Regular and Chaotic Dynamics

, Volume 23, Issue 6, pp 638–653 | Cite as

On the “Hidden” Harmonics Associated to Best Approximants Due to Quasi-periodicity in Splitting Phenomena

  • Ernest FontichEmail author
  • Carles Simó
  • Arturo Vieiro


The effects of quasi-periodicity on the splitting of invariant manifolds are examined. We have found that some harmonics that could be expected to be dominant in some ranges of the perturbation parameter actually are nondominant. It is proved that, under reasonable conditions, this is due to the arithmetic properties of the frequencies.


quasi-periodic splitting dominant harmonics hidden harmonics irrational numbers properties 


37C55 37J40 37J45 


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  1. 1.
    Barrabés, E., Ollé, M., Borondo, F., Farrelly, D., and Mondelo, J.M., Phase Space Structure of the Hydrogen Atom in a Circularly Polarized Microwave Field, Phys. D, 2012, no. 4, vol. 241, pp. 333–349.CrossRefGoogle Scholar
  2. 2.
    Batut, C., Belabas, K., Bernardi, D., Cohen, H., and Olivier, M., Users’ guide to PARI/GP, .Google Scholar
  3. 3.
    Delshams, A., Gelfreich, V., Jorba, `A., and Seara, T. M., Exponentially Small Splitting of Separatrices under Fast Quasiperiodic Forcing, Comm. Math. Phys., 1997, vol. 189, no. 1, pp. 35–71.Google Scholar
  4. 4.
    Delshams, A., Gonchenko, M., and Gutiérrez, P., Exponentially Small Lower Bounds for the Splitting of Separatrices to Whiskered Tori with Frequencies of Constant Type, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2014, vol. 24, no. 8, 1440011, 12 pp.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Delshams, A., Gonchenko, M., and Gutiérrez, P., Continuation of the Exponentially Small Transversality for the Splitting of Separatrices to a Whiskered Torus with Silver Ratio, Regul. Chaotic Dyn., 2014, vol. 19, no. 6, pp. 663–680.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Delshams, A., Gonchenko, M., and Gutiérrez, P., Exponentially Small Splitting of Separatrices and Transversality Associated to Whiskered Tori with Quadratic Frequency Ratio, SIAM J. Appl. Dyn. Syst., 2016, vol. 15, no. 2, pp. 981–1024.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Delshams, A. and Gutiérrez, P., Splitting Potential and the Poincaré–Melnikov Method for Whiskered Tori in Hamiltonian Systems, J. Nonlinear Sci., 2000, vol. 10, no. 4, pp. 433–476.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Delshams, A. and Gutiérrez, P., Exponentially Small Splitting for Whiskered Tori in Hamiltonian Systems: Continuation of Transverse Homoclinic Orbits, Discrete Contin. Dyn. Syst., 2004, vol. 11, no. 4, pp. 757–783.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Delshams, A. and Gutiérrez, P., Exponentially Small Splitting of Separatrices for Whiskered Tori in Hamiltonian Systems, J. Math. Sci. (N. Y.), 2005, vol. 128, no. 2, pp. 2726–2746MathSciNetCrossRefzbMATHGoogle Scholar
  10. 9a.
    see also: Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov (POMI), 2003, vol. 300, no. 8, pp. 87–121, 287.Google Scholar
  11. 10.
    Fontich, E., Simó, C., and Vieiro, A., Splitting of the Separatrices after a Hamiltonian–Hopf Bifurcation under Periodic Forcing, Nonlinearity, to appear.Google Scholar
  12. 11.
    Gaivão, J.P. and Gelfreich, V., Splitting of Separatrices for the Hamiltonian–Hopf Bifurcation with the Swift–Hohenberg Equation As an Example, Nonlinearity, 2011, vol. 24, no. 3, pp. 677–698.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 12.
    Khinchin, A.Ya., Continued Fractions, Chicago, Ill.: Univ. of Chicago, 1964.zbMATHGoogle Scholar
  14. 13.
