29,663 research outputs found
Domains of analyticity of Lindstedt expansions of KAM tori in dissipative perturbations of Hamiltonian systems
Many problems in Physics are described by dynamical systems that are
conformally symplectic (e.g., mechanical systems with a friction proportional
to the velocity, variational problems with a small discount or thermostated
systems). Conformally symplectic systems are characterized by the property that
they transform a symplectic form into a multiple of itself. The limit of small
dissipation, which is the object of the present study, is particularly
interesting.
We provide all details for maps, but we present also the modifications needed
to obtain a direct proof for the case of differential equations. We consider a
family of conformally symplectic maps defined on a
-dimensional symplectic manifold with exact symplectic form
; we assume that satisfies
. We assume that the family
depends on a -dimensional parameter (called drift) and also on a small
scalar parameter . Furthermore, we assume that the conformal factor
depends on , in such a way that for we have
(the symplectic case).
We study the domains of analyticity in near of
perturbative expansions (Lindstedt series) of the parameterization of the
quasi--periodic orbits of frequency (assumed to be Diophantine) and of
the parameter . Notice that this is a singular perturbation, since any
friction (no matter how small) reduces the set of quasi-periodic solutions in
the system. We prove that the Lindstedt series are analytic in a domain in the
complex plane, which is obtained by taking from a ball centered at
zero a sequence of smaller balls with center along smooth lines going through
the origin. The radii of the excluded balls decrease faster than any power of
the distance of the center to the origin
Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions
We present theorems which provide the existence of invariant whiskered tori
in finite-dimensional exact symplectic maps and flows. The method is based on
the study of a functional equation expressing that there is an invariant torus.
We show that, given an approximate solution of the invariance equation which
satisfies some non-degeneracy conditions, there is a true solution nearby. We
call this an {\sl a posteriori} approach.
The proof of the main theorems is based on an iterative method to solve the
functional equation.
The theorems do not assume that the system is close to integrable nor that it
is written in action-angle variables (hence we can deal in a unified way with
primary and secondary tori). It also does not assume that the hyperbolic
bundles are trivial and much less that the hyperbolic motion can be reduced to
constant.
The a posteriori formulation allows us to justify approximate solutions
produced by many non-rigorous methods (e.g. formal series expansions, numerical
methods). The iterative method is not based on transformation theory, but
rather on succesive corrections. This makes it possible to adapt the method
almost verbatim to several infinite-dimensional situations, which we will
discuss in a forthcoming paper. We also note that the method leads to fast and
efficient algorithms. We plan to develop these improvements in forthcoming
papers.Comment: To appear in JD
Iterative and Iterative-Noniterative Integral Solutions in 3-Loop Massive QCD Calculations
Various of the single scale quantities in massless and massive QCD up to
3-loop order can be expressed by iterative integrals over certain classes of
alphabets, from the harmonic polylogarithms to root-valued alphabets. Examples
are the anomalous dimensions to 3-loop order, the massless Wilson coefficients
and also different massive operator matrix elements. Starting at 3-loop order,
however, also other letters appear in the case of massive operator matrix
elements, the so called iterative non-iterative integrals, which are related to
solutions based on complete elliptic integrals or any other special function
with an integral representation that is definite but not a Volterra-type
integral. After outlining the formalism leading to iterative non-iterative
integrals,we present examples for both of these cases with the 3-loop anomalous
dimension and the structure of the principle solution in
the iterative non-interative case of the 3-loop QCD corrections to the
-parameter.Comment: 13 pages LATEX, 2 Figure
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
- …