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Regular and Chaotic Dynamics

, Volume 23, Issue 2, pp 161–177 | Cite as

Recent Results on the Dynamics of Higher-dimensional Hénon Maps

  • Stavros Anastassiou
  • Anastasios Bountis
  • Arnd Bäcker
Article
  • 31 Downloads

Abstract

We investigate different aspects of chaotic dynamics in Hénon maps of dimension higher than 2. First, we review recent results on the existence of homoclinic points in 2-d and 4-d such maps, by demonstrating how they can be located with great accuracy using the parametrization method. Then we turn our attention to perturbations of Hénon maps by an angle variable that are defined on the solid torus, and prove the existence of uniformly hyperbolic solenoid attractors for an open set of parameters.We thus argue that higher-dimensional Hénon maps exhibit a rich variety of chaotic behavior that deserves to be further studied in a systematic way.

Keywords

invariant manifolds parametrization method solenoid attractor hyperbolic sets 

Keywords

37D05 37D10 37D20 37D45 

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References

  1. 1.
    Friedland, Sh. and Milnor, J., Dynamical Properties of Plane Polynomial Automorphisms, Ergodic Theory Dynam. Systems, 1989, vol. 9, no. 1, pp. 67–99.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dullin, H.R. and Meiss, J. D., Generalized Hénon Maps: The Cubic Diffeomorphisms of the Plane. Bifurcations, Patterns and Symmetry, Phys. D, 2000, vol. 143, nos. 1–4, pp. 262–289.zbMATHGoogle Scholar
  3. 3.
    Gonchenko, V. S., Kuznetsov, Yu. A., and Meijer, H.G.E., Generalized Hénon Map and Bifurcations of Homoclinic Tangencies, SIAM J. Appl. Dyn. Syst., 2005, vol. 4, no. 2, pp. 407–436.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Zhang, X., Hyperbolic Invariant Sets of the Real Generalized Hénon Maps, Chaos Solitons Fractals, 2010, vol. 43, nos. 1–12, pp. 31–41.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Anastassiou, S., Bountis, T., and Bäcker, A., Homoclinic Points of 2D and 4D Maps via the Parametrization Method, Nonlinearity, 2017, vol. 30, no. 10, pp. 3799–3820.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Shilnikov, L.P., Shilnikov, A. L., and Turaev, D. V., Showcase of Blue Sky Catastrophes, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2014, vol. 24, no. 8, 1440003, 10 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bountis, T., Capel, H.W., Kollmann, M., Ross, J.C., Bergamin, J.M., and van der Weele, J.P., Multibreathers and Homoclinic Orbits in 1-Dimensional Nonlinear Lattices, Phys. Lett. A, 2000, vol. 268, nos. 1–2, pp. 50–60.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bergamin, J.M., Bountis, T., and Jung, C., A Method for Locating Symmetric Homoclinic Orbits Using Symbolic Dynamics, J. Phys. A, 2000, vol. 33, no. 45, pp. 8059–8070.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bergamin, J.M., Bountis, T., and Vrahatis, M.N., Homoclinic Orbits of Invertible Maps, Nonlinearity, 2002, vol. 15, no. 5, pp. 1603–1619.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hirsch, M.W., Pugh, C.C., and Shub, M., Invariant Manifolds, Lecture Notes in Math., vol. 583, New York: Springer, 1977.Google Scholar
  11. 11.
    Cabré, X., Fontich, E., and de la Llave, R., The Parameterization Method for Invariant Manifolds: 3. Overview and Applications, J. Differential Equations, 2005, vol. 218, no. 2, pp. 444–515.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Mireles James, J.D. and Lomelí, H., Computation of Heteroclinic Arcs with Application to the Volume Preserving Hénon Family, SIAM J. Appl. Dyn. Syst., 2010, vol. 9, no. 3, pp. 919–953.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Mireles James, J.D. and Mischaikow, K., Rigorous a Posteriori Computation of (Un)Stable Manifolds and Connecting Orbits for Analytic Maps, SIAM J. Appl. Dyn. Syst., 2013, vol. 12, no. 2, pp. 957–1006.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mireles James, J.D., Quadratic Volume-Preserving Maps: (Un)Stable Manifolds, Hyperbolic Dynamics, and Vortex-Bubble Bifurcations, J. Nonlinear Sci., 2013, vol. 23, no. 4, pp. 585–615.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mireles James, J.D., Polynomial Approximation of One Parameter Families of (Un)Stable Manifolds with Rigorous Computer Assisted Error Bounds, Indag. Math. (N. S.), 2015, vol. 26, no. 1, pp. 225–265.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Capiński, M. J. and Mireles James, J.D., Validated Computation of Heteroclinic Sets, SIAM J. Appl. Dyn. Syst., 2017, vol. 16, no. 1, pp. 375–409.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Haro, À., Canadell, M., Figueras, J.-L., Luque, A., and Mondelo, J.M., The Parameterization Method for Invariant Manifolds, Appl. Math. Sci., vol. 195, Sham: Springer, 2016.CrossRefzbMATHGoogle Scholar
