A dynamical model composed of a disk galaxy with an elliptic companion, moving in a circular orbit, is used in order to study the stellar orbits in a binary galaxy. Using the Poincare surface of section we study the evolution of the stochastic regions in the primary galaxy considering the mass of the companion or the value of the Jacobi’s integral as a parameter. Our numerical calculations suggest that the regions of stochasticity increase, as the mass of the companion or the value of the Jacobi’s integral increase. An interesting observation is that only direct orbits become stochastic.

The formation of spiral structure in a galaxy, as a result of the gravitational perturbation caused by a permanent companion, is studied. It is found that spiral structure appears only when a resonance exists between the rotational frequency of the stars in the galaxy and the rotational frequency of the companion galaxy. The number of spiral arms depends strongly on the particular resonance. In the case where the companion moves in an elliptic orbit, spiral arms are formed when a resonance, inside the galactic body, exists in almost all the parts of the orbit or, at least, in the largest part of it.

We present a map for the study of resonant motion in a potential made up of two harmonic oscillators with quartic perturbing terms. This potential can be considered to describe motion in the central parts of non-rotating elliptical galaxies. The map is based on the averaged Hamiltonian. Adding on a semi-empirical basis suitable terms in the unperturbed averaged Hamiltonian, corresponding to the 1:1 resonant case, we are able to construct a map describing motion in several resonant cases. The map is used in order to find thex − p_x Poincare phase plane for each resonance. Comparing the results of the map, with those obtained by numerical integration of the equation of motion, we observe, that the map describes satisfactorily the broad features of orbits in all studied cases for regular motion. There are cases where the map describes satisfactorily the properties of the chaotic orbits as well.

We present connections between global and local parameters in a realistic dynamical model, describing motion in a barred galaxy. Expanding the global model in the vicinity of a stable Lagrange point, we find the potential of a two-dimensional perturbed harmonic oscillator, which describes local motion near the centre of the global model. The frequencies of oscillations and the coefficients of the perturbing terms are not arbitrary but are connected to the mass, the angular rotation velocity, the scale length and the strength of the galactic bar. The local energy is also connected to the global energy. A comparison of the properties of orbits in the global and local potential is also made.

We obtain thex - p_xPoincare phase plane for a two dimensional, resonant, galactic type Hamiltonian using conventional numerical integration, a second order symplectic integrator and a map based on the averaged Hamiltonian. It is found that all three methods give good results, for small values of the perturbation parameter, while the symplectic integrator does a better job than the mapping, for large perturbations. The dynamical spectra are used to distinguish between regular and chaotic motion.

The role of the angular momentum in the regular or chaotic character of motion in an axially symmetric quasar model is examined. It is found that, for a given value of the critical angular momentumL_zc, there are two values of the mass of the nucleusM_n for which transition from regular to chaotic motion occurs. The [L_zc– M_n] relationship shows a linear dependence for the time independent model and an exponential dependence for the evolving model. Both cases are explained using theoretical arguments together with some numerical evidence. The evolution of the orbits is studied, as mass is transported from the disk to the nucleus. The results are compared with the outcomes derived for galactic models with massive nuclei.