Order and chaos in a three dimensional galaxy model

We explore the orbital dynamics of a realistic three dimensional model describing the properties of a disk galaxy with a spherically symmetric central dense nucleus and a triaxial dark matter halo component. Regions of phase space with regular and chaotic motion are identified depending on the parameter values for triaxiality of the dark matter halo and for breaking the rotational symmetry. The four dimensional Poincar\'e map of the three degrees of freedom system is analyzed by a study of its restriction to various two dimensional invariant subsets of its domain.