Jordi Villanueva

KAM theory without action-angle variables

We give a proof of a KAM theorem on existence of invariant tori with a Diophantine rotation vector for Hamiltonian systems. The method of proof is based on the use of the geometric properties of Hamiltonian systems which, in particular, do not require the Hamiltonian system either to be written in action-angle variables or to be a perturbation of an integrable one. The proposed method is also useful to compute numerically invariant tori for Hamiltonian systems. We also prove a translated torus theorem in any number of degrees of freedom.

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 Fine Geometry of the Cantor Families of Invariant Tori in Hamiltonian Systems

L_5

Quasi-Periodic Frequency Analysis Using Averaging-Extrapolation Methods

. The mass parameter

A KAM theorem without action-angle variables for elliptic lower dimensional tori

\mu

Quantitative estimates on the normal form around a non-semi-simple 1:−1 resonant periodic orbit

has been chosen such that

A dynamical systems approach to Bohmian trajectories in a 2D harmonic oscillator

L_5

Asymptotic size of Herman rings of the complex standard family by quantitative quasiconformal surgery

is unstable but the periodic orbit is still stable. This allows to show the existence of regions of effective stability near

Asymptotic behaviour of the domain of analyticity of invariant curves of the standard map

L_5

Effective Stability Around Periodic Orbits of the Spatial RTBP

for values of