11,495 research outputs found
Hyperbolic low-dimensional invariant tori and summations of divergent series
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 coefficients 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 . 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 theoryComment: 32 pages, 5 figure
Fractional Lindstedt series
The parametric equations of the surfaces on which highly resonant
quasi-periodic motions develop (lower-dimensional tori) cannot be analytically
continued, in general, in the perturbation parameter, i.e. they are not
analytic functions of the perturbation parameter. However rather generally
quasi-periodic motions whose frequencies satisfy only one rational relation
("resonances of order 1") admit formal perturbation expansions in terms of a
fractional power of the perturbation parameter, 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.Comment: 40 pages, 6 figure
Methods for the analysis of the Lindstedt series for KAM tori and renormalizability in classical mechanics
This paper consists in a unified exposition of methods and techniques of the
renormalization group approach to quantum field theory applied to classical
mechanics, and in a review of results: (1) a proof of the KAM theorem, by
studing the perturbative expansion (Lindstedt series) for the formal solution
of the equations of motion; (2) a proof of a conjecture by Gallavotti about the
renormalizability of isochronous hamiltonians, i.e. the possibility to add a
term depending only on the actions in a hamiltonian function not verifying the
anisochrony condition so that the resulting hamiltonian is integrable. Such
results were obtained first by Eliasson; however the difficulties arising in
the study of the perturbative series are very similar to the problems which one
has to deal with in quantum field theory, so that the use the methods which
have been envisaged and developed in the last twenty years exactly in order to
solve them allows us to obtain unified proofs, both conceptually and
technically. In the final part of the review, the original work of Eliasson is
analyzed and exposed in detail; its connection with other proofs of the KAM
theorem based on his method is elucidated.Comment: 58, compile with dvips to get the figure
Resummation of perturbation series and reducibility for Bryuno skew-product flows
We consider skew-product systems on T^d x SL(2,R) for Bryuno base flows close
to constant coefficients, depending on a parameter, in any dimension d, and we
prove reducibility for a large measure set of values of the parameter. The
proof is based on a resummation procedure of the formal power series for the
conjugation, and uses techniques of renormalisation group in quantum field
theory.Comment: 30 pages, 12 figure
Melnikov's approximation dominance. Some examples
We continue a previous paper to show that Mel'nikov's first order formula for
part of the separatrix splitting of a pendulum under fast quasi periodic
forcing holds, in special examples, as an asymptotic formula in the forcing
rapidity.Comment: 46 Kb; 9 pages, plain Te
Homoclinic splitting, II. A possible counterexample to a claim by Rudnev and Wiggins on Physica D
Results in the mentioned paper do not seem valid.Comment: 2 pages, plain Te
Pendulum: separatrix splitting
An exact expression for the determinant of the splitting matrix is derived:
it allows us to analyze the asympotic behaviour needed to amend the large
angles theorem proposed in Ann. Inst. H. Poincar\'e, B-60, 1, 1994. The
asymptotic validity of Melnokov's formulae is proved for the class of models
considered, which include polynomial perturbations.Comment: 30 pages, one figur
A fluctuation theorem in a random environment
A simple class of chaotic systems in a random environment is considered and
the fluctuation theorem is extended under the assumption of reversibility.Comment: 9 page
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