100 research outputs found
Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points
We study the Gevrey character of a natural parameterization of one
dimensional invariant manifolds associated to a parabolic direction of fixed
points of analytic maps, that is, a direction associated with an eigenvalue
equal to . We show that, under general hypotheses, these invariant manifolds
are Gevrey with type related to some explicit constants. We provide examples of
the optimality of our results as well as some applications to celestial
mechanics, namely, the Sitnikov problem and the restricted planar three body
problem
Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices
We construct quasi-periodic and almost periodic solutions for coupled
Hamiltonian systems on an infinite lattice which is translation invariant. The
couplings can be long range, provided that they decay moderately fast with
respect to the distance. For the solutions we construct, most of the sites are
moving in a neighborhood of a hyperbolic fixed point, but there are oscillating
sites clustered around a sequence of nodes. The amplitude of these oscillations
does not need to tend to zero. In particular, the almost periodic solutions do
not decay at infinity. We formulate an invariance equation. Solutions of this
equation are embeddings of an invariant torus on which the motion is conjugate
to a rotation. We show that, if there is an approximate solution of the
invariance equation that satisfies some non-degeneracy conditions, there is a
true solution close by. The proof of this \emph{a-posteriori} theorem is based
on a Nash-Moser iteration, which does not use transformation theory. Simpler
versions of the scheme were developed in E. Fontich, R. de la Llave,Y. Sire
\emph{J. Differential. Equations.} {\bf 246}, 3136 (2009). One technical tool,
important for our purposes, is the use of weighted spaces that capture the idea
that the maps under consideration are local interactions. Using these weighted
spaces, the estimates of iterative steps are similar to those in finite
dimensional spaces. In particular, the estimates are independent of the number
of nodes that get excited. Using these techniques, given two breathers, we can
place them apart and obtain an approximate solution, which leads to a true
solution nearby. By repeating the process infinitely often, we can get
solutions with infinitely many frequencies which do not tend to zero at
infinity.Comment: This is a revised version of the paper located at
http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=12-2
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
Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results
We study the problem of exponentially small splitting of separatrices of one
degree of freedom classical Hamiltonian systems with a non-autonomous
perturbation fast and periodic in time. We provide a result valid for general
systems which are algebraic or trigonometric polynomials in the state
variables. It consists on obtaining a rigorous proof of the asymptotic formula
for the measure of the splitting. We obtain that the splitting has the
asymptotic behavior ,
identifying the constants in terms of the system features.
We consider several cases. In some cases, assuming the perturbation is small
enough, the values of coincide with the classical Melnikov approach.
We identify the limit size of the perturbation for which this theory holds
true. However for the limit cases, which appear naturally both in averaging and
bifurcation theories, we encounter that, generically, and are not
well predicted by Melnikov theory
Differentiable invariant manifolds of nilpotent parabolic points
We consider a map of class with a fixed point of parabolic type
whose differential is not diagonalizable and we study the existence and
regularity of the invariant manifolds associated with the fixed point using the
parameterization method. Concretely, we show that under suitable conditions on
the coefficients of , there exist invariant curves of class away from
the fixed point, and that they are analytic when is analytic. The
differentiability result is obtained as an application of the fiber contraction
theorem. We also provide an algorithm to compute an approximation of a
parameterization of the invariant curves and a normal form of the restricted
dynamics of on them.Comment: 43 page
Exponentially small splitting of invariant manifolds of parabolic points
We consider families of one and a half degrees of freedom Hamiltonians with high frequency periodic dependence on time, which are perturbations of an autonomous system. We suppose that the origin is a parabolic xed point with non-diagonalizable linear part and that the unperturbed system has a homoclinic connexion associated to it. We provide a set of hypotheses under which the splitting is exponentially small and is given by the Poincaré-Melnikov function
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