Advertisement

Regular and Chaotic Dynamics

, Volume 19, Issue 6, pp 702–717 | Cite as

On bifurcations of area-preserving and nonorientable maps with quadratic homoclinic tangencies

  • Amadeu Delshams
  • Marina Gonchenko
  • Sergey V. Gonchenko
Article

Abstract

We study bifurcations of nonorientable area-preserving maps with quadratic homoclinic tangencies. We study the case when the maps are given on nonorientable two-dimensional surfaces. We consider one- and two-parameter general unfoldings and establish results related to the emergence of elliptic periodic orbits.

Keywords

area-preserving map non-orientable surface elliptic point homoclinic tangency bifurcation 

MSC2010 numbers

37C05 37C29 37E30 37G25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Gavrilov, N.K. and Shilnikov, L.P., On Three-Dimensional Dynamical Systems Close to Systems with a Structurally Unstable Homoclinic Curve: 1, Math. USSR-Sb., 1972, vol. 17, no. 4, pp. 467–485; see also: Mat. Sb. (N. S.), 1972, vol. 88(130), no. 4(8), pp. 475–492. Gavrilov, N.K. and Shilnikov, L.P., On Three-Dimensional Dynamical Systems Close to Systems with a Structurally Unstable Homoclinic Curve: 2, Math. USSR-Sb., 1973, vol. 19, no. 1, pp. 139–156; see also: Mat. Sb. (N. S.), 1973, vol. 90(132), no. 1, pp. 139–156.CrossRefGoogle Scholar
  2. 2.
    Newhouse, Sh.E., Diffeomorphisms with Infinitely Many Sinks, Topology, 1974, vol. 13, pp. 9–18.CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Gonchenko, S. V., Stable Periodic Motions in Systems Close to a Structurally Unstable Homoclinic Curve, Math. Notes, 1983, vol. 33, no. 5, pp. 384–389; see also: Mat. Zametki, 1983, vol. 33, no. 5, pp. 745–756.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Newhouse, Sh.E., Quasi-Elliptic Periodic Points in Conservative Dynamical Systems, Amer. J. Math., 1977, vol. 99, no. 5, pp. 1061–1087.CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Gonchenko, S. V., Shilnikov, L.P., and Turaev, D. V., Elliptic Periodic Orbits near a Homoclinic Tangency in Four-Dimensional Symplectic Maps and Hamiltonian Systems with Three Degrees of Freedom, Regul. Chaotic Dyn., 1998, vol. 3, no. 4, pp. 3–26.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Gonchenko, S. V., Shilnikov, L.P., and Turaev, D. V., Existence of Infinitely Many Elliptic Periodic Orbits in Four-Dimensional Symplectic Maps with a Homoclinic Tangency, Proc. Steklov Inst. Math., 2004, vol. 244, pp. 106–131; see also: Dynamical Systems and Related Problems of Geometry: Collected Papers: Dedicated to the Memory of Academician Andrei Andreevich Bolibrukh, Tr. Mat. Inst. Steklova, vol. 244, Moscow: Nauka, 2004, pp. 115–142.MathSciNetGoogle Scholar
  7. 7.
    Mora, L. and Romero, N., Moser’s Invariant Curves and Homoclinic Bifurcations, Dynam. Systems Appl., 1997, vol. 6, no. 1, pp. 29–41.zbMATHMathSciNetGoogle Scholar
  8. 8.
    Gonchenko, S. V. and Gonchenko, M. S., On Cascades of Elliptic Periodic Points in Two-Dimensional Symplectic Maps with Homoclinic Tangencies, Regul. Chaotic Dyn., 2009, vol. 14, no. 1, pp. 116–136.CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Biragov, V. S., Bifurcations in a Two-Parameter Family of Conservative Mappings That Are Close to the Hénon Mapping, Selecta Math. Soviet., 1990, vol. 9, no. 3, pp. 273–282; see also: Methods of the Qualitative Theory of Differential Equations, Gorki: GGU, 1987, pp. 10–24.zbMATHMathSciNetGoogle Scholar
  10. 10.
    Biragov, V. S. and Shilnikov, L.P., On the Bifurcation of a Saddle-Focus Separatrix Loop in a Three-Dimensional Conservative System, Selecta Math. Soviet., 1992, vol. 11, no. 4, pp. 333–340; see also: Methods of the Qualitative Theory of Differential Equations, Gorki: GGU, 1987, pp. 25–34.MathSciNetGoogle Scholar
  11. 11.
    Gonchenko, S. V. and Shilnikov, L.P., On Two-Dimensional Analytic Area-Preserving Diffeomorphisms with Infinitely Many Stable Elliptic Periodic Points, Regul. Chaotic Dyn., 1997, vol. 2, nos. 3/4, pp. 106–123.zbMATHMathSciNetGoogle Scholar
  12. 12.
    Gonchenko, S. V. and Shilnikov, L.P., On Two-Dimensional Area-Preserving Diffeomorphisms with Infinitely Many Elliptic Islands, J. Statist. Phys., 2000, vol. 101, nos. 1/2, pp. 321–356.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Gonchenko, S. V. and Shilnikov, L.P., On Two-Dimensional Area-Preserving Mappings with Homoclinic Tangencies, Dokl. Math., 2001, vol. 63, no. 3, pp. 395–399; see also: Dokl. Akad. Nauk, 2001, vol. 378, no. 6, pp. 727–732.MathSciNetGoogle Scholar
  14. 14.
    Gonchenko, S. V. and Shilnikov, L.P., On Two-Dimensional Area-Preserving Maps with Homoclinic Tangencies That Have Infinitely Many Generic Elliptic Periodic Points, J. Math. Sci., 2005, vol. 128, no. 2, pp. 2767–2773; see also: Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 2003, vol. 300, pp. 155–166.CrossRefMathSciNetGoogle Scholar
