Probing the local dynamics of periodic orbits by the generalized
  alignment index (GALI) method
  As originally formulated, the Generalized Alignment Index (GALI) method of
chaos detection has so far been applied to distinguish quasiperiodic from
chaotic motion in conservative nonlinear dynamical systems. In this paper we
extend its realm of applicability by using it to investigate the local dynamics
of periodic orbits. We show theoretically and verify numerically that for
stable periodic orbits the GALIs tend to zero following particular power laws
for Hamiltonian flows, while they fluctuate around non-zero values for
symplectic maps. By comparison, the GALIs of unstable periodic orbits tend
exponentially to zero, both for flows and maps. We also apply the GALIs for
investigating the dynamics in the neighborhood of periodic orbits, and show
that for chaotic solutions influenced by the homoclinic tangle of unstable
periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during
which their amplitudes change by many orders of magnitude. Finally, we use the
GALI method to elucidate further the connection between the dynamics of
Hamiltonian flows and symplectic maps. In particular, we show that, using for
the computation of GALIs the components of deviation vectors orthogonal to the
direction of motion, the indices of stable periodic orbits behave for flows as
they do for maps.
