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Regular and Chaotic Dynamics

, Volume 16, Issue 1–2, pp 154–184 | Cite as

Quasi-periodic bifurcations of invariant circles in low-dimensional dissipative dynamical systems

  • Renato VitoloEmail author
  • Henk Broer
  • Carles Simó
Article

Abstract

This paper first summarizes the theory of quasi-periodic bifurcations for dissipative dynamical systems. Then it presents algorithms for the computation and continuation of invariant circles and of their bifurcations. Finally several applications are given for quasiperiodic bifurcations of Hopf, saddle-node and period-doubling type.

Keywords

bifurcations invariant tori resonances KAM theory 

MSC2010 numbers

37M20 37C55 37G30 

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References

  1. 1.
    Broer, H.W., Puig, J., and Simó, C., Resonance Tongues and Instability Pockets in the Quasi-periodic Hill-Schrödinger Equation. Comm. Math.l Phys., 2003, vol. 241, pp. 467–503.zbMATHGoogle Scholar
  2. 2.
    Broer, H.W and Simó, C., Hill’s Equation with Quasi-periodic Forcing: Resonance Tongues, Instability Pockets and Global Phenomena, Boletim Sociedade Brasileira Matemática, 1998, vol. 29, pp. 253–293.zbMATHCrossRefGoogle Scholar
  3. 3.
    Broer, H.W., Hanßmann, and Wagener, F. O. O., Quasi-Periodic Bifurcation Theory, the Geometry of kam, in preparation.Google Scholar
  4. 4.
    Broer, H.W., Takens, F., and Wagener, F. O. O., Integrable and Non-Integrable Deformations of the Skew Hopf Bifurcation, Regul. Chaotic Dyn., 1999, vol. 4, no. 2, pp. 16–43.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Broer, H. W. and Wagener, F.O.O., Quasi-Periodic Stability of Subfamilies of an Unfolded Skew Hopf Bifurcation, Arch. Ration. Mech. Anal., 2000, vol. 152, pp. 283–326.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Hanßmann, H., A Survey on Bifurcations of Invariant Tori, New Advances in Celestial Mechanics and Hamiltonian systems (Guanajuato, 2001), J. Delgado et al. (Eds.), New York: Kluwer/Plenum, 2004, pp. 109–121.Google Scholar
  7. 7.
    Hanßmann, H., Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems: Results and Examples, Lecture Notes in Math., vol. 1893, Berlin: Springer, 2007.zbMATHGoogle Scholar
  8. 8.
    Takens, F. and Wagener, F. O. O., Resonances in Skew and Reducible Quasi-Periodic Hopf Bifurcations, Nonlinearity, 2000, vol. 13, no. 2, pp. 377–396.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Broer, H.W., Normal Forms in Perturbation Theory, Encyclopœia of Complexity & System Science, R. Meyers (Ed.) New York: Springer, 2009, pp. 6310–6329.Google Scholar
  10. 10.
    Braaksma, B. L. J., Broer, H.W., and Huitema, G.B., Towards a Quasi-Periodic Bifurcation Theory, Mem. Amer. Math. Soc., 1990, vol. 83, no. 421, pp. 83–175.MathSciNetGoogle Scholar
  11. 11.
    Broer, H. W., Hoo, J., and Naudot, V., Normal Linear Stability of Quasi-Periodic Tori, J. Differential Equations, 2007, vol. 232, no. 2, pp. 355–418.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Broer, H. W., Huitema, G.B., and Takens, F., Unfoldings of Quasi-Periodic Tori, Mem. Amer. Math. Soc., 1990, vol. 83, no. 421, pp. 1–82.MathSciNetGoogle Scholar
  13. 13.
    Newhouse, S.E., Palis, J., and Takens, F., Bifurcations and Stability of Families of Diffeomorphisms, Inst. Hautes Études Sci. Publ. Math., 1983, vol. 57, pp. 5–71.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hirsch, M.W., Pugh, C.C., and Shub, M., Invariant Manifolds, Lecture Notes in Math., vol. 583, Berlin-New York: Springer, 1977.zbMATHGoogle Scholar
