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Absolutely convergent series expansions for quasi periodic motions (1996)

by L H Eliasson
Venue:Math. Phys. Electron. J
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Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable hamiltonian systems. A review.

by Giovanni Gallavotti , 1993
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Abstract - Cited by 66 (11 self) - Add to MetaCart
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Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics -- A review with some applications

by G. Gentile, V. Mastropietro , 1995
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Abstract - Cited by 55 (22 self) - Add to MetaCart
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A renormalization group for Hamiltonians, with applications to KAM theory. Ergodic Theory Dynam

by H Koch - Systems , 1999
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Hyperbolic Low-Dimensional Invariant Tori and Summations of Divergent Series

by G. Gallavotti, G. Gentile - Comm. Math. Phys , 2001
"... We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance conditions, such tori exist and can be identified through th ..."
Abstract - Cited by 26 (19 self) - Add to MetaCart
We consider a class of a priori stable quasi-integrable analytic Hamiltonian systems and study the regularity of low-dimensional hyperbolic invariant tori as functions of the perturbation parameter. We show that, under natural nonresonance conditions, such tori exist and can be identified through the maxima or minima of a suitable potential. They are analytic inside a disc centered at the origin and deprived of a region around the positive or negative real axis with a quadratic cusp at the origin. The invariant tori admit an asymptotic series at the origin with Taylor coe#cients that grow at most as a power of a factorial and a remainder that to any order N is bounded by the (N + 1)-st power of the argument times a power of N !. We show the existence of a summation criterion of the (generically divergent) series, in powers of the perturbation size, that represent the parametric equations of the tori by following the renormalization group methods for the resummations of perturbative series in quantum field theory. 1.
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...s. The definition and usage of graphical tools based on tree graphs 12/luglio/2001; 13:05 4 in the context of KAM theory has been advocated recently in the literature as an interpretation of the work =-=[E]-=-; see for instance [G1], [GG], [BGGM] and [BaG]. A tree # (see Figure 2 below) is defined as a partially ordered set of points, connected by lines. The lines are oriented toward the root, which is the...

KAM Theorem and Quantum Field Theory

by Jean Bricmont, Krzysztof Gawedzki, Antti Kupiainen , 1998
"... We give a new proof of the KAM theorem for analytic Hamiltonians. The proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale. The crucial cancellations of resonances are show ..."
Abstract - Cited by 26 (1 self) - Add to MetaCart
We give a new proof of the KAM theorem for analytic Hamiltonians. The proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale. The crucial cancellations of resonances are shown to follow from the Ward identities expressing the translation invariance of the corresponding field theory.

Quasi-Periodic Solutions for Two-Level Systems

by Guido Gentile
"... We consider the Schörodinger equation for a class of two-level atoms in a quasi-periodic external field in the case in which the spacing 2 epsilon between the two unperturbed energy levels is small. We prove the existence of quasi-periodic solutions for a Cantor set E of values of epsilon around the ..."
Abstract - Cited by 24 (18 self) - Add to MetaCart
We consider the Schörodinger equation for a class of two-level atoms in a quasi-periodic external field in the case in which the spacing 2 epsilon between the two unperturbed energy levels is small. We prove the existence of quasi-periodic solutions for a Cantor set E of values of epsilon around the origin which is of positive Lebesgue measure: such solutions can be obtained from the formal power series by a suitable resummation procedure. The set E can be characterized by requesting infinitely many Diophantine conditions of Mel'nikov type.
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...direction can be found in [4] and [5], were the case of periodic external field was considered. To prove our results, we use a version of the techniques introduced in classical mechanics by Eliasson, =-=[8]-=-, in order to study KAM-type problems. Such techniques were further developed (see [10], [14], [11], [12] and papers quoted therein), by emphasizing the analogy with the methods of quantum field theor...

Periodic solutions for completely resonant nonlinear wave equations

by Guido Gentile, Vieri Mastropietro, Michela Procesi , 2004
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Abstract - Cited by 22 (9 self) - Add to MetaCart
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Scaling Properties For The Radius Of Convergence Of Lindstedt Series: Generalized Standard Maps

by Alberto Berretti, Guido Gentile - J. Math. Pures Appl , 2000
"... . For a class of symplectic two-dimensional maps which generalize the standard map by allowing more general nonlinear terms, the radius of convergence of the Lindstedt series describing the homotopically non-trivial invariant curves is proved to satisfy a scaling law as the complexied rotation numbe ..."
Abstract - Cited by 22 (13 self) - Add to MetaCart
. For a class of symplectic two-dimensional maps which generalize the standard map by allowing more general nonlinear terms, the radius of convergence of the Lindstedt series describing the homotopically non-trivial invariant curves is proved to satisfy a scaling law as the complexied rotation number tends to a rational value non-tangentially to the real axis, thus generalizing previous results of the authors. The function conjugating the dynamics to rotations by ! possesses a limit which is explicitly computed and related to hyperelliptic functions in the case of nonlinear terms which are trigonometric polynomials. The case of the standard map is shown to be non-generic. 1. Introduction In this paper we generalize the results of [1] considering maps of the kind T ";f : 8 > < > : x 0 = x + y + "f(x); y 0 = y + "f(x); (1.1) where f(x) is a 2-periodic function of x, analytic in a strip S = fj Im(x)j < g of width 2 around the real x axis. The nonlinear term f(x) can be expande...
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...d, in the specic case of the standard map (that is, f(x) = sin x); thesrst proof of existence of invariant tori for Hamiltonian systems by tree expansions of the Lindstedt series is due to Eliasson, [=-=1-=-5]. The trees are dened as in [1] (see also [2] and the references quoted therein). Note that, as our perturbation f(x) is more general than the one considered in [1], [2], the mode label u associate...

A Lecture on the Classical KAM-Theorem

by Jürgen Pöschel , 2009
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Abstract - Cited by 21 (1 self) - Add to MetaCart
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...mplicit function theorem in tame Frechet spaces [4]. Recently, Salamon and Zehnder [21] gave a proof that avoids coordinate transformations altogether and works in configuration space. Also, Eliasson =-=[5]-=- described a way of using power series expansions and majorant techniques in a very tricky way. But here we stick to the traditional method of proof, as it probably is the most transparent way to get ...

Kolmogorov theorem and classical perturbation theory

by Antonio Giorgilli, Ugo Locatelli, A. Giorgilli, U. Locatelli - J. of App. Math. and Phys. (ZAMP , 1997
"... Abstract. We reconsider the original proof of Kolmogorov’s theorem in the light of classical perturbation methods based on expansions in some parameter. This produces quasiperiodic solutions on invariant tori in the form of power series in a small parameter, that we prove to be absolutely convergent ..."
Abstract - Cited by 21 (16 self) - Add to MetaCart
Abstract. We reconsider the original proof of Kolmogorov’s theorem in the light of classical perturbation methods based on expansions in some parameter. This produces quasiperiodic solutions on invariant tori in the form of power series in a small parameter, that we prove to be absolutely convergent.
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...to Cauchy. However, Moser shows that an indirect proof of convergence can be given using the method of Kolmogorov. This complicated state of affairs has been clarified in the recent works of Eliasson =-=[10]-=-[11][12], Gallavotti[13][14], and Chierchia and Falcolini[15] (see also [16], [17], [18]). Indeed, Eliasson proved that in Lindstedt’s expansions there are cancellations of critical terms. In slightly...

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