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September  2020, 40(9): 5325-5345. doi: 10.3934/dcds.2020229

Reducibility of quasi-periodically forced circle flows

Saša Kocić 1,, and  João Lopes Dias 2,

1. 

Department of Mathematics, University of Mississippi, Hume Hall 305, University, MS 38677, USA
2. 

Departamento de Matemática, ISEG, Universidade de Lisboa, Rua do Quelhas 6, 1200-781 Lisboa, Portugal

* Corresponding author: Saša Kocić

Received  October 2019 Published  June 2020

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.

Keywords: Circle flows, renormalization, quasi-periodic forcing.
Mathematics Subject Classification: Primary: 37E10, 37F25.
Citation: Saša Kocić, João Lopes Dias. Reducibility of quasi-periodically forced circle flows. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5325-5345. doi: 10.3934/dcds.2020229
References:
[1]

V. I. Arnol'd, Small denominators. Ⅰ: On the mapping of a circle into itself, Izv. Akad. Nauk. Math. Serie, 25 (1961), 21-86.   Google Scholar

[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 (1963), 13-40.   Google Scholar

[3]

A. D. Brjuno, Analytic form of differential equations. Ⅰ, Trudy Moskov. Mat. Obšč., 25 (1971), 119-262.  Google Scholar

[4]

A. D. Brjuno, Analytic form of differential equations. Ⅱ, Trudy Moskov. Mat. Obšč., 26 (1972), 199-239.  Google Scholar

[5]

L. Corsi and G. Gentile, Oscillator synchronisation under arbitrary quasi-periodic forcing, Commun. Math. Phys., 316 (2012), 489-529.  doi: 10.1007/s00220-012-1548-2.  Google Scholar

[6]

G. Gentile, Resummation of perturbation series and reducibility for Bryuno skew-product flows, J. Stat. Phys., 125 (2006), 321-361.  doi: 10.1007/s10955-006-9127-6.  Google Scholar

[7]

G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory and Dynam. Systems, 27 (2007), 427-457.  doi: 10.1017/S0143385706000757.  Google Scholar

[8]

M.-R. Herman, Sur la conjugasion differentiable des difféomorphismes du cercle a de rotations, Inst. Hautes études Sci. Publ. Math., 49 (1979), 5-233.   Google Scholar

[9]

R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438.  doi: 10.1007/BF01208484.  Google Scholar

[10]

K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamic renormalization and KAM theory, Commun. Math. Phys., 270 (2007), 197-231.  doi: 10.1007/s00220-006-0125-y.  Google Scholar

[11]

H. Koch and S. Kocić, Renormalization of vector fields and Diophantine invariant tori, Ergodic Theory Dynam. Systems, 28 (2008), 1559-1585.  doi: 10.1017/S0143385707000892.  Google Scholar

[12]

H. Koch and S. Kocić, A renormalization group approach to quasiperiodic motion with Brjuno frequencies, Ergodic Theory Dynam. Systems, 30 (2010), 1131-1146.  doi: 10.1017/S014338570900042X.  Google Scholar

[13]

H. Koch and S. Kocić, A renormalization approach to lower-dimensional tori with Brjuno frequency vectors, J. Differential Equations, 249 (2010), 1986-2004.  doi: 10.1016/j.jde.2010.05.004.  Google Scholar

[14]

S. Kocić, Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18 (2005), 2513-2544.  doi: 10.1088/0951-7715/18/6/006.  Google Scholar

[15]

S. Kocić, Reducibility of skew-product systems with multidimensional Brjuno base flows, Discrete Contin. Dyn. Syst. A, 29 (2011), 261-283.  doi: 10.3934/dcds.2011.29.261.  Google Scholar

[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 (1954), 527-530.   Google Scholar

[17]

J. Lopes Dias, A normal form theorem for Brjuno skew-systems through renormalization, J. Differential Equations, 230 (2006), 1-23.  doi: 10.1016/j.jde.2006.07.021.  Google Scholar

[18]

J. Lopes Dias, Local conjugacy classes for analytic torus flows, J. Differential Equations, 245 (2008), 468-489.  doi: 10.1016/j.jde.2008.04.006.  Google Scholar

