Statistical versus optimal partitioning for block entropies

Mistakidis, I.S.; Karamanos, Konstantinos; Mistakidis, S. I.

doi:10.1108/03684921311295466

Cited by 1 publication

(2 citation statements)

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“…It is of importance the comparison and identification of the relation of N to the mean time of recurrence knl, and to the escape time distribution for different control parameter values r. The escape time statistics for the L-L succession of the logistic map with the R-L generating partition have been investigated in a recent manuscript (Mistakidis et al, 2011), The comparison for two other "typical" cells is shown in Figures 2 and 3.…”

Section: K 415/6mentioning

confidence: 99%

“…In previous work we have restricted ourselves to a "version" of Poincare theorem tailored for symbolic dynamics and especially for the symbolic dynamics of the generating R-L partition of the logistic map (Mistakidis et al, 2011). It should be very interesting in the immediate future to "escape" from this "static" and "unhydrated" picture of the Poincare recurrence time theorem and pass to a dynamic investigation changing also the initial points and diameters of the cells, point by point.…”

Section: Introductionmentioning

confidence: 99%

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The many facets of Poincare recurrence theorem of the logistic map

Karamanos

Mistakidis

2012

Kybernetes

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Purpose -The purpose of this paper is to illustrate the many aspects of Poincare recurrence time theorem for an archetype of a complex system, the logistic map. Design/methodology/approach -At the beginning of the twentieth century, Poincare's recurrence theorem had revolutionized modern mechanics and statistical physics. However, this theorem did not attract considerable attention, at least from a numerical and computational point of view. In a series of relatively recent papers, Balakrishnan, Nicolis and Nicolis have addressed the recurrence time problem in a firm basis, introducing notation, theory, and numerical studies. Motivated by this call, the paper proposes to illustrate the many aspects of Poincare recurrence time theorem for an archetype of a complex system, the logistic map. The authors propose here in different tests and computations, each one illuminating the many aspects of the problem of recurrence. The paper ends up with a short discussion and conclusions. Findings -In this paper, the authors obtain new results on computations, each one illuminating the many aspects of the problem of recurrence. One striking aspect of this detailed work, is that when the sizes of the cells in the phase space became considerable, then the recurrence times assume ordinary values. Originality/value -The paper extends previous results on chaotic maps to the logistic map, enhancing comprehension, making possible connections with number theory, combinatorics and cryptography. IntroductionIn its classical version, Poincare recurrence theorem refers to a one-parameter family F 0 of one-to-one measure-preserving transformations. It states that if C is a subset of the phase space G such that m(C) . 0 (m is a completely additive measure with m(G) ¼ 1), then for almost every point P [ C there exists a sufficiently large t such that F 0 P [ C.

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Section: K 415/6mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%