  15. 15.
    Poincaré, H., Les méthodes nouvelles de la mécanique céleste: T. 3, Paris: Gauthier-Villars, 1899.Google Scholar
  16. 16.
    Pugh, Ch.C. and Robinson, C., The C 1 Closing Lemma, including Hamiltonians, Ergodic Theory Dynam. Systems, 1983, vol. 3, no. 2, pp. 261–313.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Takens, F., Hamiltonian Systems: Generic Properties of Closed Orbits and Local Perturbations, Math. Ann., 1970, vol. 188, no. 4, pp. 304–312.CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Takens, F., Homoclinic Points in Conservative Systems, Invent. Math., 1972, vol. 18, pp. 267–292.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Rüssmann, H., Kleine Nenner: 1. Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1970, pp. 67–105.Google Scholar
  20. 20.
    Vasiliev, A.A., Neishtadt, A.I., Simó, C., and Treschev, D.V., Stability Islands in Domains of Separatrix Crossings in Slow-Fast Hamiltonian Systems, Proc. Steklov Inst. Math., 2007, vol. 259, pp. 236–247; see also: Analysis and Singularities: Part 2: Collected Papers: Dedicated to Academician Vladimir Igorevich Arnold on the Occasion of His 70th Birthday, Tr. Mat. Inst. Steklova, vol. 259, Moscow: Nauka, 2007, pp. 243–255.CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Lamb, J. S.W. and Quispel, G.R.W., Reversible k-Symmetries in Dynamical Systems, Phys. D, 1994, vol. 73, no. 4, pp. 277–304.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Dmitriev, A. S., Komlev, Yu. A., and Turaev, D.V., Bifurcation Phenomena in the 1: 1 Resonant Horn for the Forced van der Pol -Duffing Equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1992, vol. 2, no. 1, pp. 93–100.CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Delshams, A., Gonchenko, S. V., Gonchenko, V. S., Lázaro, J. T., and Sten’kin, O.V., Abundance of Attracting, Repelling and Elliptic Orbits in Two-Dimensional Reversible Maps, Nonlinearity, 2013, vol. 26, no. 1, pp. 1–35.CrossRefzbMATHGoogle Scholar
  24. 24.
    Gonchenko, S. V. and Shilnikov, L.P., Invariants of Ω-Conjugacy of Diffeomorphisms with a Structurally Unstable Homoclinic Trajectory, Ukrainian Math. J., 1990, vol. 42, no. 2, pp. 134–140; see also: Ukraïn. Mat. Zh., 1990, vol. 42, no. 2, pp. 153–159.CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Gonchenko, S. V., Shilnikov, L.P., and Turaev, D. V., Homoclinic Tangencies of Arbitrarily High Orders in Conservative and Dissipative Two-Dimensional Maps, Nonlinearity, 2007, vol. 20, no. 2, pp. 241–275.CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Shilnikov, L.P., Shilnikov, A. L., Turaev, D.V., and Chua L.O., Methods of Qualitative Theory in Nonlinear Dynamics: Part 1, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, vol. 4, River Edge,NJ: World Scientific, 1998.CrossRefzbMATHGoogle Scholar
  27. 27.
    Moser, J., The Analytic Invariants of an Area-Preserving Mapping near a Hyperbolic Fixed Point, Comm. Pure Appl. Math., 1956, vol. 9, no. 4, pp. 673–692.CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Gonchenko, S.V. and Shilnikov, L.P., Arithmetic Properties of Topological Invariants of Systems with a Structurally Unstable Homoclinic Trajectory, Ukrainian Math. J., 1987, vol. 39, no. 1, pp. 15–21; see also: Ukraïn. Mat. Zh., 1987, vol. 39, no. 1, pp. 21–28.CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Gonchenko, S. V., Sten’kin, O. V., and Turaev, D.V., Complexity of Homoclinic Bifurcations and Ω-Moduli, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1996, vol. 6, no. 6, pp. 969–989.CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Turaev, D. and Rom-Kedar, V., Elliptic Islands Appearing in Near-Ergodic Flows, Nonlinearity, 1998, vol. 11, no. 3, pp. 575–600.CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Devaney, R. and Nitecki, Z., Shift Automorphisms in the Hénon Mapping, Comm. Math. Phys., 1979, vol. 67, pp. 137–146.CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Afraĭmovich, V. S. and Shil’nikov, L.P., Strange Attractors and Quasiattractors, in Nonlinear Dynamics and Turbulence, G. I. Barenblatt, G. Iooss, D. D. Joseph (Eds.), Interaction Mech. Math. Ser., Boston, Mass.: Pitman, 1983, pp. 1–34.Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  • Amadeu Delshams
    • 1
  • Marina Gonchenko
    • 2
  • Sergey V. Gonchenko
    • 3
  1. 1.Departament de Matemàtica Aplicada IUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Institute of Applied Mathematics and CyberneticsNizhny Novgorod UniversityNizhny NovgorodRussia

Personalised recommendations