Rotation numbers for quasiperiodically forced circle maps-mode-locking vs. strict monotonicity

Bjerklöv, Kristian; Jäger, Tobias

doi:10.1090/S0894-0347-08-00627-9

Rotation numbers for quasiperiodically forced circle maps-mode-locking vs. strict monotonicity
HTML articles powered by AMS MathViewer

by Kristian Bjerklöv and Tobias Jäger

J. Amer. Math. Soc. 22 (2009), 353-362

DOI: https://doi.org/10.1090/S0894-0347-08-00627-9

Published electronically: October 21, 2008

PDF | Request permission

Abstract:

We describe the relation between the dynamical properties of a quasiperiodically forced orientation-preserving circle homeomorphism $f$ 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 $f$ 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

Michael-R. Herman, Une méthode pour minorer les exposants de Lyapounov et quelques exemples montrant le caractère local d’un théorème d’Arnol′d et de Moser sur le tore de dimension $2$, Comment. Math. Helv. 58 (1983), no. 3, 453–502 (French). MR 727713, DOI 10.1007/BF02564647
Tobias H. Jäger and Gerhard Keller, The Denjoy type of argument for quasiperiodically forced circle diffeomorphisms, Ergodic Theory Dynam. Systems 26 (2006), no. 2, 447–465. MR 2218770, DOI 10.1017/S0143385705000477
Tobias H. Jäger and Jaroslav Stark, Towards a classification for quasiperiodically forced circle homeomorphisms, J. London Math. Soc. (2) 73 (2006), no. 3, 727–744. MR 2241977, DOI 10.1112/S0024610706022782
F. Delyon and B. Souillard, The rotation number for finite difference operators and its properties, Comm. Math. Phys. 89 (1983), no. 3, 415–426. MR 709475, DOI 10.1007/BF01214663
R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys. 84 (1982), no. 3, 403–438. MR 667409, DOI 10.1007/BF01208484
Awadhesh Prasad, Surendra Singh Negi, and Ramakrishna Ramaswamy, Strange nonchaotic attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 11 (2001), no. 2, 291–309. MR 1830343, DOI 10.1142/S0218127401002195

Physical Review A

Ulrike Feudel, Jürgen Kurths, and Arkady S. Pikovsky, Strange non-chaotic attractor in a quasiperiodically forced circle map, Phys. D 88 (1995), no. 3-4, 176–186. MR 1360883, DOI 10.1016/0167-2789(95)00205-I
P. Glendinning, U. Feudel, A. S. Pikovsky, and J. Stark, The structure of mode-locked regions in quasi-periodically forced circle maps, Phys. D 140 (2000), no. 3-4, 227–243. MR 1757089, DOI 10.1016/S0167-2789(99)00235-3
J. Stark, U. Feudel, P. A. Glendinning, and A. Pikovsky, Rotation numbers for quasi-periodically forced monotone circle maps, Dyn. Syst. 17 (2002), no. 1, 1–28. MR 1888695, DOI 10.1080/14689360110073641
Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
Kristian Bjerklöv, Positive Lyapunov exponent and minimality for a class of one-dimensional quasi-periodic Schrödinger equations, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1015–1045. MR 2158395, DOI 10.1017/S0143385704000999
J. Stark, Transitive sets for quasi-periodically forced monotone maps, Dyn. Syst. 18 (2003), no. 4, 351–364. MR 2021504, DOI 10.1080/14689360310001610155
H. Furstenberg, Strict ergodicity and transformation of the torus, Amer. J. Math. 83 (1961), 573–601. MR 133429, DOI 10.2307/2372899
Ludwig Arnold, Random dynamical systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. MR 1723992, DOI 10.1007/978-3-662-12878-7
Russell A. Johnson, On a Floquet theory for almost-periodic, two-dimensional linear systems, J. Differential Equations 37 (1980), no. 2, 184–205. MR 587221, DOI 10.1016/0022-0396(80)90094-7