  We propose to construct cross and joint ordinal pattern transition networks
from multivariate time series for two coupled systems, where synchronizations
are often present. In particular, we focus on phase synchronization, which is
one of prototypical scenarios in dynamical systems. We systematically show that
cross and joint ordinal patterns transition networks are sensitive to phase
synchronization. Furthermore, we find that some particular missing ordinal
patterns play crucial roles in forming the detailed structures in the parameter
space whereas the calculations of permutation entropy measures often do not. We
conclude that cross and joint ordinal partition transition network approaches
provide complementary insights to the traditional symbolic analysis of
synchronization transitions.
