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

Bulletin of the Brazilian Mathematical Society

Volumn 29, Issue 2, 1998, Pages 253-293

  Broer, Henk ^a   Simó, Carles ^b  

^a UNIVERSITY OF GRONINGEN   (Netherlands)

^b UNIVERSITY OF BARCELONA   (Spain)

Author keywords

(Non ) reducibility to Floquet form; Collapse of resonance tongues and breakdown of tori; Positive Lyapunov exponent; Quasiperiodic resonance tongues and unstability pockets; Schr dinger equation with quasi periodic potential

Indexed keywords

EID: 27144516452     PISSN: 01003569     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF01237651     Document Type: Article

References (23)

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.
2. Broer, H.W., Hoveijn, I. and van Noort, M.: A reversible bifurcation analysis of the inverted pendulum, Physica D, 112, (1997), 50-63.
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.
4. Broer, H.W., Huitema, G.B. and Sevryuk, M.B.: Quasi-periodic motions in families of dynamical systems, order amidst chaos. LNM 1645 Springer-Verlag, 1996.
5. Broer, H.W. and Levi, M.: Geometrical aspects of stability theory for Hill's equations. Arch. Rat. Mech. An. 131 (1995), 225-240.
6. Broer, H.W. and Simó, C.: Resonance tongues in Hill's equations: a geometric approach, preprint, 1997.
7. Broer, H.W. and Vegter, G.: Bifurcational aspects of parametric resonance. Dynamics Reported, New Series 1 (1992), 1-51.
8. Broer, H.W., Takens, F. and Wagener, F.O.O.: Unfolding the skew Hopf bifurcation, preprint, 1998.
9. Broer, H.W. and Wagener, F.O.O.: Quasi-periodic stability of subfamilies of an unfolded skew Hopf bifurcation, preprint, 1998.
10. Eliasson. L. H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146 (1991), 447-482.
11. d × SO(3, ℝ), preprint, 1996.
12. Fabbri, R., Johnson, R. and Pavani, R: On the spectrum of the quasi-periodic Schrödinger operator. In preparation.
13. Giorgilli, A. and Galgani, L.: Formal integrals for an autonomous Hamiltonian system near an equilibrium point. Celest. Mech. 17 (1978), 267-280.
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.
15. Johnson, R.: Cantor spectrum for the quasi-periodic Schrödinger operator. J. Diff. Eqns. 91 (1991), 88-110.
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.
17. Jorba, À. and Simó, C.: On the reducibility of linear differential equations with quasiperiodic coefficients. J. Diff. Eq. 98 (1992), 111-124.
18. Jorba, À. and Simó, C.: On quasiperiodic perturbations of elliptic equilibrium points, SIAM J. of Math. Anal. 27 (1996), 1704-1737.
19. Krikorian, R.: Réductibilité presque partout des flots fibrés quasi-périodiques à valeurs dans des groupes compacts, preprint, 1996.
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.
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.
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.
23. Wagener. F.O.O.: On a.e. reducibility of quasi-periodically perturbed twodimensional Floquet systems. In preparation.