Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-22T02:13:46.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Normal forms for perturbations of systems possessing a Diophantine invariant torus

Published online by Cambridge University Press:  12 December 2017

JESSICA ELISA MASSETTI
JESSICA ELISA MASSETTI*
Affiliation:
Università degli Studi Roma Tre, Dipartimento di Matematica e Fisica, Largo San L. Murialdo 1, 00146 Roma, Italy email jmassetti@mat.uniroma3.it
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 give a new proof of Moser’s 1967 normal-form theorem for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. The proposed approach, based on an inverse function theorem in analytic class, is flexible and can be adapted to several contexts. This allows us to prove in a unified framework the persistence, up to finitely many parameters, of Diophantine quasi-periodic normally hyperbolic reducible invariant tori for vector fields originating from dissipative generalizations of Hamiltonian mechanics. As a byproduct, generalizations of Herman’s twist theorem and Rüssmann’s translated curve theorem are proved.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 8 , August 2019 , pp. 2176 - 2222
Copyright
© Cambridge University Press, 2017 

References

Abraham, R. and Robbin, J.. Transversal Mappings and Flows. W. A. Benjamin, New York, 1967, with an appendix by Al Kelley.Google Scholar
Arnol’d, V. I.. Small denominators. I. Mapping the circle onto itself. Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961), 21–86.Google Scholar
Bost, J.-B.. Tores invariants des systèmes dynamiques Hamiltoniens [d’après Kolmogorov, Arnold, Moser, Rüssmann, Zender, Herman, Pöschel, …]. Séminaires Bourbaki, Vol. 1984/85, Exp. No. 639. Astérisque 133–134 (1986), 113–157.Google Scholar
Broer, H. W., Huitema, G. B., Takens, F. and Braaksma, B. L. J.. Unfoldings and bifurcations of quasi-periodic tori. Mem. Amer. Math. Soc. 83(421) (1990).Google Scholar
Celletti, A. and Chierchia, L.. Quasi-periodic attractors in celestial mechanics. Arch. Ration. Mech. Anal. 191 (2009), 311–345.Google Scholar
Chenciner, A.. Bifurcations de points fixes elliptiques. I. Courbes invariantes. Publ. Math. Inst. Hautes Études Sci. 61 (1985), 67–127.Google Scholar
Chenciner, A.. Bifurcations de points fixes elliptiques. II. Orbites periodiques et ensembles de Cantor invariants. Invent. Math. 80(1) (1985), 81–106.Google Scholar
Chierchia, L.. KAM lectures. Dynamical Systems. Part I (Publ. Centro di Ricerca Matematica Ennio De Giorgi) . Scuola Normale Superiore, Pisa, 2003, pp. 1–55.Google Scholar
Correia, A. C. M., Laskar, J. and Dotson, R.. Tidal evolution of exoplanets. Exoplanets. University of Arizona Press, Tucson, AZ, 2010, pp. 239–266.Google Scholar
Féjoz, J.. Démonstration du ‘théorème d’Arnol’d sur la stabilité du système planétaire (d’après Michael Herman). Ergod. Th. & Dynam. Syst. 24(5) (2004), 1521–1582.Google Scholar
Féjoz, J.. Mouvements périodiques et quasi-périodiques dans le problème des $n$ corps, Mémoire d’habilitation à diriger des recherches, 2010. UPMC.Google Scholar
Féjoz, J.. A proof of the invariant torus theorem of Kolmogorov. Regul. Chaotic Dyn. 17(1) (2012), 1–5.Google Scholar
Féjoz, J.. Introduction to KAM theory with a view to celestial mechanics. Variational Methods (Radon Series on Computatinal and Applied Mathematics, 18) . De Gruyter, Berlin, 2017, pp. 387–433.Google Scholar
Fritzsche, K. and Grauert, H.. From Holomorphic Functions to Complex Manifolds (Graduate Texts in Mathematics, 213) . Springer, New York, 2002.Google Scholar
Hamilton, R.. The implicit function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.) 7(1) (1982), 65–222.Google Scholar
Herman, M. R. and Sergeraert, F.. Sur un théorème d’Arnold et Kolmogorov. C. R. Acad. Sci. Paris Sér. A 273 (1971), 409–411.Google Scholar
Massetti, J. E.. A normal form à la Moser for diffeomorphisms and a generalization of Rüssmann’s translated curve theorem to higher dimensions. Anal. PDE 11(1) (2018), 149–170.Google Scholar
Meyer, K. R.. The implicit function theorem and analytic differential equations. Dynamical Systems—Warwick 1974 (Proc. Symp. Appl. Topology and Dynamical Systems, University of Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday) (Lecture Notes in Mathematics, 468) . Springer, Berlin, 1975, pp. 191–208.Google Scholar
Moser, J.. Convergent series expansions for quasi-periodic motions. Math. Annal. 169 (1967), 136–176.Google Scholar
Pöschel, J.. On elliptic lower-dimensional tori in Hamiltonian systems. Math. Z. 202(4) (1989), 559–608.Google Scholar
Pöschel, J.. A lecture on the classical KAM theorem. Smooth Ergodic Theory and its Applications (Seattle, WA, 1999) (Proceedings of Symposia in Pure Mathematics, 69) . American Mathematical Society, Providence, RI, 2001, pp. 707–732.Google Scholar
Pöschel, J.. KAM à la R. Regul. Chaotic Dyn. 16(1–2) (2011), 17–23.Google Scholar
Rüssmann, H.. Kleine Nenner. I. Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes. Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1970 (1970), 67–105.Google Scholar
Rüssmann, H.. On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus. Dynamical Systems, Theory and Applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974) (Lecture Notes in Physics, 38) . Springer, Berlin, 1975, pp. 598–624.Google Scholar
Sevryuk, M. B.. The lack-of-parameters problem in the KAM theory revisited. Hamiltonian Systems with Three or More Degrees of Freedom (S’Agaró, 1995) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533) . Kluwer, Dordrecht, 1999, pp. 568–572.Google Scholar
Sevryuk, M. B.. Partial preservation of frequencies in KAM theory. Nonlinearity 19(5) (2006), 1099–1140.Google Scholar
Stefanelli, L. and Locatelli, U.. Kolmogorov’s normal form for equations of motion with dissipative effects. Discrete Contin. Dyn. Syst. Ser. B 17(7) (2012), 2561–2593.Google Scholar
Wagener, F.. A parametrised version of Moser’s modifying terms theorem. Discrete Contin. Dyn. Syst. Ser. S 3(4) (2010), 719–768.Google Scholar
Yoccoz, J-C.. Travaux de Herman sur les tores invariants. Séminaire Bourbaki, Vol. 1991/92, Exp. No. 754. Astérisque 206 (1992), 311–344.Google Scholar
Zehnder, E.. Generalized implicit function theorem with applications to some small divisor problems, I. Comm. Pure Appl. Math. 28 (1975), 91–140.Google Scholar
Zehnder, E.. Generalized implicit function theorem with applications to some small divisor problems, II. Comm. Pure Appl. Math. 29 (1976), 49–111.Google Scholar