  Swing amplification is a model of spiral arm formation in disk galaxies.
Previous $N$-body simulations show that the epicycle phases of stars in spiral
arms are synchronized. However, the elementary process of the phase
synchronization is not well understood. In order to investigate phase
synchronization, we investigate the orbital evolution of stars due to
gravitational scattering by a perturber under the epicycle approximation and
its dependence on orbital elements and a disk parameter. We find that
gravitational scattering by the perturber can cause phase synchronization of
stellar orbits. The epicycle phases are better synchronized for smaller initial
epicycle amplitudes of stars and larger shear rates of galactic disks. The
vertical motion of stars does not affect the phase synchronization. The phase
synchronization forms trailing dense regions, which may correspond to spiral
arms.
