  The main objective of this work is to determine the character of orbits of
stars moving in the meridional $(R,z)$ plane of an axially symmetric
time-independent disk galaxy model with a central massive nucleus and an
additional spherical dark matter halo component. In particular, we try to
reveal the influence of the scale length of the dark matter halo on the
different families of orbits of stars, by monitoring how the percentage of
chaotic orbits, as well as the percentages of orbits of the main regular
resonant families evolve when this parameter varies. The smaller alignment
index (SALI) was computed by numerically integrating the equations of motion as
well as the variational equations to extensive samples of orbits in order to
distinguish safely between ordered and chaotic motion. In addition, a method
based on the concept of spectral dynamics that utilizes the Fourier transform
of the time series of each coordinate is used to identify the various families
of regular orbits and also to recognize the secondary resonances that bifurcate
from them. Our numerical computations reveal that when the dark matter halo is
highly concentrated, that is when the scale length has low values the vast
majority of star orbits move in regular orbits, while on the other hand in less
concentrated dark matter halos the percentage of chaos increases significantly.
We also compared our results with early related work.
