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We propose to construct cross and joint ordinal pattern transition networksfrom multivariate time series for two coupled systems, where synchronizationsare often present. In particular, we focus on phase synchronization, which isone of prototypical scenarios in dynamical systems. We systematically show thatcross and joint ordinal patterns transition networks are sensitive to phasesynchronization. Furthermore, we find that some particular missing ordinalpatterns play crucial roles in forming the detailed structures in the parameterspace whereas the calculations of permutation entropy measures often do not. Weconclude that cross and joint ordinal partition transition network approachesprovide complementary insights to the traditional symbolic analysis ofsynchronization transitions.

Cross and joint ordinal partition transition networks for multivariate time series analysis