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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)

- Amadeu Delshams, Vassili Gelfreich, Angel Jorba, Tere M Seara, A Delshams, V Gelfreich +2 others
- 1997

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)

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)

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)

- Angel Jorba
- 2003

In this paper we give a numerical description of the neighbourhood of a xed point of a symplectic map undergoing a transition from linear stability to complex instability, i.e., the so called Hamiltonian-Hopf bifurcation. We have considered both the direct and inverse cases. The study is based on the numerical computation of the Lyapunov families of… (More)

We study the dynamics of the Forced Logistic Map in the cylinder. We compute a bifurcation diagram in terms of the dynamics of the attracting set. Different properties of the attracting set are considered, as the Lyapunov exponent and, in the case of having a periodic invariant curve, its period and its reducibility. This reveals that the parameter values… (More)

The purpose of this paper is to make an explicit analysis of the nonlinear dynamics around a two-dimensional invariant torus of an analytic Hamiltonian system. The study is based on normal form techniques and the computation of an approximated first integral around the torus. One of the main novel aspects of the current work is the implementation of the… (More)