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Astron. Astrophys. 349, 70-76 (1999)

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1. Introduction

The application of modern methods in the domain of Galactic Dynamics has been proved very fruitful during the last decade. Among them is the use of maps to describe galactic motion (Caranicolas 1990, 1994), and the invariant spectra in galactic Hamiltonian systems (see Contopoulos et al. 1995, Patsis et al. 1997).

Maps, derived from galactic type Hamiltonians, are useful in the study of galactic orbits because they are faster, in general, than numerical integration and allow a quick visualisation of the corresponding phase plane. Maps also can give the stability conditions for the periodic orbits. On the other hand, our experience based on previous work shows that the results given by the maps are in good agreement with those given by numerical integration at least for small perturbations (see Caranicolas & Karanis 1999). On this basis, it seems that one has some good reasons to use a map for the study of galactic motion.

In the present work we consider that the local (i.e. near an equilibrium point) galactic motion is described by the potential


where [FORMULA] are the unperturbed frequencies of oscillation along the x and y axis respectively, [FORMULA] is the perturbation strength while [FORMULA] are parameters.We shall study the case where [FORMULA]. Without the loss of generality we can take [FORMULA], that is the 1:1 resonance case. Then the Hamiltonian to the potential (1) is


where [FORMULA] are the momenta per unit mass conjugate to x, y and h is the numerical value of H.

Our aim is to study the various types of periodic orbits, their stability and the kind of non periodic motion (regular or chaotic) in the Hamiltonian (2) for various values of parameters [FORMULA] and [FORMULA] using the map corresponding to the Hamiltonian (2), as well as numerical integration. This resonance case is also known as the perturbed elliptic oscillators (Deprit 1991, Deprit & Elipe 1991, Caranicolas & Innanen 1992, Caranicolas 1993). The [FORMULA] Poincare phase plane derived by the map and the numerical integration will be compared in each case. Of a special interest is the study of the chaotic motion. We shall try to find an answer to questions such as:

  1. Does the map describe in a satisfactory way the chaotic layers in the [FORMULA] plane and, if so, how this behavior evolves by increasing [FORMULA]?

  2. What are the differences, if any, in the Lyapunov Characteristic Number (LCN) found by the map and the numerical integration in the regular and the chaotic area?

  3. Are there any similarities in the invariant spectra derived by the map and the numerical integration?

The map and the stability conditions of the periodic orbits are given in Sect. 2. In the same section we compare the [FORMULA] phase plane found by the map and numerical integration for some of the main different cases. In Sect. 3 we compare the LCNs and the spectra of orbits, derived using the map and numerical integration. Sect. 4 is devoted to a discussion and the conclusions of this work.

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© European Southern Observatory (ESO) 1999

Online publication: August 25, 1999