Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T03:21:41.520Z Has data issue: false hasContentIssue false
  • English
  • Français

A renormalization group approach to quasiperiodic motion with Brjuno frequencies

Published online by Cambridge University Press:  17 July 2009

HANS KOCH  and
SAŠA KOCIĆ
HANS KOCH
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA (email: [email protected], [email protected])
SAŠA KOCIĆ
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA (email: [email protected], [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce a renormalization group scheme that applies to vector fields on 𝕋d×ℝm with frequency vectors that satisfy a Brjuno condition. Earlier approaches were restricted to Diophantine frequencies, owing to a limited control of multidimensional continued fractions. We get around this restriction by avoiding the use of a continued-fractions expansion. Our results concerning invariant tori generalize those of [H. Koch and S. Kocić, Renormalization of vector fields and Diophantine invariant tori. Ergod. Th. & Dynam. Sys.28 (2008), 1559–1585] from Diophantine- to Brjuno-type frequency vectors. In particular, each Brjuno vector ω∈ℝd determines an analytic manifold 𝒲 of infinitely renormalizable vector fields, and each vector field on 𝒲 is shown to have an elliptic invariant d-torus with frequencies ω1,ω2,…,ωd.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 30 , Issue 4 , August 2010 , pp. 1131 - 1146
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Escande, D. F. and Doveil, F.. Renormalisation method for computing the threshold of the large scale stochastic instability in two degree of freedom Hamiltonian systems. J. Stat. Phys. 26 (1981), 257–284.CrossRefGoogle Scholar
[2]Shenker, S. J. and Kadanoff, L. P.. Critical behavior of a KAM surface. I. Empirical results. J. Stat. Phys. 27 (1982), 631–656.CrossRefGoogle Scholar
[3]MacKay, R. S.. Renormalisation in Area-Preserving Maps. World Scientific, London, 1993, (PhD Thesis, Princeton University, 1982).CrossRefGoogle Scholar
[4]Stirnemann, A.. Towards an existence proof of MacKay’s fixed point. Comm. Math. Phys. 188 (1997), 723–735.CrossRefGoogle Scholar
[5]Chandre, C., Govin, M. and Jauslin, H. R.. KAM–renormalization group analysis of stability in Hamiltonian flows. Phys. Rev. Lett. 79 (1997), 3881–3884.Google Scholar
[6]Abad, J. J., Koch, H. and Wittwer, P.. A renormalization group for Hamiltonians: numerical results. Nonlinearity 11 (1998), 1185–1194.CrossRefGoogle Scholar
[7]Chandre, C. and Jauslin, H. R.. Renormalization-group analysis for the transition to chaos in Hamiltonian systems. Phys. Rep. 365 (2002), 1–64.CrossRefGoogle Scholar
[8]Koch, H.. A renormalization group for Hamiltonians, with applications to KAM tori. Ergod. Th. & Dynam. Sys. 19 (1999), 1–47.CrossRefGoogle Scholar
[9]Koch, H.. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete Contin. Dyn. Syst. Ser. A 8 (2002), 633–646.CrossRefGoogle Scholar
[10]Koch, H.. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete Contin. Dyn. Syst. 11 (2004), 881–909.CrossRefGoogle Scholar
[11]Koch, H.. Existence of critical invariant tori. Ergod. Th. & Dynam. Sys. 28 (2008), 1879–1894.CrossRefGoogle Scholar
[12]Gaidashev, D. G.. Renormalization of isoenergetically degenerate Hamiltonian flows and associated bifurcations of invariant tori. Discrete Contin. Dyn. Syst. Ser. A 13 (2005), 63–102.CrossRefGoogle Scholar
