Josep-Maria Mondelo

We consider autonomous Hamiltonian systems and present an algorithm to compute at the same time partially hyperbolic invariant tori (whiskered tori), as well as high-order expansions of their stable and unstable manifolds. Such whiskered tori have been shown to be important for transport phenomena in phase space. For instance, by following their invariant manifolds one could obtain zero-cost trajectories in space mission design. We present in detail the case when the (un)stable directions are one-dimensional. The strategy to compute tori and their invariant manifolds is based on the parameterization method. We formulate a functional equation for a parameterization of both the torus and its whiskers expressing that they are invariant. This equation is naturally discretized in Fourier-Taylor series or, equivalently, in a grid of Taylor series. Using a return map, we are reduced to study functions of n - 1 variables where n is the number of degrees of freedom (the phase space is 2n dimensional). Then, we implement a Newton-like method that converges quadratically. They key advantage of our approach is that, using geometric identities coming from the Hamiltonian nature of the problem, the algorithm has small storage requirements and a low operation count per step which is highly efficient. The simultaneous computation of the torus and the whiskers improves the efficiency and the stability of the algorithm. We present implementations and extensive numerical experiments in the Circular Restricted Three Body Problem.

The purpose of this paper is to present a method to compute parameterizations of invariant tori and bundles in non-autonomous quasi-periodic Hamiltonian systems. We generalize flow map parameterization methods to the quasi-periodic setting. To this end, we introduce the notion of fiberwise isotropic tori and sketch definitions and results on fiberwise symplectic deformations and their moment maps. These constructs are vital to work in a suitable setting and lead to the proofs of magic cancellations that guarantee the existence of solutions of cohomological equations. We apply our algorithms in the Elliptic Restricted Three Body Problem and compute non-resonant 3-dimensional invariant tori and their invariant bundles around the L1 point.

In this paper, we show a mechanism to explain transport from the outer to the inner Solar system. Such a mechanism is based on dynamical systems theory. More concretely, we consider a sequence of uncoupled bicircular restricted four-body problems –BR4BP –(involving the Sun, Jupiter, a planet and an infinitesimal mass), being the planet Neptune, Uranus and Saturn. For each BR4BP, we compute the dynamical substitutes of the collinear equilibrium points of the corresponding restricted three-body problem (Sun, planet and infinitesimal mass), which become periodic orbits. These periodic orbits are unstable, and the role that their invariant manifolds play in relation with transport from exterior planets to the inner ones is discussed.

We estimate numerically the regularities of a family of Strange Non--Chaotic Attractors related with one of the models studied in C. Grebogy et al. (1984) (see also G. Keller (1996)). To estimate these regularities we use wavelet analysis in the spirit of R. de la Llave et al. (2002) together with some ad-hoc techniques that we develop to overcome the theoretical difficulties that arise in the application of the method to the particular family that we consider. These difficulties are mainly due to the facts that we do not have an explicit formula for the attractor and it is discontinuous almost everywhere for some values of the parameters. Concretely we propose an algorithm based on the Fast Wavelet Transform. Also a quality check of the wavelet coefficients and regularity estimates is done.

In a previous paper [6], a numerical procedure for the Fourier analysis of quasi-periodic functions was developed, allowing for an accurate determination of frequencies and amplitudes from equally-spaced samples of the input function on a finite time interval. This paper is devoted to a complete error analysis of that procedure, from which computable bounds are deduced. These bounds are verified and further discussed in examples.