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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 and 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. 

[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. 

[3]

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

[4]

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

[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.

[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.

[7]

G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory and Dynam. Systems, 27 (2007), 427-457.  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 (1979), 5-233. 

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[19]

J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176.  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 (1980), 90-107. 

[21]

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

[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.

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. 

[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. 

[3]

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

[4]

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

[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.

[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.

[7]

G. Gentile, Degenerate lower-dimensional tori under the Bryuno condition, Ergodic Theory and Dynam. Systems, 27 (2007), 427-457.  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 (1979), 5-233. 

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[19]

J. Moser, Convergent series expansions for quasi-periodic motions, Mathematische Annalen, 169 (1967), 136-176.  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 (1980), 90-107. 

[21]

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

[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.

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