Detecting Recurrence Domains of Dynamical Systems by Symbolic Dynamics

We propose an algorithm for the detection of recurrence domains of complex dynamical systems from time series. Our approach exploits the characteristic checkerboard texture of recurrence domains exhibited in recurrence plots. In phase space, recurrence plots yield intersecting balls around sampling points that could be merged into cells of a phase space partition. We construct this partition by a rewriting grammar applied to the symbolic dynamics of time indices. A maximum entropy principle defines the optimal size of intersecting balls. The final application to high-dimensional brain signals yields an optimal symbolic recurrence plot revealing functional components of the signal.

Figure 1

Recurrence-based symbolic dynamics of the Lorenz attractor [4]. (a) Time series ${\mathit{x}}_t$ (upper panel) and optimal encoding $s^{\prime }$ (color bar beneath). (b) $\epsilon$-recurrence plot [Eq. (1)] for $\epsilon =5.0$ and Euclidian norm; black pixels denote $R_{ij}=1$, white ones $R_{ij}=0$. (c) Symbolic dynamics $s^{\prime }$ for range $\epsilon \in [0.2,3]$. (d) Dependence of entropy ratio [Eq. (5)] from $\epsilon$. (e) Phase space partition into recurrence domains for optimal encoding ${\epsilon }^{*}=1.9$. (f) Symbolic recurrence plot [Eq. (2)] of optimal encoding.

Figure 2

Recurrence-based symbolic dynamics of a 17-dimensional ERP data set of a single subject [19]. (a) Single subject ERP time series (upper panel) and symbolic encoding $s^{\prime }$ with ${\epsilon }^{*}=3.5\mu \mathrm{V}$ (color bar beneath) for $d=17$ scalp channels. (b) $\epsilon$ recurrence plot [Eq. (1)] for $\epsilon =8.0\mu \mathrm{V}$. (c) Utility function $h(\epsilon )$ [Eq. (5)]. (d) Symbolic recurrence plot [Eq. (2)] for optimal encoding ${\epsilon }^{*}=3.5\mu \mathrm{V}$.