Resonances and bifurcations in axisymmetric scale-free potentials
  We investigate an analytical treatment of bifurcations of families of
resonant `thin' tubes in axisymmetric galactic potentials. We verify that the
most relevant bifurcations are due to the (1:1) resonance producing the
`inclined' orbits through two different mechanisms: from the disk orbit and
from the `thin' tube associated to the vertical oscillation. The closest
resonances occurring after these are the (4:3) resonance in the oblate case and
the (2:1) resonance in the prolate case. The (1:1) resonances are treated in a
straightforward way using a 2nd-order truncated normal form. The higher-order
resonances are instead cumbersome to investigate, because the normal form has
to be truncated to a high degree and the number of terms grows very rapidly. We
therefore adopt a further simplification giving analytic formulas for the
values of the parameters at which bifurcations ensue and compare them with
selected numerical results. Thanks to the asymptotic nature of the series
involved, the predictions are reliable well beyond the convergence radius of
the original series.
