Geometrical representation of dynamical symmetries in ordinal pattern analysis of time series

Using ordinal patterns for the time-series analysis of complex and chaotic dynamics in systems across a wide variety of physical, biological, and other phenomena has proven extremely powerful, with recent work exploiting metrics that also characterize symmetries and correlations underlying the complex dynamics. For that purpose we introduce here a geometrical representation for the ordinal pattern description of observed time series. We show that specific projections of this space, emphasizing time reversal symmetry, reveals information about the phase-space dynamics and identifies families of chaos and periodicity via the trajectories created as a function of parameter across this (approximate) dynamical-symmetry space. We discuss general consequences and specific applications of these observations.