    Kuzmin, R.O., On a Problem of Gauss, Dokl. Akad. Nauk SSSR. Ser. A, 1928, pp. 375–380 (Russian).Google Scholar
  15. 14.
    Lahiri, A. and Roy, M. S., The Hamiltonian Hopf Bifurcation: An Elementary Perturbative Approach, Internat. J. Non-Linear Mech., 2001, vol. 36, no. 5, pp. 787–802.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 15.
    Lévy, P., Théorie de l’addition des variables aléatoires, 2nd ed., Paris: Gauthier-Villars, 1954.zbMATHGoogle Scholar
  17. 16.
    Lochak, P., Marco, J.-P., and Sauzin, D., On the Splitting of Invariant Manifolds in Multidimensional Near-Integrable Hamiltonian Systems, Mem. Amer. Math. Soc., vol. 163, no. 775, Providence,R.I.: AMS, 2003.Google Scholar
  18. 17.
    McSwiggen, P.D. and Meyer, K.R., The Evolution of Invariant Manifolds in Hamiltonian–Hopf Bifurcations, J. Differential Equations, 2003, vol. 189, no. 2, pp. 538–555.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 18.
    Meyer, K.R. and Hall, G.R., Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Appl. Math. Sci., vol. 90, New York: Springer, 1992.CrossRefzbMATHGoogle Scholar
  20. 19.
    Moser, J., On Invariant Curves of Area-Preserving Mappings of an Annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962, vol. 1962, pp. 1–20.MathSciNetzbMATHGoogle Scholar
  21. 20.
    Moser, J., Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics, Ann. of Math. Stud., No. 77, Princeton,N.J.: Princeton Univ. Press, 1973.zbMATHGoogle Scholar
  22. 21.
    Ollé, M. and Pacha, J.R., Hopf Bifurcation for the Hydrogen Atom in a Circularly Polarized Microwave Field, Commun. Nonlinear Sci. Numer. Simul., 2018, vol. 62, pp. 27–60.MathSciNetCrossRefGoogle Scholar
  23. 22.
    Rudnev, M. and Wiggins, S., Existence of Exponentially Small Separatrix Splitting and Homoclinic Connections between Whiskered Tori in Weakly Hyperbolic Near-Integrable Hamiltonian Systems, Phys. D, 1998, vol. 114, nos. 1–2, pp. 3–80.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 23.
    Rudnev, M. and Wiggins, S., On the Dominant Fourier Modes in the Series Associated with Separatrix Splitting for an a-priori Stable, Three Degree-of-Freedom Hamiltonian System, in The Arnoldfest: Proc. of a conference in Honour of V. I.Arnold for His Sixtieth Birthday (Toronto,ON, 1997), E. Bierstone et al. (Eds.), Fields Inst. Commun., vol. 24, Providence,R.I.: AMS, 1999, pp. 415–449.CrossRefzbMATHGoogle Scholar
  25. 24.
    Simó, C., Averaging under Fast Quasiperiodic Forcing, in Hamiltonian Mechanics: Proc. of the NATOARW “Integrable and Chaotic Behavior in Hamiltonian Systems” (Torún, 1993), I. Seimenis (Ed.), NATO Adv. Sci. Inst. Ser. B Phys., vol. 331, New York: Plenum, 1994, pp. 13–34.Google Scholar
  26. 25.
    Simó, C. and Valls, C., A Formal Approximation of the Splitting of Separatrices in the Classical Arnold’s Example of Diffusion with Two Equal Parameters, Nonlinearity, 2001, vol. 14, no. 6, pp. 1707–1760.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 26.
    Sokol’skiĭ, A. G., On the Stability of an Autonomous Hamiltonian System with Two Degrees of Freedom in the Case of Equal Frequencies, J. Appl. Math. Mech., 1974, vol. 38, no. 5, pp. 741–749; see also: Prikl. Mat. Mekh., 1974, vol. 38, no. 5, pp. 791–799 (Russian).MathSciNetCrossRefGoogle Scholar
  28. 27.
    van der Meer, J.-C., Nonsemisimple 1: 1 Resonance at an Equilibrium, Celestial Mech., 1982, vol. 27, no. 2, pp. 131–149.MathSciNetCrossRefzbMATHGoogle Scholar

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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Departament de Matemàtiques i InformàticaUniversitat de Barcelona, BGSMathBarcelona, CataloniaSpain

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