Abstract
We study the problem of existence of response solutions for a real-analytic one-dimensional system, consisting of a rotator subject to a small quasi-periodic forcing with Bryuno frequency vector. We prove that at least one response solution always exists, without any assumption on the forcing besides smallness and analyticity. This strengthens the results available in the literature, where generic non-degeneracy conditions are assumed. The proof is based on a diagrammatic formalism and relies on renormalisation group techniques, which exploit the formal analogy with problems of quantum field theory; a crucial role is played by remarkable identities between classes of diagrams.
Similar content being viewed by others
References
Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maĭer, A.G.: Theory of bifurcations of dynamic systems on a plane. Jerusalem-London: Halsted Press/Israel Program for Scientific Translations, 1973
Berretti A., Gentile G.: Bryuno function and the standard map. Commun. Math. Phys. 220(3), 623–656 (2001)
Blekman I.I.: Synchronization in Science and Technology. ASME Press, New York (1988)
Boccaletti S., Kurths J., Osipov G., Valladares D.L., Zhou C.S.: The synchronization of chaotic systems. Phys. Rep. 366, 1–101 (2002)
Bricmont J., Gawe¸dzki K., Kupiainen A.: KAM theorem and quantum field theory. Commun. Math. Phys. 201(3), 699–727 (1999)
Broer H.W., Hanssmann H., You J.: Bifurcations of normally parabolic tori in Hamiltonian systems. Nonlinearity 18(4), 1735–1769 (2005)
Bryuno, A.D.: Analytic form of differential equations. I, II. Trudy Moskov. Mat. Obšč. 25, 119–262 (1971), ibid. 26, 199–239 (1972). English translations: Trans. Moscow Math. Soc. 25(1971), 131–288 (1973); ibid. 26(1972), 199–239 (1974)
Cheng Ch.-Q.: Birkhoff-Kolmogorov-Arnold-Moser tori in convex Hamiltonian systems. Commun. Math. Phys. 177(3), 529–559 (1996)
Chow, S.-N., Hale, J.K.: Methods of bifurcation theory. Grundlehren der Mathematischen Wissenschaften 251, New York-Berlin: Springer-Verlag, 1982
Corsi L., Gentile G.: Melnikov theory to all orders and Puiseux series for subharmonic solutions. J. Math. Phys. 49(11), 112701 (2008)
Davie A.M.: The critical function for the semistandard map. Nonlinearity 7(1), 219–229 (1994)
De Simone E., Kupiainen A.: The KAM theorem and renormalization group. Erg. Th. Dyna. Sys. 29(2), 419–431 (2009)
Eliasson, L.H.: Absolutely convergent series expansions for quasi periodic motions. Math. Phys. Electron. J. 2, Paper 4, 33 pp. (1996) (electronic)
Franz A.: Hausdorff dimension estimates for non-injective maps using the cardinality of the pre-image sets. Nonlinearity 13(5), 1425–1438 (2000)
Gallavotti G.: Twistless KAM tori. Commun. Math. Phys. 164(1), 145–156 (1994)
Gallavotti, G., Bonetto, F., Gentile, G.: Aspects of ergodic, qualitative and statistical theory of motion. Texts and Monographs in Physics, Berlin: Springer-Verlag, 2004
Gallavotti G., Gentile G.: Hyperbolic low-dimensional invariant tori and summation of divergent series. Commun. Math. Phys. 227(3), 421–460 (2002)
Gallavotti, G., Gentile, G., Giuliani, A.: Fractional Lindstedt series. J. Math. Phys. 47(1), 012702, 33 pp. (2006)
Gallavotti G., Gentile G., Giuliani A.: Resonances within chaos. Chaos 22(2), 026108 (2012)
Gallavotti, G., Gentile, G., Mastropietro, V.: Field theory and KAM tori. Math. Phys. Electron. J. 1, Paper 5, 13 pp. (1995) (electronic)
Gentile G.: Quasi-periodic solutions for two level systems. Commun. Math. Phys. 242(1-2), 221–250 (2003)
Gentile G.: Resummation of perturbation series and reducibility for Bryuno skew-product flows. J. Stat. Phys. 125(2), 321–361 (2006)
Gentile G.: Degenerate lower-dimensional tori under the Bryuno condition. Erg. Th. Dyna. Sys. 27(2), 427–457 (2007)
Gentile G.: Quasi-periodic motions in strongly dissipative forced systems. Erg. Th. Dyna. Sys. 30(5), 1457–1469 (2010)
Gentile G.: Construction of quasi-periodic response solutions in forced strongly dissipative systems. Forum Math. 24(4), 791–808 (2012)
Gentile G., Cortes D.A., Barata J.C.A.: Stability for quasi-periodically perturbed Hill’s equations. Commun. Math. Phys. 260(2), 403–443 (2005)
Gentile G., Gallavotti G.: Degenerate elliptic resonances. Commun. Math. Phys. 257(2), 319–362 (2005)
Guckenheimer, J., Holmes, Ph.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied Mathematical Sciences 42, New York: Springer-Verlag, 1990
Lochak P.: Tores invariants à torsion évanescente dans les systèmes hamiltoniens proches de l’intégrable. C. R. Acad. Sci. Paris Sér. I Math. 327(9), 833–836 (1998)
Mastropietro V.: Non-perturbative renormalization. World Scientific Publishing, Hackensack, NJ (2008)
Pöschel J.: On invariant manifolds of complex analytic mappings near fixed points. Exposition. Math. 4(2), 97–109 (1986)
Pikovsky, A., Rosenblum, M., Kurths, J.: Synchronization. A universal concept in nonlinear sciences. Cambridge Nonlinear Science Series 12, Cambridge: Cambridge University Press, 2001
Reitmann, V.: Dimension estimates for invariant sets of dynamical systems. In: Ergodic theory, analysis, and efficient simulation of dynamical systems, Ed. B. Fiedler, Berlin: Springer, 2001, pp. 585–615
Ruelle D., Wilkinson A.: Absolutely singular dynamical foliations. Commun. Math. Phys. 219, 481–487 (2001)
Shub M., Wilkinson A.: Pathological foliations and removable zero exponents. Invent. Math. 139, 495–508 (2000)
Siegel C.L.: Iteration of analytic functions. Ann. Math. 43, 607–612 (1942)
Yoccoz J.-C.: Théorème de Siegel, nombres de Bruno et polinômes quadratiques. Astérisque 231, 3–88 (1995)
Zhang Zh.F., Li B.Y.: High order Melnikov functions and the problem of uniformity in global bifurcation. Ann. Mat. Pura Appl. (4) 161, 181–212 (1992)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Gallavotti
Rights and permissions
About this article
Cite this article
Corsi, L., Gentile, G. Oscillator Synchronisation under Arbitrary Quasi-periodic Forcing. Commun. Math. Phys. 316, 489–529 (2012). https://doi.org/10.1007/s00220-012-1548-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-012-1548-2