Detecting Recurrence Domains of Dynamical Systems by Symbolic Dynamics

Abstract

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.

Sequences by Metastable Attractors: Interweaving Dynamical Systems and Experimental Data

“…For finding an optimal partition and hence an optimal value for ε, beim Graben and Hutt [12,26] and beim Graben et al [27] have suggested several entropybased criteria. To gain an optimal segmentation, the idea is to assume an underlying stochastic model for the symbolic sequences.…”

“…Recurrent events (R ij = 1) of the dynamics lead to intersecting ε-balls B ε (x i ) ∩ B ε (x j ) = ∅ which can be merged together into equivalence classes of phase space X. By merging balls together into a set A j = B ε (x i ) ∪ B ε (x j ) when states x i and x j are recurrent and when i > j, we simply replace the larger time index i in the recurrence plot R by the smaller one j, symbolized as a rewriting rule i → j of a recurrence grammar [12,26,27].…”

“…Other methods consider MS as recurrence domains and transients between them as non-recurrent regimes. Then the recurrent and nonrecurrent states partition the underlying phase space into disjoint cells, thereby yielding a symbolic dynamics where all nonrecurrent transients are mapped to a single symbol, i.e., they are undistinguishable [12,26,27]. To determine such a symbolic dynamics, an underlying stochastic model on the temporal distribution of MS is mandatory.…”

“…Since a HO exhibits a temporal sequence of MS, in a first step these states are extracted by data analysis techniques identifying their location in phase space, their onset times and their durations as essential features [8,12,22,[25][26][27]. These features are assumed to fully describe the HO and typically are rather invariant in experimental repetitions.…”

Complex and Nonlinear Analysis of Heart Rate Variability in the Assessment of Fetal and Neonatal Wellbeing

“…In the field of neuroscience, symbolic dynamics was applied, e.g. to characterized brain microstates [7], to study the recurrence of microstates during stimulus [8], to detect determinism in periictal intracranial electroencephalographic signals [9], to predict epileptic seizures [10], to assess causal relations between brain signals under different levels of consciousness [11], to detect the development of epileptic seizure [12,13], and to investigate the central-autonomic coupling in mental disorder [14,15].…”