Absolutely convergent series expansions for quasi-periodic motions,

"... 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 ..."

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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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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 ..."

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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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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 ..."

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. 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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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 ..."

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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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