Modeling Ordinal First Return Dynamics from Time Series

We examine the family of ordinal First Return Maps (FRMs) of a dynamical system time series to model the system's dynamics. We consider two distinct modeling approaches using the same modeling method and compare their outcomes to ascertain the prerequisites for each to function and which approach is more effective in different circumstances. The first approach involves short-term modeling of the entire flow, followed by identifying the ordinal sections on the predicted flow. The second approach relies on locating the ordinal sections within the reference time series and using the data from each section independently to build a model that can produce a long-term prediction for each section's FRM. We demonstrate the effectiveness of these approaches on two well-known dynamical systems, the Lorenz and Rössler equations, emulating data deficiency by selecting a low sampling rate. We consider the effect of downsampling on the shape of the FRMs to choose suitable ordinal sequences to use in creating Poincaré sections from the time series. Following that, we propose a dissimilarity measure between the reference and predicted FRMs to assess the model quality. We then compare the reference and predicted FRMs using the introduced dissimilarity criterion to determine which modeling approach performs better in each situation. Ultimately, we classify the FRMs situation into continuous and piecewise continuous scenarios. Based on our observations from the Lorenz and Rössler dynamical systems, we found that when the system's FRMs are piecewise continuous, it is more effective to model the entire flow. However, when the FRMs are continuous, our choice of modeling approach depends on the criterion that the predicted FRM is more similar to the reference.