Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

  • Henk Broer
  • Carles Simó


A simple example is considered of Hill's equation\(\ddot x + (a^2 + bp(t))x = 0\), where the forcing termp, instead of periodic, is quasi-periodic with two frequencies. A geometric exploration is carried out of certain resonance tongues, containing instability pockets. This phenomenon in the perturbative case of small |b|, can be explained by averaging. Next a numerical exploration is given for the global case of arbitraryb, where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.


Schrödinger equation with quasi-periodic potential (non-) reducibility to Floquet form quasiperiodic resonance tongues and unstability pockets positive Lyapunov exponent collapse of resonance tongues and breakdown of tori 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Braaksma, B.L.J. and Broer, H.W.,On a quasi-periodic Hopf bifurcation, Ann. Institut Henri Poincaré, Analyse non Linéaire,4(2): (1987), 115–168.Google Scholar
  2. [2]
    Broer, H.W., Hoveijn, I., and van Noort, M.: A reversible bifurcation analysis of the inverted pendulum,Physica D,112, (1997), 50–63.Google Scholar
  3. [3]
    Broer, H.W., Huitema, G.B., Takens, F. and Braaksma, B.L.J.: Unfoldings and bifurcations of quasi-periodic tori.Mem AMS 83 (421), 1990.Google Scholar
  4. [4]
    Broer, H.W., Huitema, G.B. and Sevryuk, M.B.:Quasi-periodic motions in families of dynamical systems, order amidst chaos. LNM1645 Springer-Verlag, 1996.Google Scholar
  5. [5]
    Broer, H.W. and Levi, M.: Geometrical aspects of stability theory for Hill's equations.Arch. Rat. Mech. An. 131, (1995), 225–240.Google Scholar
  6. [6]
    Broer, H.W. and Simó, C.: Resonance tongues in Hill's equations: a geometric approach, preprint, 1997.Google Scholar
  7. [7]
    Broer, H.W. and Vegter, G.: Bifurcational aspects of parametric resonance.Dynamics Reported, New Series 1 (1992), 1–51.Google Scholar
  8. [8]
    Broer, H.W., Takens, F. and Wagener, F.O.O.: Unfolding the skew Hopf bifurcation, preprint, 1998.Google Scholar
  9. [9]
    Broer, H.W. and Wagener, F.O.O.: Quasi-periodic stability of subfamilies of an unfolded skew Hopf bifurcation, preprint, 1998.Google Scholar
  10. [10]
    Eliasson, L. H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation.Commun. Math. Phys. 146 (1991), 447–482.Google Scholar
  11. [11]
    Eliasson, L. H.: Ergodic skew systems on\(\mathbb{T}^d \times SO(3,\mathbb{R})\), preprint, 1996.Google Scholar
  12. [12]
    Fabbri, R., Johnson, R. and Pavani, R.: On the spectrum of the quasi-periodic Schrödinger operator. In preparation.Google Scholar
  13. [13]
    Giorgilli, A. and Galgani, L.: Formal integrals for an autonomous Hamiltonian system near an equilibrium point.Celest. Mech. 17 (1978), 267–280.Google Scholar
  14. [14]
    Gómez, G., Llibre, J., Martínez, R. and Simó, C.:Study on orbits near the triangular libration point in the perturbed restricted three-body problem. ESA Technical Report, 1987, 270p.Google Scholar
  15. [15]
    Johnson, R.: Cantor spectrum for the quasi-periodic Schrödinger operator.J. Diff. Eqns. 91 (1991), 88–110.Google Scholar
  16. [16]
    Jorba, À., Ramírez-Ros, R. and Villanueva, J.: Effective Reducibility of Quasiperiodic Linear Equations Close to Constant Coefficients.SIAM J. on Math. Anal. 28 (1996), 178–188.Google Scholar
  17. [17]
    Jorba, À. and Simó, C.: On the reducibility of linear differential equations with quasiperiodic coefficients.J. Diff. Eq. 98 (1992), 111–124.Google Scholar
  18. [18]
    Jorba, À. and Simó, C.: On quasiperiodic perturbations of elliptic equilibrium points.SIAM J. of Math. Anal. 27 (1996), 1704–1737.Google Scholar
  19. [19]
    Krikorian, R.: Réductibilité presque partout des flots fibrés quasi-périodiques à valeurs dans des groupes compacts, preprint, 1996.Google Scholar
  20. [20]
    Laskar, J.: The chaotic motion of the solar system. A numerical estimate of the size of the chaotic zones,Icarus 88 (1990), 266–291.Google Scholar
  21. [21]
    Laskar, J., Froeschlé, C., Celletti, A.: The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping,Physica D 56, (1992), 253–269.Google Scholar
  22. [22]
    Moser, J. and Pöschel, J.: An extension of a result by Dinaburg and Sinai on quasi-periodic potentials,Comment. Math. Helvetici 59 (1984), 39–85.Google Scholar
  23. [23]
    Wagener, F.O.O.: On a.e. reducibility of quasi-periodically perturbed two-dimensional Floquet systems. In preparation.Google Scholar

Copyright information

© Sociedade Brasileira de Matemática 1998

Authors and Affiliations

  • Henk Broer
    • 1
  • Carles Simó
    • 2
  1. 1.Dept. of Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands
  2. 2.Dept. de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

Personalised recommendations