89 research outputs found
A new approach to the parameterization method for Lagrangian tori of hamiltonian systems
We compute invariant Lagrangian tori of analytic Hamiltonian systems by the parameterization method. Under Kolmogorovâs non-degeneracy condition, we look for an invariant torus of the system carrying quasi-periodic motion with fixed frequencies. Our approach consists in replacing the invariance equation of the parameterization of the torus by three conditions which are altogether equivalent to invariance. We construct a quasi-Newton method by solving, approximately, the linearization of the functional equations defined by these three conditions around an approximate solution. Instead of dealing with the invariance error as a single source of error, we consider three different errors that take account of the Lagrangian character of the torus and the preservation of both energy and frequency. The condition of convergence reflects at which level contributes each of these errors to the total error of the parameterization. We do not require the system to be nearly integrable or to be written in action-angle variables. For nearly integrable Hamiltonians, the Lebesgue measure of the holes between invariant tori predicted by this parameterization result is of O(e1/2)O(e1/2) , where ee is the size of the perturbation. This estimate coincides with the one provided by the KAM theorem.Peer ReviewedPostprint (author's final draft
A parameterization method for Lagrangian tori of exact symplectic maps of R2r
We are concerned with analytic exact symplectic maps of endowed with the standard symplectic form. We study the existence of a real analytic torus of dimension , invariant by the map and carrying quasi-periodic motion with a prefixed Diophantine rotation vector. Therefore, this torus is a Lagrangian manifold. We address the problem by the parameterization method in KAM theory. The main aspect of our approach is that we do not look for the parameterization of the torus as a solution of the corresponding invariance equation. Instead, we consider a set of three equations that, all together, are equivalent to the invariance equation. These equations arise from the geometric and dynamical properties of the map and the torus. Suppose that an approximate solution of these equations is known and that a suitable nondegeneracy (twist) condition is satisfied. Then, this system of equations is solved by a quasi-Newton-like method, provided that the initial error is sufficiently small. By âquasi-Newton-likeâ we mean that the convergence is almost quadratic but that at each iteration we have to solve a nonlinear equation. Although it is straightforward to build a quasi-Newton method for the selected set of equations, proceeding in this way we improve the convergence condition. The selected definition of error reflects the level at which the error associated with each of these three equations contributes to the total error. The map is not required to be close to integrable or expressed in action-angle variables. Suppose the map is -close to an integrable one, and consider the portion of the phase space not filled up by Lagrangian invariant tori of the map. Then, the upper bound for the Lebesgue measure of this set that we may predict from the result is of . In light of the classical KAM theory for exact symplectic maps, an upper bound of for this measure is the expected estimate. The result also has some implications for finitely differentiable mapsPeer ReviewedPostprint (author's final draft
Kolmogorov Theorem Revisited
Kolmogorov Theorem on the persistence of invariant tori of real analytic Hamiltonian
systems is revisited. In this paper we are mainly concerned with the lower bound on the
constant of the Diophantine condition required by the theorem. From the existing proofs in the literature, this lower bound turns to be of O(Δ1/4), where Δ is the size of the perturbation.
In this paper, by means of careful (but involved) estimates on Kolmogorovâsmethod, we show that this lower bound can be weakened to be of O(Δ1/2). This condition coincides with the optimal one of KAM Theorem. Moreover, we also obtain optimal estimates for the distance between the actions of the perturbed and unperturbed tori
Numerical computation of normal forms around some periodic orbits of the restricted three body problem
In this paper we introduce a general methodology for computing
(numerically) the normal form around a periodic orbit of an autonomous
analytic Hamiltonian system. The process follows two steps. First,
we expand the Hamiltonian in suitable coordinates around the orbit and
second, we perform a standard normal form scheme, based on the Lie
series method. This scheme is carried out up to some finite order
and, neglecting the remainder, we obtain an accurate description of
the dynamics in a (small enough) neighbourhood of the orbit. In
particular, we obtain the invariant tori that generalize the elliptic
directions of the periodic orbit. On the other hand, bounding the
remainder one obtains lower estimates for the diffusion time around
the orbit.
