Determinism Analysis with Iso-Directional Recurrence Plots

Abstract

In this paper, we propose the Iso-directional Recurrence Plots(IDRP) and its techniques. In IDRP, characterization of vectors on orbits reconstructed from the time series data is represented on a twodimensional plane, similarly to the Recurrence Plots(RP). In addition, we also propose Iso-directional Neigh bors Plots(IDNP) which is the product set of RP and IDRP. IDNP shows the set of points that are neighbors and possess similar directions. We apply the proposed methods to various data such as chaotic and stochastic time series, and compare IDNP with the method proposed by Wayland et al. for recognizing determinism. The property of the results is similar to that of Wayland method, and thus IDNP is applicable to the analysis of determinism. Since the proposed methods in this paper are based on RP, not only quantitative analysis of determinism but also qualitative observation of the behavior of time series are possible with time series data.

Recurrence plots for the analysis of complex systems

“…This kind of RP is more efficient for estimating invariants and is more robust for the detection of UPOs (if they exist). • The iso-directional recurrence plot, introduced by Horai and Aihara [67],…”

“…Here a recurrence is related to neighboured trajectories which run parallel and in the same direction. The authors introduced an additional iso-directional neighbours plot, which is simply the product between the common RP and the iso-directional RP [67] …”

Recurrence-Based Time Series Analysis by Means of Complex Network Methods

“…Further definitions of recurrences add dynamical aspects, such as local rank orders or strictly parallel evolution of states (parallel segments of phase-space trajectory considered in iso-directional RPs [Horai et al, 2002]). For a more detailed overview, we refer to [Marwan et al, 2007b;Bandt et al, 2008].…”

Local Minima-Based Recurrence Plots for Continuous Dynamical Systems

“…The advantages are that the local minima-based thresholding produces RPs with thin lines (thickness not larger than two), requires less computational effort than perpendicular or iso-directional RPs [Choi et al, 1999;Horai & Aihara, 2002], and the threshold parameter has less impact on the results than it does with the standard method (Fig. 5).…”

“…For example, in perpendicular RPs two points x i and x j are recurrent if the jth point is within the neighborhood of i and lies on a plane that is perpendicular to the tangential flow at the ith vector point [Choi et al, 1999]. In an iso-directional RP recurrences are related to trajectories which run parallel and in the same direction [Horai & Aihara, 2002]. Such approaches are refinements of the standard method that uses a definitional criteria to further restrict the set of points initially qualified as recurrent using a distance threshold in order to exclude recurrences coming from tangential motion.…”