Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T21:26:09.493Z 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: koch@math.utexas.edu, kocic@math.utexas.edu)
SAŠA KOCIĆ
Affiliation:
Department of Mathematics, University of Texas at Austin, 1 University Station C1200, Austin, TX 78712, USA (email: koch@math.utexas.edu, kocic@math.utexas.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

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.

Information

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

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