k large, J. London Math. Soc., 20, 180-189.
  12. [12]  N. Levinson (1949), A second order differential equation with singular solutions, Ann. of Math., 50, 127-153.
  13. [13]  Levi,M. (1981), Qualitative Analysis of the Periodically Forced RelaxationOscillations,Memoirs of the American Mathematical Society, United States of America.
  14. [14]  Thompson, J.M.T. and Stewart, H.B. (2002), Nonlinear Dynamics And Chaos, JohnWiley, West Sussex.
  15. [15]  Moon, F.C. (2004), Chaotic Vibrations: An Introduction For Applied Scientists and Engineers, JohnWiley & Sons, Hoboken & New Jersey.
  16. [16]  Luo, A.C.J. (2012), Regularity and Complexity in Dynamical Systems, Springer, New York.
  17. [17]  Devaney, R. (1987), An Introduction to Chaotic Dynamical Systems, Addison-Wesley, United States of America.
  18. [18]  Li, T.Y. and Yorke, J.A. (1975), Period three implies chaos, The American Mathematical Monthly, 82, 985-992.
  19. [19]  Čiklová, M. (2006), Li-Yorke sensitive minimal maps, Nonlinearity, 19, 517-529.
  20. [20]  Kloeden, P. and Li, Z. (2006), Li-Yorke chaos in higher dimensions: a review, Journal of Difference Equations and Applications, 12, 247-269.
  21. [21]  Akin, E. and Kolyada, S. (2003), Li-Yorke sensitivity, Nonlinearity, 16, 1421-1433.
  22. [22]  Palmer, K. (2000), Shadowing in Dynamical Systems, Kluwer Academic Publishers, Dordrecht, The Netherlands.
  23. [23]  Hale, J.K. (1980), Ordinary Differential Equations, Krieger Publishing Company, Malabar, Florida.
  24. [24]  Pyragas, K. (1992), Continuous control of chaos by self-controlling feedback, Phys. Rev. A, 170, 421-428.
  25. [25]  Gonzales-Miranda, J.M. (2004), Synchronization and Control of Chaos, Imperial College Press, London.
  26. [26]  Schöll, E. and Schuster, H.G. (2008), Handbook of Chaos Control, Wiley-Vch,Weinheim.
  27. [27]  Fradkov, A.L. (2007), Cybernetical Physics, Springer-Verlag, Berlin & Heidelberg.