  15. 15.
    Braaksma, B. L. J. and Broer, H.W., On a Quasi-Periodic Hopf Bifurcation, Ann. Inst. H.Poincaré Anal. Non Linéaire, 1987, vol. 4, no. 2, pp. 115–168.zbMATHMathSciNetGoogle Scholar
  16. 16.
    Chow, S.-N. and Hale, J. K., Methods of Bifurcation Theory, New York: Springer, 1982.zbMATHGoogle Scholar
  17. 17.
    Wagener, F. O. O., On the Quasi-Periodic d-fold Degenerate Bifurcation, J. Differential Equations, 2005, vol. 216, no. 2, pp. 261–281.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Wagener, F. O. O., A Parametrised Version of Moser’s Modifying Terms Theorem, Discrete Contin. Dyn. Syst. Ser. S, 2010, vol. 3, no. 4, pp. 719–768.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Chenciner, A., Bifurcations de points fixes elliptiques: I. Courbes invariantes, Inst. Hautes Études Sci. Publ. Math., 1985, vol. 61, pp. 67–127.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Chenciner, A., Bifurcations de points fixes elliptiques: II. Orbites périodiques et ensembles de Cantor invariants, Invent. Math., 1985, vol. 80, no. 1, pp. 81–106.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Chenciner, A., Bifurcations de points fixes elliptiques: III. Orbites périodiques de “petites” périodes et élimination résonante des couples de courbes invariantes, Inst. Hautes Études Sci. Publ. Math., 1987, vol. 66, pp. 5–91.CrossRefGoogle Scholar
  22. 22.
    Hanßmann, H., The Quasi-Periodic Centre-Saddle Bifurcation, J. Differential Equations, 1998, vol. 142, pp. 305–370.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Broer, H. W., Hanßmann, H., and You, J., Bifurcations of Normally Parabolic Tori in Hamiltonian Systems, Nonlinearity, 2005, vol. 18, no. 4, pp. 1735–1769.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Broer, H. W., Hanßmann, H., and You, J., Umbilical Torus Bifurcations in Hamiltonian Systems, J. Differential Equations, 2006, vol. 222, no. 1, pp. 233–262.zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Broer, H.W., Hanßmann, H., and You, J., On the Destruction of Resonant Lagrangean Tori in Hamiltonian Systems, Preprint, Mathematisch Instituut, Universiteit Utrecht, Utrecht, 2008.Google Scholar
  26. 26.
    Broer, H.W. and Takens, F., Dynamical Systems and Chaos, Appl. Math. Sci., vol. 172, New York: Springer, 2011.zbMATHCrossRefGoogle Scholar
  27. 27.
    Ciocci, M.C., Litvak-Hinenzon, A., and Broer, H. W., Survey on Dissipative kam Theory Including Quasi-Periodic Bifurcation Theory: Based on Lectures by Henk Broer, Geometric Mechanics and Symmetry: The Peyresq Lectures, J. Montaldi and T. Ratiu (Eds.), London Math. Soc. Lecture Note Ser., vol. 306, Cambridge: Cambridge Univ. Press, 2005, pp. 303–355.CrossRefGoogle Scholar
  28. 28.
    Broer, H.W. and Takens, F., Mixed Spectrum and Rotational Symmetry, Arch. Ration. Mech. Anal., 1993, vol. 124, pp. 13–42.zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Wagener, F.O.O., On the skew Hopf bifurcation, PhD Thesis, University of Groningen, 1998.Google Scholar
  30. 30.
    Broer, H.W., Ciocci, M.C., and Hanßann, H., The Quasi-Periodic Reversible Hopf Bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2007, vol. 17, no. 8, pp. 2605–2623.zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Broer, H. W., Ciocci, M.C., Hanßann, H., and Vanderbauwhede, A., Quasi-Periodic Stability of Normally Resonant Tori, Phys. D, 2009, vol. 238, no. 3, pp. 309–318.zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Broer, H.W., Hanßann, H., and Hoo, J., The Quasi-Periodic Hamiltonian Hopf Bifurcation, Nonlinearity, 2007, vol. 20, no. 2, pp. 417–460.zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Arnold, V. I., Geometrical Method in the Theory of Ordinary Differential Equations, New York-Berlin: Springer, 1983.Google Scholar
