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

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Abstract

This chapter is devoted to functional analytical methods for showing chaos in discrete dynamical systems involving difference equations, diffeomorphisms, regular and singular ODEs with impulses, and inflated mappings as well.

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References

  1. K.R. MEYER & G. R. SELL: Melnikov transforms, Bernoulli bundles, and almost periodic perturbations, Trans. Amer. Math. Soc. 314 (1989), 63–105.

    MathSciNet  MATH  Google Scholar 

  2. D. STOFFER: Transversal homoclinic points and hyperbolic sets for non-autonomous maps I, II, Zeit. Ang. Math. Phys. (ZAMP) 39 (1988), 518–549, 783–812.

    Article  MathSciNet  MATH  Google Scholar 

  3. S. WIGGINS: Chaotic Transport in Dynamical Systems, Springer-Verlag, New York, 1992.

    Book  MATH  Google Scholar 

  4. M.L. GLASSER, V.G. PAPAGEORGIOU & T.C. BOUNTIS: Mel’nikov’s function for two dimensional mappings, SIAM J. Appl. Math. 49 (1989), 692–703.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. DELSHAMS & R. RAMÍREZ-ROS: Poincarè-Melnikov-Arnold method for analytic planar maps, Nonlinearity. 9 (1996), 1–26.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. F. BATTELLI: Perturbing diffeomorphisms which have heteroclinic orbits with semihyperbolic fixed points, Dyn. Sys. Appl. 3 (1994), 305–332.

    MathSciNet  MATH  Google Scholar 

  7. T.C. BOUNTIS, A. GORIELY & M. KOLLMANN: A Mel’nikov vector for Ndimensional mappings, Phys. Lett. A 206 (1995), 38–48.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. M. FEČKAN: On the existence of chaotic behaviour of diffeomorphisms, Appl. Math. 38 (1993), 101–122.

    MathSciNet  MATH  Google Scholar 

  9. F. BATTELLI & C. LAZZARI: On the bifurcation from critical homoclinic orbits in Ndimensional maps, Disc. Cont. Dyn. Syst. 3 (1997), 289–303.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. FEČKAN: Topological Degree Approach to Bifurcation Problems, Springer, Berlin, 2008.

    Book  MATH  Google Scholar 

  11. J. GRUENDLER: Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations, J. Differential Equations 122 (1995), 1–26.

    Article  MathSciNet  MATH  Google Scholar 

  12. M. MEDVEĎ: Fundamentals of Dynamical Systems and Bifurcation Theory, Adam Hilger, Bristol, 1992.

    MATH  Google Scholar 

  13. F. BATTELLI & M. FEČKAN: Chaos arising near a topologically transversal homoclinic set, Top. Meth. Nonl. Anal. 20 (2002), 195–215.

    MATH  Google Scholar 

  14. M.U. AKHMET & O. YILMAZ: Positive solutions of linear impulsive differential equations, Nonlinear Oscillations, 8 (2005), 291–297.

    Article  MathSciNet  MATH  Google Scholar 

  15. M.U. AKHMET & R. SEJILOVA: On the control of a boundary value problem for a system of linear impulsive differential equations with impulse action, Differential Equations 36 (2000), 1512–1520.

    Article  MathSciNet  Google Scholar 

  16. D. BAINOV & P. S. SIMEONOV: Impulsive Differential Equations: Asymptotic Properties of the Solutions, World Scientific Publishing Co., Singapore, 1995.

    MATH  Google Scholar 

  17. A. HALANAY & D. WEXLER: Qualitative Theory of Impulsive Systems, Editura Academiei Republicii Socialiste Romania, Bucharest, 1968.

    Google Scholar 

  18. V. LAKSHMIKANTHAM: Trends in the theory of impulsive differential equations, in “Differential Equations and Applications”, Vols. I,II, pp. 76–87, Ohio Univ. Press, Athens, OH, 1989.

    Google Scholar 

  19. V. LAKSHMIKANTHAM, D. BAINOV & P.S. SIMEONOV: Theory of Impulsive Differential Equations, World Scientific Publishing Co., Singapore, 1989.

    Book  MATH  Google Scholar 

  20. A.M. SAMOILENKO & N.A. PERESTYUK: Impulsive Differential Equations, World Scientific Publishing Co., Singapore, 1995.

    MATH  Google Scholar 

  21. M.U. AKHMET: Li-Yorke chaos in the impact system, J. Math. Anal. Appl. 351 (2009), 804–810.

    Article  MathSciNet  MATH  Google Scholar 

  22. S. CSÖRGÖ & L. HATVANI: Stability properties of solutions of linear second order differential equations with random coefficients, J. Differential Equations. 248 (2010), 21–49.

    Article  MathSciNet  MATH  Google Scholar 

  23. Á. ELBERT: Stability of some difference equations, in “Advances in Difference Equations”, Proc. Second Int. Conf. Difference Eqns., Veszprém, Hungary, August 7–11, 1995, Gordon and Breach Science Publ., London, 1997, 165–187.