[19]

J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176.  doi: 10.1007/BF01399536.  Google Scholar

[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 (1980), 90-107.   Google Scholar

[21]

J.-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995). Google Scholar

[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 (2002), 125-173.  doi: 10.1007/978-3-540-47928-4_3.  Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Small denominators. Ⅰ: On the mapping of a circle into itself, Izv. Akad. Nauk. Math. Serie, 25 (1961), 21-86.   Google Scholar

[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 (1963), 13-40.   Google Scholar

[3]

A. D. Brjuno, Analytic form of differential equations. Ⅰ, Trudy Moskov. Mat. Obšč., 25 (1971), 119-262.  Google Scholar

[4]

A. D. Brjuno, Analytic form of differential equations. Ⅱ, Trudy Moskov. Mat. Obšč., 26 (1972), 199-239.  Google Scholar

[5]

L. Corsi and G. Gentile, Oscillator synchronisation under arbitrary quasi-periodic forcing, Commun. Math. Phys., 316 (2012), 489-529.  doi: 10.1007/s00220-012-1548-2.  Google Scholar

[6]

G. Gentile, Resummation of perturbation series and reducibility for Bryuno skew-product flows, J. Stat. Phys., 125 (2006), 321-361.  doi: 10.1007/s10955-006-9127-6.  Google Scholar

[7]

G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory and Dynam. Systems, 27 (2007), 427-457.  doi: 10.1017/S0143385706000757.  Google Scholar

[8]

M.-R. Herman, Sur la conjugasion differentiable des difféomorphismes du cercle a de rotations, Inst. Hautes études Sci. Publ. Math., 49 (1979), 5-233.   Google Scholar

[9]

R. Johnson and J. Moser, The rotation number for almost periodic potentials, Comm. Math. Phys., 84 (1982), 403-438.  doi: 10.1007/BF01208484.  Google Scholar

[10]

K. Khanin, J. Lopes Dias and J. Marklof, Multidimensional continued fractions, dynamic renormalization and KAM theory, Commun. Math. Phys., 270 (2007), 197-231.  doi: 10.1007/s00220-006-0125-y.  Google Scholar

[11]

H. Koch and S. Kocić, Renormalization of vector fields and Diophantine invariant tori, Ergodic Theory Dynam. Systems, 28 (2008), 1559-1585.  doi: 10.1017/S0143385707000892.  Google Scholar

[12]

H. Koch and S. Kocić, A renormalization group approach to quasiperiodic motion with Brjuno frequencies, Ergodic Theory Dynam. Systems, 30 (2010), 1131-1146.  doi: 10.1017/S014338570900042X.  Google Scholar

[13]

H. Koch and S. Kocić, A renormalization approach to lower-dimensional tori with Brjuno frequency vectors, J. Differential Equations, 249 (2010), 1986-2004.  doi: 10.1016/j.jde.2010.05.004.  Google Scholar

[14]

S. Kocić, Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori, Nonlinearity, 18 (2005), 2513-2544.  doi: 10.1088/0951-7715/18/6/006.  Google Scholar

[15]

S. Kocić, Reducibility of skew-product systems with multidimensional Brjuno base flows, Discrete Contin. Dyn. Syst. A, 29 (2011), 261-283.  doi: 10.3934/dcds.2011.29.261.  Google Scholar

[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 (1954), 527-530.   Google Scholar

[17]

J. Lopes Dias, A normal form theorem for Brjuno skew-systems through renormalization, J. Differential Equations, 230 (2006), 1-23.  doi: 10.1016/j.jde.2006.07.021.  Google Scholar

[18]

J. Lopes Dias, Local conjugacy classes for analytic torus flows, J. Differential Equations, 245 (2008), 468-489.  doi: 10.1016/j.jde.2008.04.006.  Google Scholar

[19]

J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176.  doi: 10.1007/BF01399536.  Google Scholar

[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 (1980), 90-107.   Google Scholar

[21]

J.-C. Yoccoz, Petits diviseurs en dimension 1, Astérisque, 231 (1995). Google Scholar

[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 (2002), 125-173.  doi: 10.1007/978-3-540-47928-4_3.  Google Scholar

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