  34. 34.
    Broer, H.W., Simó, C., Tatjer, J.C., Towards Global Models Near Homoclinic Tangencies of Dissipative Diffeomorphisms, Nonlinearity, 1998, vol. 11, no. 3, pp. 667–770.zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Broer, H.W., Hanßann, H., Jorba, À., Villanueva, J., and Wagener, F.O.O., Normal-Internal Resonances in Quasi-Periodically Forced Oscillators: A Conservative Approach, Nonlinearity, 2003, vol. 16, no. 5, pp. 1751–1791.zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Broer, H.W., Holtman, S. J., Vegter, G., and Vitolo, R., Dynamics and Geometry Near Resonant Bifurcations, Regul. Chaotic Dyn., 2011, vol. 16, nos. 1–2, pp. 39–50.CrossRefGoogle Scholar
  37. 37.
    Broer, H.W. and Vegter, G., Generic Hopf-Neĭmark-Sacker Bifurcations in Feed Forward Systems, Nonlinearity, 2008, vol. 21, pp. 1547–1578.zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Castellà, E., Sobre la dinàmica prop dels punts de Lagrange del sistema Terra-Luna, PhD thesis, Univ. of Barcelona, 2003.Google Scholar
  39. 39.
    Castellà, E. and Jorba, À., On the Vertical Families of Two-Dimensional Tori Near the Triangular Points of the Bicircular Problem, Celestial Mech. Dynam. Astronom., 2000, vol. 76, no. 1, pp. 35–54.zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    Jorba, À., Numerical Computation of the Normal Behaviour of Invariant Curves of n-Dimensional Maps, Nonlinearity, 2001 vol. 14, no. 5, pp. 943–976.zbMATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Gómez, G., Mondelo, J.-M., and Simó, C., A Collocation Method for the Numerical Fourier Analysis of Quasi-Periodic Functions: I.Numerical Tests and Examples, Discrete Contin. Dyn. Syst. Ser. B, 2010, vol. 14, no. 1, pp. 41–74.zbMATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Gómez, G., Mondelo, J.-M., and Simó, C., A Collocation Method for the Numerical Fourier Analysis of Quasi-Periodic Functions: II.Analytical Error Estimates, Discrete Contin. Dyn. Syst. Ser. B, 2010, vol. 14, no. 1, pp. 75–109.zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Mondelo, J.M., Contribution to the Study of Fourier Methods for Quasi-Periodic Functions and the Vicinity of the Collinear Libration Points, PhD thesis, Univ. of Barcelona, 2001.Google Scholar
  44. 44.
    Haro, À. and de la Llave, R., A Parameterization Method for the Computation of Invariant Tori and Their Whiskers in Quasi-Periodic Maps: Rigorous Results, J. Differential Equations, 2006, vol. 228, no. 2, pp. 530–579.zbMATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Haro, À. and de la Llave, R., A Parameterization Method for the Computation of Invariant Tori and Their Whiskers in Quasi-Periodic Maps: Numerical Algorithms, Discrete Contin. Dyn. Syst. Ser. B, 2006, vol. 6, no. 6, pp. 1261–1300.zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Haro, À. and de la Llave, R., A ParameterizationMethod for the Computation of Invariant Tori and Their Whiskers in Quasi-Periodic Maps: Explorations and Mechanisms for the Breakdown of Hyperbolicity, SIAM J. Appl. Dyn. Syst., 2007, vol. 6, no. 1, pp. 142–207.zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Schilder, F., Osinga, H.M., and Vogt, W., Continuation of Quasi-Periodic Invariant Tori, SIAM J. Appl. Dyn. Syst., 2005, vol. 4, no. 3, pp. 459–488.zbMATHCrossRefMathSciNetGoogle Scholar
