The many facets of Poincare recurrence theorem of the logistic map

Abstract

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.

Managing service capacity with boundedly rational consumers under social interactions

“…Hao and Zheng (1998) point out the importance of symbolic dynamics, which provides almost the only rigorous way to understand global systematics of periodic and, especially, chaotic motion in dynamical systems. Karamanos et al (2012) illustrate many aspects of Poincare recurrence time theorem for an archetype of a complex system, e.g. the logistic map.…”

Violating the second law of thermodynamics in a dynamical system through equivalence closure via mutual information carriers of a 5-tuple measure space

“…Which is closed through equivalence with the infinite iteration state occurring but the generates the states with the entropy being conserved thus violating the second law of thermodynamics according to the second law of thermodynamics, entropy of a closed system always tends to increase with time. [10].…”

Violating the second law of thermodynamics in a dynamical system through equivalence closure via mutual information carriers of a 5-tuple measure space

“…Which is closed through equivalence with the infinite iteration state occurring but the generates the states with the entropy being conserved thus violating the second law of thermodynamics according to the second law of thermodynamics, entropy of a closed system always tends to increase with time. [10].…”