[13]Dias, J. L.. Renormalisation scheme for vector fields on T 2 with a Diophantine frequency. Nonlinearity 15 (2002), 665–679.CrossRefGoogle Scholar
[14]Kocić, S.. Renormalization of Hamiltonians for Diophantine frequency vectors and KAM tori. Nonlinearity 18 (2005), 2513–2544.CrossRefGoogle Scholar
[15]Khanin, K., Dias, J. L. and Marklof, J.. Multidimensional continued cractions, dynamical renormalization and KAM theory. Comm. Math. Phys. 270 (2007), 197–231.CrossRefGoogle Scholar
[16]Koch, H.. Renormalization of vector fields. Holomorphic Dynamics and Renormalization (Fields Institute Communications). Eds. Lyubich, M. and Yampolsky, M.. American Mathematical Society, Providence, RI, 2008, pp. 269–330.Google Scholar
[17]Koch, H. and Kocić, S.. Renormalization of vector fields and Diophantine invariant tori. Ergod. Th. & Dynam. Sys. 28 (2008), 1559–1585.CrossRefGoogle Scholar
[18]Koch, H. and Dias, J. L.. Renormalization of Diophantine skew flows, with applications to the reducibility problem. Discrete Contin. Dyn. Syst. Ser. A 21 (2008), 477–500.CrossRefGoogle Scholar
[19]Dias, J. L.. Brjuno condition and renormalization for Poincaré flows. Discrete Contin. Dyn. Syst. 15 (2006), 641–656.CrossRefGoogle Scholar
[20]Dias, J. L.. A normal form theorem for Brjuno skew-systems through renormalization. J. Differential Equations 230 (2006), 1–23.CrossRefGoogle Scholar
[21]Gentile, G. and Mastropietro, V.. Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics. A review with some applications. Rev. Math. Phys. 8 (1996), 393–444.CrossRefGoogle Scholar
[22]Brjuno, A. D.. Analytic form of differential equations. I. Tr. Mosk. Mat. Obs. 25 (1971), 119–262. Trans. Moscow Math. Soc. 25 (1973), 131–288.Google Scholar
[23]Brjuno, A. D.. Analytic form of differential equations. II. Tr. Mosk. Mat. Obs. 26 (1972), 199–239. Trans. Moscow Math. Soc. 26 (1974), 199–239.Google Scholar
[24]Pöschel, J.. Integrability of Hamiltonian systems on Cantor sets. Comm. Pure Appl. Math. 35 (1989), 653–696.CrossRefGoogle Scholar
[25]Rüssmann, H.. On the frequencies of quasi periodic solutions of analytic nearly integrable Hamiltonian systems. Seminar on Dynamical Systems: Euler International Mathematical Institute, St. Petersburg, 1991 (Progress in Nonlinear Differential Equations and Their Applications, 12). Eds. Kuksin, S., Lazutkin, V. and Pöschel, J.. Birkhäuser, Basel, 1994, pp. 160–183.CrossRefGoogle Scholar
[26]Ecalle, J. and Valet, B.. Correction and linearization of resonant vector fields and diffeomorphisms. Math. Z. 229 (1998), 249–318.CrossRefGoogle Scholar
[27]Berretti, A. and Gentile, G.. Bryuno function and the standard map. Comm. Math. Phys. 220 (2001), 623–656.CrossRefGoogle Scholar
[28]Rüssmann, H.. Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Regul. Chaotic Dyn. 6 (2001), 119–204.CrossRefGoogle Scholar
[29]Gentile, G.. Degenerate lower-dimensional tori under the Bryuno condition. Ergod. Th. & Dynam. Sys. 27 (2007), 427–457.CrossRefGoogle Scholar
[30]Gentile, G., Bartuccelli, M. V. and Deane, J. H. B.. Quasi-periodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems. J. Math. Phys. 47 (2006), 072702.1–072702.10.CrossRefGoogle Scholar
[31]Lagarias, J. C.. Geodesic multidimensional continued fractions. Proc. London Math. Soc. 69 (1994), 464–488.CrossRefGoogle Scholar
[32]Kleinbock, D. Y. and Margulis, G. A.. Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann. of Math. (2) 148 (1998), 339–360.CrossRefGoogle Scholar