  18. 18.
    Robinson, C., Dynamical Systems: Stability, Symbolic Dynamics and Chaos, 2nd ed., Stud. Adv. Math., vol. 28, Boca Raton, Fla.: CRC, 1998.zbMATHGoogle Scholar
  19. 19.
    Katok, A. and Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems, Encyclopedia Math. Appl., vol. 54, Cambridge: Cambridge Univ. Press, 1995.CrossRefzbMATHGoogle Scholar
  20. 20.
    Mac Lane, S., Categories for the Working Mathematician, 2nd ed., Grad. Texts in Math., vol. 5, New York: Springer, 1998.zbMATHGoogle Scholar
  21. 21.
    Williams, R. F., Expanding Attractors, Inst. Hautes Études Sci. Publ. Math., 1974, no. 43, pp. 169–203.MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gonchenko, V. S., Kuznetsov, Yu. A., and Meijer, H.G.E., Generalized Hénon Map and Bifurcations of Homoclinic Tangencies, SIAM J. Appl. Dyn. Syst., 2005, vol. 4, no. 2, pp. 407–436.MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Burns, K. and Wilkinson, A., Stable Ergodicity of Skew Products, Ann. Sci. École Norm. Sup. (4), 1999, vol. 32, no. 6, pp. 859–889.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Jalnine, A.Yu., Kuznetsov, S.P., and Osbaldestin, A. H., Dynamics of Small Perturbations of Orbits on a Torus in a Quasiperiodically Forced 2D Dissipative Map, Regul. Chaotic Dyn., 2006, vol. 11, no. 1, pp. 19–30.MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Broer, H.W., Simó, C., and Vitolo, R., Chaos and Quasi-Periodicity in Diffeomorphisms of the Solid Torus, Discrete Contin. Dyn. Syst. Ser. B, 2010, vol. 14, no. 3, pp. 871–905.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sosnovtseva, O., Feudel, U., Kurths, J., and Pikovsky, A., Multiband Strange Nonchaotic Attractors in Quasiperiodically Forced Systems, Phys. Lett. A, 1996, vol. 218, nos. 3–6, pp. 255–267.CrossRefGoogle Scholar
  27. 27.
    Gorodetski, A. S. and Ilyashenko, Yu. S., Some Properties of Skew Products over a Horseshoe and a Solenoid, Proc. Steklov Inst. Math., 2000, no. 4(231), pp. 90–112; see also: Tr. Mat. Inst. Steklova, 2000, vol. 231, pp. 96–118.MathSciNetzbMATHGoogle Scholar
  28. 28.
    Kuznetsov, S.P. and Seleznev, E.P., Strange Attractor of Smale–Williams Type in the Chaotic Dynamics of a Physical System, J. Exp. Theor. Phys., 2006, vol. 102, no. 2, pp. 355–364; see also: Zh. Èksper. Teoret. Fiz., 2006, vol. 129, no. 2, pp. 400–412.MathSciNetCrossRefGoogle Scholar
  29. 29.
    Kuznetsov, S.P. and Sataev, I.R., Hyperbolic Attractor in a System of Coupled Non-Autonomous van derPol Oscillators: Numerical Test for Expanding and Contracting Cones, Phys. Lett. A, 2007, vol. 365, nos. 1–2, pp. 97–104.CrossRefzbMATHGoogle Scholar
  30. 30.
    Wilczak, D., Uniformly Hyperbolic Attractor of the Smale–Williams Type for a Poincaré Map in the Kuznetsov System: With Online Multimedia Enhancements, SIAM J. Appl. Dyn. Syst., 2010, vol. 9, no. 4, pp. 1263–1283.MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kuznetsov, S.P., Hyperbolic Chaos: A Physicist’s View, Berlin: Springer, 2012.CrossRefzbMATHGoogle Scholar
  32. 32.
    Isaeva, O.B., Kuznetsov, S.P., and Sataev, I.R., A “Saddle–Node” Bifurcation Scenario for Birth or Destruction of a Smale–Williams Solenoid, Chaos, 2012, vol. 22, no. 4, 043111, 7 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kruglov, V.P., Kuznetsov, S.P., and Pikovsky, A., Attractor of Smale–Williams Type in an Autonomous Distributed System, Regul. Chaotic Dyn., 2014, vol. 19, no. 4, pp. 483–494.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  • Stavros Anastassiou
    • 1
  • Anastasios Bountis
    • 2
  • Arnd Bäcker
    • 3
    • 4
  1. 1.Center of Research and Applications of Nonlinear Systems (CRANS), University of PatrasDepartment of MathematicsRionGreece
  2. 2.Department of Mathematics, School of Science and TechnologyNazarbayev UniversityAstanaKazakhstan
  3. 3.Technische Universität DresdenInstitut für Theoretische Physik and Center for DynamicsDresdenGermany
  4. 4.Max-Planck-Institut für Physik komplexer SystemeDresdenGermany

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