Abstract. In a first part we discuss the well-known problem of the motion of a star in a general non-axisymmetric 2D galactic potential by means of a very simple but almost universal system: the pendulum model. It is shown that both loop and box families of orbits arise as a natural consequence of the dynamics of the pendulum. An approximate invariant of motion is derived. A critical value of the latter sharply separates the domains of loops and boxes and a very simple computation allows to get a clear picture of the distribution of orbits on a given energy surface. Besides, a geometrical representation of the global phase space using the natural surface of section for the problem, the 2D sphere, is presented. This provides a better visualization of the dynamics.In a second part we introduce a new indicator of the basic dynamics, the Mean Exponential Growth factor of Nearby Orbits (MEGNO), that is suitable to investigate the phase space structure associated to a general Hamiltonian. When applied to the 2D logarithmic potential it is shown to be effective to obtain a picture of the global dynamics and, also, to derive good estimates of the largest Lyapunov characteristic number in realistic physical times. Comparisons with other techniques reveal that the MEGNO provides more information about the dynamics in the phase space than other wide used tools.Finally, we discuss the structure of the phase space associated to the 2D logarithmic potential for several values of the semiaxis ratio and energy. We focus our attention on the stability analysis of the principal periodic orbits and on the chaotic component. We obtain critical energy values for which connections between the main stochastic zones take place. In any case, the whole chaotic domain appears to be always confined to narrow filaments, with a Lyapunov time about three characteristic periods.Send offprint requests to: P.M. Cincotta

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A representative model of a return map near homoclinic bifurcation is studied. This model is the so-called fattened Arnold map, a diffeomorphism of the annulus. The dynamics is extremely rich, involving periodicity, quasiperiodicity and chaos.The method of study is a mixture of analytic perturbation theory, numerical continuation, iteration to an attractor and experiments, in which the guesses are inspired by the theory. In turn the results lead to fine-tuning of the theory. This approach is a natural paradigm for the study of complicated dynamical systems.By following generic bifurcations, both local and homoclinic, various routes to chaos and strange attractors are detected. Here, particularly, the 'large' strange attractors which wind around the annulus are of interest. Furthermore, a global phenomenon regarding Arnold tongues is important. This concerns the accumulation of tongues on lines of homoclinic bifurcation. This phenomenon sheds some new light on the occurrence of infinitely many sinks in certain cases, as predicted by the theory.

Given a complex analytical Hamiltonian system, we prove that a necessary condition for meromorphic complete integrability is that the identity component of the Galois group of each variational equation of arbitrary order along each integral curve must be commutative. This was conjectured by the first author based on a suggestion made by the third author due to numerical and analytical evidences concerning higher order variational equations. This non-integrability criterion extends to higher orders a non-integrability criterion (Morales-Ramis criterion), using only the first order variational equation, obtained by the first and the second author. Using our result (at order two, three or higher) it is possible to solve important open problems of integrability which escaped to Morales-Ramis criterion.

We discuss a rather new phenomenon in chaotic dynamics connected with the fact that some three-dimensional diffeomorphisms can possess wild Lorenz-type strange attractors. These attractors persist for open domains in the parameter space. In particular, we report on the existence of such domains for a three-dimensional Hénon map (a simple quadratic map with a constant Jacobian which occurs in a natural way in unfoldings of several types of homoclinic bifurcations). Among other observations, we have evidence that there are different types of Lorenz-like attractor domains in the parameter space of the 3D Hénon map. In all cases the maximal Lyapunov exponent, Λ 1 , is positive. Concerning the next Lyapunov exponent, Λ 2 , there are open domains where it is definitely positive, others where it is definitely negative and, finally, domains where it cannot be distinguished numerically from zero (i.e. |Λ 2 | < ρ, where ρ is some tolerance ranging between 10 −5 and 10 −6 ). Furthermore, several other types of interesting attractors have been found in this family of 3D Hénon maps.