  Invariant curves are generally closed curves in the Poincare's surface of
section. Here we study an interesting dynamical phenomenon, first discovered by
Binney et al. (1985) in a rotating Kepler potential, where an invariant curve
of the surface of section can split into two disconnected line segments under
certain conditions, which is distinctively different from the islands of
resonant orbits. We first demonstrate the existence of split invariant curves
in the Freeman bar model, where all orbits can be described analytically. We
find that the split phenomenon occurs when orbits are nearly tangent to the
minor/major axis of the bar potential. Moreover, the split phenomenon seems
necessary to avoid invariant curves intersecting with each other. Such a
phenomenon appears only in rotating potentials, and we demonstrate its
universal existence in other general rotating bar potentials. It also implies
that actions are no longer proportional to the area bounded by an invariant
curve if the split occurs, but they can still be computed by other means.
