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1: Fractional Lindstedt series Fractional Lindstedt series
, 2005
"... Abstract. The parametric equations of the surfaces on which highly resonant quasiperiodic motions develop (lowerdimensional tori) cannot be analytically continued, in general, in the perturbation parameter ε, i.e. they are not analytic functions of ε. However rather generally quasiperiodic motions ..."
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Abstract. The parametric equations of the surfaces on which highly resonant quasiperiodic motions develop (lowerdimensional tori) cannot be analytically continued, in general, in the perturbation parameter ε, i.e. they are not analytic functions of ε. However rather generally quasiperiodic motions whose frequencies satisfy only one rational relation (“resonances of order 1”) admit formal perturbation expansions in terms of a fractional power of ε depending on the degeneration of the resonance. We find conditions for this to happen, and in such a case we prove that the formal expansion is convergent after suitable resummation. 1.
Borel summability and Lindstedt series
 Comm. Math. Phys
"... Abstract. Resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation. Such series may be, sometimes, summed via suitable sum rules defining C ∞ functions ..."
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Cited by 11 (4 self)
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Abstract. Resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation. Such series may be, sometimes, summed via suitable sum rules defining C
Lindstedt series and Kolmogorov theorem
, 2995
"... the KAM theorem from a combinatorial viewpoint. ..."
Lindstedt Series for Lower Dimensional Tori
"... Introduction One of the first methods to compute quasiperiodic orbits (i. e. invariant tori with linear motions on them) was the Lindstedt method (see [12] Vol. 2) which produces an expansion of the quasiperiodic orbit with a fixed frequency in powers of a small parameter measuring the distance t ..."
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Cited by 16 (5 self)
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Diophantine condition). A more recent development is the proof of KAM theorem using directly the compensations in the Lindstedt series (see [3], [5], [2] and the lectures of Gallavotti in this proceedings). When the number of independent frequencies is less than the number of degrees of freedom the situation
KAM theory in configuration space and cancellations in the Lindstedt series
"... The KAM theorem for analytic quasiintegrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for any quasiperiodic solution with Diophantine frequency vector converges. If one studies the Lindstedt series by following a perturbation theory approach, one ..."
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Cited by 2 (1 self)
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The KAM theorem for analytic quasiintegrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for any quasiperiodic solution with Diophantine frequency vector converges. If one studies the Lindstedt series by following a perturbation theory approach, one
The Shape Of Analyticity Domains Of Lindstedt Series: The Standard Map
, 2000
"... . The analyticity domains of the Lindstedt series for the standard map are studied numerically using Pade approximants to model their natural boundaries. We show that if the rotation number is a Diophantine number close to a rational value p/q, then the radius of convergence of the Lindstedt ser ..."
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Cited by 5 (4 self)
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. The analyticity domains of the Lindstedt series for the standard map are studied numerically using Pade approximants to model their natural boundaries. We show that if the rotation number is a Diophantine number close to a rational value p/q, then the radius of convergence of the Lindstedt
Convergence of Lindstedt series for the nonlinear wave equation
 Commun. Pure Appl. Anal
"... (Communicated by Vieri Mastropietro) Abstract. We prove the existence of oscillatory solutions of the nonlinear wave equation, under irrationality conditions stronger than the usual Diophantine one, by perturbative techniques inspired by the Lindstedt series method originally introduced in classica ..."
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Cited by 2 (2 self)
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(Communicated by Vieri Mastropietro) Abstract. We prove the existence of oscillatory solutions of the nonlinear wave equation, under irrationality conditions stronger than the usual Diophantine one, by perturbative techniques inspired by the Lindstedt series method originally introduced
Lindstedt Series Solutions of the FermiPastaUlam Lattice
, 2007
"... We apply the Lindstedt method to the one dimensional FermiPastaUlam β lattice to find fully general solutions to the complete set of equations of motion. The pertubative scheme employed uses ǫ as the expansion parameter, where ǫ is the coefficient of the quartic coupling between nearest neighbors. ..."
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We apply the Lindstedt method to the one dimensional FermiPastaUlam β lattice to find fully general solutions to the complete set of equations of motion. The pertubative scheme employed uses ǫ as the expansion parameter, where ǫ is the coefficient of the quartic coupling between nearest neighbors
Lindstedt series for perturbations of isochronous systems. I. General theory
, 2000
"... . We give a proof of the persistence of invariant tori for analytic perturbations of isochronous systems by using the Lindstedt series expansion for the solutions. With respect to the case of anisochronous systems, there is the additional problem to find the set of allowed rotation vectors for the i ..."
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Cited by 11 (9 self)
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. We give a proof of the persistence of invariant tori for analytic perturbations of isochronous systems by using the Lindstedt series expansion for the solutions. With respect to the case of anisochronous systems, there is the additional problem to find the set of allowed rotation vectors
Renormalization Group And Field Theoretic Techniques For The Analysis Of The Lindstedt Series
"... The Lindstedt series were introduced in the XIX century in Astronomy to study perturbatively quasiperiodic motions in Celestial Mechanics. In Mathematical Physics, after getting the attention of Poincare, who studied them widely by pursuing to all orders the analysis of Lindstedt and Newcomb, th ..."
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Cited by 1 (1 self)
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The Lindstedt series were introduced in the XIX century in Astronomy to study perturbatively quasiperiodic motions in Celestial Mechanics. In Mathematical Physics, after getting the attention of Poincare, who studied them widely by pursuing to all orders the analysis of Lindstedt and Newcomb
Results 1  10
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