Effective Stability for Periodically Perturbed Hamiltonian Systems

Abstract

In this work we present a method to bound the di usion near an elliptic equilibrium point of a periodically time-dependent Hamiltonian system. The method is based on the computation of the normal form (up to a certain degree) of that Hamiltonian, in order to obtain an adequate number of (approximate) rst integrals of the motion. Then, bounding the variation of those integrals with respect to time provides estimates of the di usion of the motion.The example used to illustrate the method is the Elliptic Spatial Restricted Three Body Problem, in a neighbourhood of the points L 4 5 . The mass parameter and the eccentricity are the ones corresponding to the Sun-Jupiter case.

A Methodology for the Numerical Computation of Normal Forms, Centre Manifolds and First Integrals of Hamiltonian Systems

“…Finally, you have to modify the input/output routines accordingly. This is what we have done in [28] or [50], for the case of a periodically perturbed Hamiltonian system.…”

“…Of course, in order to produce realistic diusion times one needs to have H 1 as small as it can be. Astandard way of producing the splitting (1) is by means of a normal form calculation: H 0 is the normal form and H 1 the corresponding remainder (see [14] and also [43] and [28]).…”

“…This will produce a high order approximation to the results wanted that, in many cases, are goodenough for pratical purposes. On the other hand, it is possible to derive rigorous estimates on the size of this remainder so one can obtain bounds on the error of the results obtained with the truncated series (see [43], [28] or [32] for numerical examples of this).…”

Computer Assisted Existence Proofs of Lyapunov Orbits at $L_2$ and Transversal Intersections of Invariant Manifolds in the Jupiter--Sun PCR3BP

“…The model has applications in space mission design [14,15], explains symbolic dynamics phenomena observed in trajectories of comets [18], and can be used for the study of diffusion estimates [16,17]. All of the above are associated with dynamics along invariant manifolds of the system.…”

Hyperbolicity and averaging for the Srzednicki–Wójcik equation

“…Our approach to averaging differs from the dominant one [AKN97,Nei84,GH,Hale,SVM,JS,L], which following the pioneering work of ( [BM, BZ]) used suitable coordinates changes to control the influence of the rapidly oscillating perturbation.…”