  48. 48.
    Simó, C., Effective Computations in Hamiltonian Dynamics, Mécanique celeste, A. Chenciner and C. Simó (Eds.), SMF Journ. Annu., vol. 1996, Paris: Soc. Math. France, 1996.Google Scholar
  49. 49.
    C. Simó, Effective Computations in Celestial Mechanics and Astrodynamics. In Modern Methods of Analytical Mechanics and their Applications (Udine, 1997), V.V. Rumyantsev and A. V. Karapetyan (Eds.), CISM Courses and Lectures, vol. 387, Vienna: Springer, 1998, pp. 55–102. Available also at http://www.maia.ub.es/dsg/1997/.Google Scholar
  50. 50.
    Sánchez, J., Net, M., and Simó, C., Computation of Invariant Tori by Newton-Krylov Methods in Large-Scale Dissipative Systems, Phys. D, 2010, vol. 239, pp. 123–133.zbMATHGoogle Scholar
  51. 51.
    Broer, H.W., Simó, C., and Vitolo, R., Bifurcations and Strange Attractors in the Lorenz-84 Climate Model with Seasonal Forcing, Nonlinearity, 2002, vol. 15, no. 4, pp. 1205–1267.zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Vitolo, R., Broer, H. W., and Simó, C., Routes to Chaos in the Hopf-Saddle-Node Bifurcation for Fixed Points of 3D-diffeomorphisms, Nonlinearity, 2010, vol. 23, no. 8, pp. 1919–1947.zbMATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Broer, H.W., Simó, C., and Vitolo, R., The Hopf-Saddle-Node Bifurcation for Fixed Points of 3Ddiffeomorphisms: Analysis of a resonance “bubble”, Phys. D, 2008, vol. 237, no. 13, pp. 1773–1799.zbMATHCrossRefMathSciNetGoogle Scholar
  54. 54.
    Broer, H.W., Simó, C., and Vitolo, R., The Hopf-Saddle-Node Bifurcation for Fixed Points of 3Ddiffeomorphisms: The Arnol’d Resonance Web, Bull. Belg. Math. Soc. Simon Stevin, 2008, vol. 15, no. 5, pp. 769–787.zbMATHMathSciNetGoogle Scholar
  55. 55.
    Jorba, À. and Tatjer, J. C., A Mechanism for the Fractalization of Invariant Curves in Quasi-Periodically Forced 1 — D maps, Discrete Contin. Dyn. Syst. Ser. B, 2008, vol. 10, nos. 2–3, pp. 537–567.zbMATHMathSciNetGoogle Scholar
  56. 56.
    Iooss, G. and Los, J.E., Quasi-Genericity of Bifurcations to High-Dimensional Invariant Tori for Maps, Comm. Math. Phys., 1988, vol. 119, no. 3, pp. 453–500.zbMATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Chenciner, A. and Iooss, G., Bifurcations de tores invariants, Arch. Ration. Mech. Anal., 1979, vol. 69, no. 2, pp. 109–198.zbMATHCrossRefMathSciNetGoogle Scholar
  58. 58.
    Los, J., Dédoublement de courbes invariantes sur le cylindre: Petit diviseurs, Ann. Inst. H.Poincaré Anal. Non Linéaire, 1988, vol. 5, pp. 37–95.zbMATHMathSciNetGoogle Scholar
  59. 59.
    Simó, C., On the Analytical and Numerical Computation of Invariant Manifolds, Modern Methods in Celestial Mechanics, D. Benest and C. Froeschlé (Eds.), Paris: Frontières, 1990, pp. 285–330. Available also at http://www.maia.ub.es/dsg/2004/.Google Scholar
  60. 60.
    Vitolo, R., Bifurcations of Attractors in 3D Diffeomorphisms, PhD thesis, University of Groningen, 2003.Google Scholar
  61. 61.
    Guckenheimer, J. and Holmes, Ph., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci., vol. 42, New York: Springer, 1983.zbMATHGoogle Scholar
  62. 62.
    Los, J., Non-Normally Hyperbolic Invariant Curves for Maps in ℝ3 and Doubling Bifurcation, Nonlinearity, 1989, vol. 2, no. 1, pp. 149–174.zbMATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    van Veen, L., The Quasi-Periodic Doubling Cascade in the Transition to Weak Turbulence, Phys. D, 2005, vol. 210, nos. 3–4, pp. 249–261.zbMATHCrossRefMathSciNetGoogle Scholar
