  We study the motion of test particle in static axisymmetric vacuum spacetimes
and discuss two criteria for strong chaos to occur: (1) a local instability
measured by the Weyl curvature, and (2) a tangle of a homoclinic orbit, which
is closely related to an unstable periodic orbit in general relativity. We
analyze several static axisymmetric spacetimes and find that the first
criterion is a sufficient condition for chaos, at least qualitatively. Although
some test particles which do not satisfy the first criterion show chaotic
behavior in some spacetimes, these can be accounted for the second criterion.
