Reducibility of quasi-periodically forced circle flows. (English) Zbl 1445.37020
Summary: We develop a renormalization group approach to the problem of reducibility of quasi-periodically forced circle flows. We apply the method to prove a reducibility theorem for such flows.
MSC:
| 37C55 | Periodic and quasi-periodic flows and diffeomorphisms |
| 37E10 | Dynamical systems involving maps of the circle |
| 70K40 | Forced motions for nonlinear problems in mechanics |
References:
| [1] | V. I. Arnol’d, Small denominators. Ⅰ: On the mapping of a circle into itself, Izv. Akad. Nauk. Math. Serie, 25, 21-86 (1961) · Zbl 0135.42603 |
| [2] | V. I. Arnol’d, Proof of A. N. Kolmogorov’s theorem on the preservation of quasiperiodic motions under small perturbations of the Hamiltonian, Uspehi Mat. Nauk, 18, 13-40 (1963) · Zbl 0129.16606 |
| [3] | A. D. Brjuno, Analytic form of differential equations. Ⅰ, Trudy Moskov. Mat. Obšč., 25 (1971), 119-262. · Zbl 0263.34003 |
| [4] | A. D. Brjuno, Analytic form of differential equations. Ⅱ, Trudy Moskov. Mat. Obšč., 26 (1972), 199-239. · Zbl 0283.34013 |
| [5] | L. Corsi; G. Gentile, Oscillator synchronisation under arbitrary quasi-periodic forcing, Commun. Math. Phys., 316, 489-529 (2012) · Zbl 1262.34058 · doi:10.1007/s00220-012-1548-2 |
| [6] | G. Gentile, Resummation of perturbation series and reducibility for Bryuno skew-product flows, J. Stat. Phys., 125, 321-361 (2006) · Zbl 1107.82033 · doi:10.1007/s10955-006-9127-6 |
| [7] | G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory and Dynam. Systems, 27, 427-457 (2007) · Zbl 1110.37048 · doi:10.1017/S0143385706000757 |
| [8] | M.-R. Herman, Sur la conjugasion differentiable des difféomorphismes du cercle a de rotations, Inst. Hautes études Sci. Publ. Math., 49, 5-233 (1979) · Zbl 0448.58019 |
| [9] | R. Johnson; J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84, 403-438 (1982) · Zbl 0497.35026 · doi:10.1007/BF01208484 |
| [10] | K. Khanin; J. Lopes Dias; J. Marklof, Multidimensional continued fractions, dynamic renormalization and KAM theory, Commun. Math. Phys., 270, 197-231 (2007) · Zbl 1114.37009 · doi:10.1007/s00220-006-0125-y |
| [11] | H. Koch; S. Kocić, Renormalization of vector fields and Diophantine invariant tori, Ergodic Theory Dynam. Systems, 28, 1559-1585 (2008) · Zbl 1170.37015 · doi:10.1017/S0143385707000892 |
| [12] | H. Koch; S. Kocić, A renormalization group approach to quasiperiodic motion with Brjuno frequencies, Ergodic Theory Dynam. Systems, 30, 1131-1146 (2010) · Zbl 1201.34003 · doi:10.1017/S014338570900042X |
| [13] | H. Koch; S. Kocić, A renormalization approach to lower-dimensional tori with Brjuno frequency vectors, J. Differential Equations, 249, 1986-2004 (2010) · Zbl 1214.37043 · doi:10.1016/j.jde.2010.05.004 |
| [14] | S. Kocić, Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18, 2513-2544 (2005) · Zbl 1125.37041 · doi:10.1088/0951-7715/18/6/006 |
| [15] | S. Kocić, Reducibility of skew-product systems with multidimensional Brjuno base flows, Discrete Contin. Dyn. Syst. A, 29, 261-283 (2011) · Zbl 1214.37032 · doi:10.3934/dcds.2011.29.261 |
| [16] | A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR (N.S.), 98, 527-530 (1954) · Zbl 0056.31502 |
| [17] | J. Lopes Dias, A normal form theorem for Brjuno skew-systems through renormalization, J. Differential Equations, 230, 1-23 (2006) · Zbl 1104.37029 · doi:10.1016/j.jde.2006.07.021 |
| [18] | J. Lopes Dias, Local conjugacy classes for analytic torus flows, J. Differential Equations, 245, 468-489 (2008) · Zbl 1151.37026 · doi:10.1016/j.jde.2008.04.006 |
| [19] | J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169, 136-176 (1967) · Zbl 0149.29903 · doi:10.1007/BF01399536 |
| [20] | H. Rüssmann, On the one-dimensional Schrödinger equation with a quasiperiodic potential, Nonlinear Dynamics, Ann. New York Acad. Sci., New York Acad. Sci., New York, 357, 90-107 (1980) · Zbl 0477.34007 |
| [21] | J.-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995). · Zbl 0836.30001 |
| [22] | J.-C. Yoccoz, Analytic linearization of circle diffeomorphisms, Dynamical Systems and Small Divisors, Lecture Notes in Mathematics, Fond. CIME/CIME Found. Subser., Springer, Berlin, 1784, 125-173 (2002) · Zbl 1417.37153 · doi:10.1007/978-3-540-47928-4_3 |
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