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Rotation numbers for quasiperiodically forced circle maps-mode-locking vs. strict monotonicity
Author(s):
Kristian
Bjerklöv;
Tobias
Jäger
Journal:
J. Amer. Math. Soc.
22
(2009),
353-362.
MSC (2000):
Primary 37E45, 37B55
Posted:
October 21, 2008
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Abstract:
We describe the relation between the dynamical properties of a quasiperiodically forced orientation-preserving circle homeomorphism  and the behaviour of the fibred rotation number with respect to strictly monotone perturbations. Despite the fact that the dynamics in the forced case can be considerably more complicated, the result we obtain is in perfect analogy with the one-dimensional situation. In particular, the fibred rotation number behaves strictly monotonically whenever the rotation vector of  is irrational, which answers a question posed by Herman (1983). In addition, we obtain the continuous structure of the Arnold tongues in parameter families such as the quasiperiodically forced Arnold circle map.
References:
-
- 1.
- M. Herman.
Une méthode pour minorer les exposants de Lyapunov et quelques exemples montrant le caractère local d'un théorème d'Arnold et de Moser sur le tore de dimension 2.
Commentarii Mathematici Helvetici, 58:453-502, 1983. MR 727713 (85g:58057) - 2.
- T. Jäger and G. Keller.
The Denjoy type-of argument for quasiperiodically forced circle diffeomorphisms.
Ergodic Theory and Dynamical Systems, 26(2):447-465, 2006. MR 2218770 (2007b:37081) - 3.
- T. Jäger and J. Stark.
Towards a classification of quasiperiodically forced circle homeomorphisms.
Journal of the LMS, 73(3):727-744, 2005. MR 2241977 (2008f:37087) - 4.
- F. Delyon and B. Souillard.
The rotation number for finite difference operators and its properties.
Commununications in Mathematical Physics, 89:415-426, 1983. MR 709475 (85d:60123) - 5.
- R. Johnson and J. Moser.
The rotation number for almost periodic potentials.
Communications in Mathematical Physics, 84:403-438, 1982. MR 667409 (83h:34018) - 6.
- A. Prasad, S. Negi, and R. Ramaswamy.
Strange nonchaotic attractors.
International Journal of Bifurcation and Chaos, 11(2):291-309, 2001. MR 1830343 (2002b:37044) - 7.
- M. Ding, C. Grebogi and E. Ott.
Evolution of attractors in quasiperiodically forced systems: From quasiperiodic to strange nonchaotic to chaotic.
Physical Review A, 39(5):2593-2598, 1989. - 8.
- U. Feudel, J. Kurths and A. Pikovsky.
Strange nonchaotic attractor in a quasiperiodically forced circle map.
Physica D, 88:176-186, 1995. MR 1360883 (96j:58146) - 9.
- P. Glendinning, U. Feudel, A.S. Pikovsky, and J. Stark.
The structure of mode-locked regions in quasi-periodically forced circle maps.
Physica D, 140:227-243, 2000. MR 1757089 (2001b:37060) - 10.
- J. Stark, U. Feudel, P. Glendinning, and A. Pikovsky.
Rotation numbers for quasi-periodically forced monotone circle maps.
Dynamical Systems, 17(1):1-28, 2002. MR 1888695 (2003h:37058) - 11.
- T. Jäger.
Strange non-chaotic attractors in quasiperiodically forced circle maps.
Preprint 2007. - 12.
- A. Katok and B. Hasselblatt.
Introduction to the Modern Theory of Dynamical Systems.
Cambridge University Press, 1997. MR 1326374 (96c:58055) - 13.
- K. Bjerklöv.
Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations.
Ergodic Theory and Dynamical Systems, 25:1015-1045, 2005. MR 2158395 (2007c:47035) - 14.
- J. Stark.
Transitive sets for quasiperiodically forced monotone maps.
Dynamical Systems, 18(4):351-364, 2003. MR 2021504 (2004k:37086) - 15.
- H. Furstenberg.
Strict ergodicity and transformation of the torus.
American Journal of Mathematics, 83:573-601, 1961. MR 0133429 (24:A3263) - 16.
- L. Arnold.
Random Dynamical Systems.
Springer, 1998. MR 1723992 (2000m:37087) - 17.
- R. Johnson.
On a Floquet theory for almost-periodic, two-dimensional linear systems.
J. Differential Equations, 37(2):184-205, 1980. MR 587221 (81j:58069)
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Additional Information:
Kristian
Bjerklöv
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario, Canada M5S 3G4
Email:
bjerklov@math.utoronto.ca
Tobias
Jäger
Affiliation:
Department of Mathematics, Collège de France, 3 rue d'Ulm, 75005 Paris, France
Email:
tobias.jager@college-de-france.fr
DOI:
10.1090/S0894-0347-08-00627-9
PII:
S 0894-0347(08)00627-9
Keywords:
Rotation numbers,
mode-locking,
quasiperiodically forced systems.
Received by editor(s):
August 10, 2006
Posted:
October 21, 2008
Copyright of article:
Copyright
2008,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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