    Google Scholar 

  24. Á. ELBERT: On asymptotic stability of some Sturm-Liouville differential equations, General Seminar of Mathematics, Univ. Patras 22–23 (1997), 57–66.

    Google Scholar 

  25. M. FEČKAN: Existence of almost periodic solutions for jumping discontinuous systems, Acta Math. Hungarica. 86 (2000), 291–303.

    Article  MATH  Google Scholar 

  26. J.R. GRAEF & J. KARSAI: On irregular growth and impulses in oscillator equations, in “Advances in Difference Equations”, Proc. Second Int. Conf. Difference Eqns., Veszprém, Hungary, August 7–11, 1995, Gordon and Breach Science Publ., London, 1997, 253–262.

    Google Scholar 

  27. L. HATVANI: On the existence of a small solution to linear second order differential equations with step function coefficients, Dynam. Contin. Discrete Impuls. Systems. 4 (1998), 321–330.

    Article  MathSciNet  MATH  Google Scholar 

  28. L. HATVANI & L. STACHÓ: On small solutions of second order differential equations with random coefficients, Arch. Math. (EQUADIFF 9, Brno, 1997) 34 (1998), 119–126.

    MATH  Google Scholar 

  29. M. FEČKAN: Bifurcations of heteroclinic orbits for diffeomorphisms, Appl. Math. 36 (1991), 355–367.

    MathSciNet  MATH  Google Scholar 

  30. J.K. HALE: Introduction to dynamic bifurcation, in “Bifurcation Theory and Applications”, L. Salvadori, Ed., LNM 1057, Springer-Verlag, 1984, 106–151.

    Google Scholar 

  31. F. BATTELLI & K.J. PALMER: Chaos in the Duffing equation, J. Differential Equations. 101 (1993), 276–301.

    Article  MathSciNet  MATH  Google Scholar 

  32. M. FEČ KAN: Chaos in singularly perturbed impulsive O.D.E., Bollettino U.M.I. 10-B (1996), 175–198.

    Google Scholar 

  33. K.J. PALMER & D. STOFFER: Chaos in almost periodic systems, Zeit. Ang. Math. Phys. (ZAMP) 40 (1989), 592–602.

    Article  MathSciNet  MATH  Google Scholar 

  34. G. ALEFELD & G. MAYER: Interval analysis: theory and applications, J. Comp. Appl. Math. 121 (2000), 421–464.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. L. GRÜNE and P.E. KLOEDEN: Discretization, inflation and perturbation of attractors, in: “Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems”, Springer, Berlin, 2001, 399–416.

    Chapter  Google Scholar 

  36. P.E. KLOEDEN & V.S. KOZYAKIN: The inflation of attractors and their discretization: the autonomous case, Nonl. Anal., Th. Meth. Appl. 40 (2000), 333–343.

    Article  MathSciNet  MATH  Google Scholar 

  37. R.E. MOORE, R.B. KEARFOTT & M.J. CLOUD: Introduction to Interval Analysis, SIAM, Philadelphia, 2009.

    Book  MATH  Google Scholar 

  38. L. GRÜNE: Asymptotic Behaviour of Dynamical and Control Systems under Perturbation and Discretization, Springer, Berlin, 2002.

    Book  Google Scholar 

  39. G. COLOMBO, M. FEČKAN & B.M. GARAY: Multivalued perturbations of a saddle dynamics, Differential Equations & Dynamical Systems. 18 (2010), 29–56.

    Article  MATH  Google Scholar 

  40. G.E. IVANOV & M.V. BALASHOV: Lipschitz parameterizations of multivalued mappings with weakly convex values, Izv. Ross. Akad. Nauk Ser. Mat. 71 (2007), 47–68, (in Russian; translation in Izv. Math. 71 (2007), 1123–1143).

    Article  MathSciNet  Google Scholar 

  41. J.P. AUBIN & A. CELLINA: Differential Inclusions: Set-Valued Maps and Viability Theory, Springer-Verlag, Berlin, 1984.

    Book  MATH  Google Scholar 

  42. M.C. IRWIN: Smooth Dynamical Systems, Academic Press, London, 1980.

    MATH  Google Scholar 

  43. M. FEČKAN: Chaos in nonautonomous differential inclusions, Int. J. Bifur. Chaos. 15 (2005), 1919–1930.

    Article  MATH  Google Scholar 

  44. S. WIGGINS: Chaos in the dynamics generated by sequences of maps, with applications to chaotic advection in flows with aperiodic time dependence, Z. Angew. Math. Phys. (ZAMP) 50 (1999), 585–616.

    Article  MathSciNet  MATH  Google Scholar 

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© 2011 Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg

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Fečkan, M. (2011). Chaos in Discrete Dynamical Systems. In: Bifurcation and Chaos in Discontinuous and Continuous Systems. Nonlinear Physical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18269-3_3

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