Various Aspects of the Theory of Dynamical Systems

Vries, J. de

doi:10.1007/978-94-015-8171-4_1

Cited by 1 publication

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“…The main object in RQA is the so-called recurrence matrix, which is a binary matrix that encodes all possible information regarding the recurrent states of a finite orbit, at a given scale. Recall that if X is a Hausdorff space and f : X → X a function, then a point x ∈ X is recurrent for f provided that for every neighborhood U of x there exists a > 0 such that f a (x) ∈ U, while if in addition, X is compact and metrizable and f is continuous, then an invariant measure exists and almost all points in the support of that measure are recurrent for f [3]. In RQA, the underlying metric structure of R m is exploited and neighborhoods are understood as open balls.…”

Section: Introductionmentioning

confidence: 99%

Critical recurrence scale thresholds

Kasolis¹,

Clemens²

2023

Preprint

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<p>Recurrence-based studies are employed for determining dynamical properties and for analyzing and processing time-series. Such studies rely on the so-called recurrence matrix, which is an orbit-specific binary representation of a proximity relation on the phase space. Customarily, this state proximity relation is deduced from the underlying metric structure via thresholding, and hence, a single user parameter is required for forming a recurrence matrix, unless embeddings are used. Since most of the prescriptions that can be found in the literature for identifying a suitable recurrence threshold are either heuristic or task-specific, a scaling-analysis framework, which identifies critical scales that emerge during the evolution based on the entropy of maximum sojourn time probability vectors, is introduced and validated here.</p>

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Section: Introductionmentioning

confidence: 99%