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This paper revisits the Taylor method for the numerical integration of initial value problems of Ordinary Differential Equations (ODEs). The main goal is to show that the Taylor method can be competitive, both in speed and accuracy, with the standard methods. To this end, we present a computer program that outputs an specific numerical integrator for a(More)
We consider fast quasiperiodic perturbations with two frequencies (1="; =") of a pendulum, where is the golden mean number. The complete system has a two-dimensional invariant torus in a neighbourhood of the saddle point. We study the splitting of the three-dimensional invariant manifolds associated to this torus. Provided that the perturbation amplitude is(More)
In this paper, we make a systematic study of the global dynamical structure of the Sun– Jupiter L 4 tadpole region. The results are based on long-time simulations of the Trojans in the Sun, Jupiter, Saturn system and on the frequency analysis of these orbits. We give some initial results in the description of the resonant structure that guides the long-term(More)
We explore different two-parametric families of quasi-periodically Forced Logistic Maps looking for universality and self-similarity properties. In the bifurcation diagram of the one-dimensional Logistic Map, it is well known that there exist parameter values s n where the 2 n-periodic orbit is superattracting. Moreover, these parameter values lay between(More)
This paper focuses on the dynamics near the collinear equilibrium points L 1;2;3 of the spatial Restricted Three Body Problem. It is well known that the linear behaviour of these three points is of the type centercentersaddle. To obtain an accurate description of the dynamics in an extended neighbourhood of those points, two diierent (but complementary)(More)
This paper deals with the eective computation of normal forms, centre mani-folds and rst integrals in Hamiltonian mechanics. These kind of calculations are very useful since they allow, for instance, to give explicit estimates on the diu-sion time or to compute invariant tori. The approach presented here is based on using algebraic manipulation for the(More)
We perform a bifurcation analysis of normal–internal resonances in parametrised families of quasi–periodically forced Hamiltonian oscillators, for small forcing. The unforced system is a one degree of freedom oscillator, called the 'back-bone' system; forced, the system is a skew–product flow with a quasi–periodic driving with Ò basic frequencies. The(More)
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(More)
We focus on the continuation with respect to parameters of smooth invariant curves of quasi-periodically forced 1-D systems. In particular, we are interested in mechanisms leading to the destruction of the curve. One of these mechanisms is the so-called fractalization: the curve gets increasingly wrinkled until it stops being a smooth curve. Here we show(More)
In this paper we focus on the stability of the Trojan asteroids for the planar Restricted Three-Body Problem (RTBP), by extending the usual techniques for the neighbourhood of an elliptic point to derive results in a larger vicinity. Our approach is based on the numerical determination of the frequencies of the asteroid and the effective computation of the(More)