  64. 64.
    Puig, J. and Simó, C., Resonance Tongues and Spectral Gaps in Quasi-Periodic Schrödinger Operators with One or More Frequencies: A Numerical Exploration, J. Dynam. Differential Equations, 2011 (to appear).Google Scholar
  65. 65.
    Puig, J. and Simó, C., Resonance Tongues in the Quasi-Periodic Hill-Schrödinger Equation with Three Frequencies, Regul. Chaotic Dyn., 2011, vol. 16, nos. 1–2, pp. 61–78.CrossRefGoogle Scholar
  66. 66.
    Baesens, C., Guckenheimer, J., Kim, S., and MacKay, R. S., Three Coupled Oscillators: Mode-locking, Global Bifurcations and Toroidal Chaos, Phys. D, 1991, vol. 49, no. 3, pp. 387–475.zbMATHCrossRefMathSciNetGoogle Scholar
  67. 67.
    Simó, C., Perturbations of Translations in the Two-Dimensional Torus: The Case near Resonance, Proceedings VI CEDYA (Universidad de Zaragoza, Jaca, Spain, 1983), Zaragoza, 1984.Google Scholar
  68. 68.
    Broer, H.W., van Dijk, R., and Vitolo, R., Survey of Strong Normal-Internal k: l Resonances in Quasi-Periodically Driven Oscillators for l = 1, 2, 3, Proceedings SPT 2007, Internat. Conf. on Symmetry and Perturbation Theory (Otranto (Italy), 2–9 June 2007), G. Gaeta, R. Vitolo, S. Walcher (Eds.), Hackensack, NJ: World Sci. Publ., 2008, pp. 45–55.CrossRefGoogle Scholar
  69. 69.
    Wagener, F. O. O., Semi-Local Analysis of the k: 1 and k: 2 Resonances in Quasi-Periodically Forced Systems, Global Analysis of Dynamical Systems: Festschrift Dedicated to Floris Takens for His 60th Birthday, H.W. Broer, B. Krauskopf, G. Vegter (Eds.), Bristol: Inst. Phys., 2001, pp. 113–129.Google Scholar
  70. 70.
    de la Llave, R., González, A., Jorba, À., and Villanueva, J., KAM theory without Action-Angle Variables, Nonlinearity, 2005, vol. 18, no. 2, pp. 855–895.zbMATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    Lucarini, V., Speranza, A., and Vitolo, R., Parametric Smoothness and Self-Scaling of the Statistical Properties of a Minimal Climate Model: What beyond the Mean Field Theories? Phys. D, 2007, vol. 234, no. 2, pp. 105–123.zbMATHCrossRefMathSciNetGoogle Scholar
  72. 72.
    Randriamampianina, A., Früh, W.-G., Maubert, P., and Read, P.L., Direct Numerical Simulations of Bifurcations in an Air-Filled Rotating Baroclinic Annulus J. Fluid Mech., 2006, vol. 561, pp. 359–389.zbMATHCrossRefMathSciNetGoogle Scholar
  73. 73.
    Young, R.M.B. and Read, P.L., Flow Transitions Resembling Bifurcations of the Logistic Map in Simulations of the Baroclinic Rotating Annulus, Phys. D, 2008, vol. 237, no. 18, pp. 2251–2262.zbMATHCrossRefMathSciNetGoogle Scholar
  74. 74.
    Jorba, À. and Olmedo, E., On the Computation of Reducible Invariant Tori on a Parallel Computer, SIAM J. Appl. Dyn. Syst., 2009, vol. 8, no. 4, pp. 1382–1404.zbMATHCrossRefMathSciNetGoogle Scholar
  75. 75.
    Broer, H.W., Formal Normal Forms Theorems for Vector Fields and Some Consequences for Bifurcations in the Volume Preserving Case, Dynamical Systems and Turbulence, Warwick 1980 (Coventry, 1979/1980), A. Dold and B. Eckmann (Eds.), Lecture Notes in Math., vol. 898, Berlin-New York: Springer, 1980, pp. 54–74.CrossRefGoogle Scholar

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© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.College of Engineering, Mathematics and Physical SciencesUniversity of Exeter, Harrison BuildingExeterUK
  2. 2.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands
  3. 3.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

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