520 research outputs found
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
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
The revision and extension of the R-MS ring for time delay systems
This paper is aimed at reviewing the ring of retarded quasipolynomial meromorphic functions (R-MS) that was recently introduced as a convenient control design tool for linear, time-invariant time delay systems (TDS). It has been found by the authors that the original definition does not constitute a ring and has some essential deficiencies, and hence it could not be used for an algebraic control design without a thorough reformulation which i.e. extends the usability to neutral TDS and to those with distributed delays. This contribution summarizes the original definition of RMS, simply highlights its deficiencies via examples, and suggests a possible new extended definition. Hence, the new ring of quasipolynomial meromorphic functions (R-QM) is established to avoid confusion. The paper also investigates and introduces selected algebraic properties supported by some illustrative examples and concisely outlines its use in controller design.European Regional Development Fund under the project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19
On a Frobenius Problem for Polynomials
We extend the famous diophantine Frobenius problem to a ring of polynomials over a field~k. Similar to the classical problem we show that the n = 2 case of the Frobenius problem for polynomials is easy to solve. In addition, we translate a few results from the Frobenius problem over ℤ to k[t] and give an algorithm to solve the Frobenius problem for polynomials over a field k of sufficiently large size
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