  The gravitational potentials of realistic galaxy models are in general
non-integrable, in the sense that they admit orbits that do not have three
independent isolating integrals of motion and are therefore chaotic. However,
if chaotic orbits are a small minority in a stellar system, it is expected that
they have negligible impact on the main dynamical properties of the system. In
this paper we address the question of quantifying the importance of chaotic
orbits in a stellar system, focusing, for simplicity, on axisymmetric systems.
Chaotic orbits have been found in essentially all (non-St\"ackel) axisymmetric
gravitational potentials in which they have been looked for. Based on the
analysis of the surfaces of section, we add new examples to those in the
literature, finding chaotic orbits, as well as resonantly trapped orbits among
regular orbits, in Miyamoto-Nagai, flattened logarithmic and shifted Plummer
axisymmetric potentials. We define the fractional contributions in mass of
chaotic ($\xi_{\rm c}$) and resonantly trapped ($\xi_{\rm t}$) orbits to a
stellar system of given distribution function, which are very useful
quantities, for instance in the study of the dispersal of stellar streams of
galaxy satellites. As a case study, we measure $\xi_{\rm c}$ and $\xi_{\rm t}$
in two axisymmetric stellar systems obtained by populating flattened
logarithmic potentials with the Evans ergodic distribution function, finding
$\xi_{\rm c}\sim 10^{-4}-10^{-3}$ and $\xi_{\rm t}\sim 10^{-2}-10^{-1}$.