This procedure is applied to an elliptic periodic orbit of the spatial
Restricted Three Body Problem. The selected orbit belongs to the
Lyapunov family associated to the vertical oscillation of the
equilibrium point . The mass parameter has been chosen such
that is unstable but the periodic orbit is still
stable. This allows to show the existence of regions of effective
stability near for values of bigger that the Routh
critical value. The computations have been done using formal
expansions with numerical coefficients
On the persistence of lower dimensional invariant tori under quasiperiodic perturbations
In this work we consider time dependent quasiperiodic perturbations of autonomous Hamiltonian systems. We focus on the effect that this kind of perturbations has on lower dimensional invariant tori. Our results show that, under standard conditions of analyticity, nondegeneracy and nonresonance, most of these tori survive, adding the frequencies of the perturbation to the ones they already have. The paper also contains estimates on the amount of surviving tori. The worst situation happens when the initial tori are normally elliptic. In this case, a torus (identified by the vector of intrinsic frequencies) can be continued with respect to a perturbative parameter , except for a set of of measure exponentially small with . In case that is fixed (and sufficiently small), we prove the existence of invariant tori for every vector of frequencies close to the one of the initial torus, except for a set of frequencies of measure exponentially small with the distance to the unperturbed torus. As a particular case, if the perturbation is autonomous, these results also give the same kind of estimates on the measure of destroyed tori. Finally, these results are applied to some problems of celestial mechanics, in order to help in the description of the phase space of some concrete models
Effective Stability Around Periodic Orbits of the Spatial RTBP
In this work we study the dynamics around an elliptic periodic orbit of Hamiltonian systems. To this end we have developped an algorithm to compute a normal form (up to a finite order) around this orbit, that gives an accurate description of the dynamics close to it. If the remainder of this normal form can be bounded, it is not difficult to produce explicit bounds of the diffusion time of trajectories starting near the periodic orbit. In order to discuss the effectivity of the method, it will be explained at the same time that it is applied to a concrete example. The one used here has been the Spatial Restricted Three Body Problem (RTBP
On the normal behaviour of partially elliptic lower dimensional tori of hamiltonian systems
The purpose of this paper is to study the dynamics near a reducible
lower dimensional invariant tori of a finite-dimensional autonomous
Hamiltonian system with degrees of freedom. We will focus in
the case in which the torus has (some) elliptic directions.
First, let us assume that the torus is totally elliptic. In this
case, it is shown that the diffusion time (the time to move away from
the torus) is exponentially big with the initial distance to the
torus. The result is valid, in particular, when the torus is of
maximal dimension and when it is of dimension 0 (elliptic point). In
the maximal dimension case, our results coincide with previous
ones. In the zero dimension case, our results improve the existing
bounds in the literature.
Let us assume now that the torus (of dimension , ) is
partially elliptic (let us call to the number of these
directions). In this case we show that, given a fixed number of
elliptic directions (let us call to this number), there
exist a Cantor family of invariant tori of dimension , that
generalize the linear oscillations corresponding to these elliptic
directions. Moreover, the Lebesgue measure of the complementary of
this Cantor set (in the frequency space \RR^{r+m_1}) is proven to
be exponentially small with the distance to the initial torus. This
is a sort of ``Cantorian central manifold'' theorem, in which the
central manifold is completely filled up by invariant tori and it is
uniquely defined.
The proof of these results is based on the construction of suitable
normal forms around the initial torus
Numerical study of rotation numbers and its variationals for parametric families of circle diffeomorphisms
We present a numerical method to compute derivatives of the rotation number for parametric families of circle diffeomorphisms with high accuracy. Our methodology is an extension of an existing approach to compute rotation numbers, that it is based on suitable averages of the iterates of the map and Richardson extrapolation. In order to justify the method, we require the family of maps to be differentiable with respect to the parameters and the rotation number to be Diophantine. The method is used to compute the Taylor expansions of Arnold tongues with high precision
On the numerical computation of Diophantine rotation numbers of analytic circle maps
In this paper we present a numerical method to compute Diophantine rotation numbers of circle maps with high accuracy. We mainly focus on analytic circle diffeomorphisms, but the method also works in the case of (enough) finite differentiability. The keystone of the method is that, under these conditions, the map is conjugate to a rigid rotation of the circle. Moreover, albeit it is not fully justified by our construction, the method turns to be quite efficient for computing rational rotation numbers. We discuss the method through several numerical examples
Effective reducibility of quasiperiodic linear equations close to constant coefficients
Let us consider the differential equation
where is an elliptic constant matrix and depends on time in a
quasiperiodic (and analytic) way. It is also assumed that the eigenvalues
of and the basic frequencies of satisfy a diophantine condition.
Then it is proved that this system can be reduced to
where is exponentially small in , and
the linear change of variables that performs such reduction is
also quasiperiodic with the same basic frequencies than .
The results are illustrated and discussed in a practical example
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