AN ALGORITHM TO COMPUTE THE TOPOLOGICAL ENTROPY OF A UNIMODAL MAP

Interval maps reveal precious information about the chaotic behavior of general nonlinear systems. If an interval map f:I→I is chaotic, then its iterates fnwill display heightened oscillatory behavior or profiles as n→∞. This manifestation is quite intuitive and is, here in this paper, studied analytically in terms of the total variations of fnon subintervals. There are four distinctive cases of the growth of total variations of fnas n→∞:(i) the total variations of fnon I remain bounded;(ii) they grow unbounded, but not exponentially with respect to n;(iii) they grow with an exponential rate with respect to n;(iv) they grow unbounded on every subinterval of I.We study in detail these four cases in relations to the well-known notions such as sensitive dependence on initial data, topological entropy, homoclinic orbits, nonwandering sets, etc. This paper is divided into three parts. There are eight main theorems, which show that when the oscillatory profiles of the graphs of fnare more extreme, the more complex is the behavior of the system.

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TOPOLOGICAL ENTROPY FOR MULTIDIMENSIONAL PERTURBATIONS OF ONE-DIMENSIONAL MAPS

International Journal of Bifurcation and Chaos ◽

10.1142/s021812740100281x ◽

2001 ◽

Vol 11 (05) ◽

pp. 1443-1446 ◽

Cited By ~ 10

Author(s):

MICHAŁ MISIUREWICZ ◽

PIOTR ZGLICZYŃSKI

Keyword(s):

Topological Entropy ◽

Positive Entropy ◽

One Dimensional ◽

Interval Map ◽

One Dimensional Maps

We prove that if an interval map of positive entropy is perturbed to a compact multidimensional map then the topological entropy cannot drop down considerably if the perturbation is small.

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Entropy of snakes and the restricted variational principle

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700007100 ◽

1992 ◽

Vol 12 (4) ◽

pp. 791-802

Author(s):

M. Misiurewicz ◽

J. Tolosa

Keyword(s):

Variational Principle ◽

Periodic Orbit ◽

Topological Entropy ◽

Periodic Orbits ◽

Symmetric Case ◽

Easy Method ◽

Interval Maps ◽

Interval Map ◽

Ergodic Measures

AbstractFor interval maps, we define the entropy of a periodic orbit as the smallest topological entropy of a continuous interval map having this orbit. We consider the problem of computing the limit entropy of longer and longer periodic orbits with the same ‘pattern’ repeated over and over (one example of such orbits is what we call ‘snakes’). We get an answer in the form of a variational principle, where the supremum of metric entropies is taken only over those ergodic measures for which the integral of a certain function is zero. In a symmetric case, this gives a very easy method of computing this limit entropy. We briefly discuss applications to topological entropy of countable chains.

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One-Dimensional Semiconjugacy Revisited

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127403007503 ◽

2003 ◽

Vol 13 (07) ◽

pp. 1657-1663 ◽

Cited By ~ 4

Author(s):

J. F. Alves ◽

J. Sousa Ramos

Keyword(s):

Topological Entropy ◽

Piecewise Linear ◽

Linear Map ◽

One Dimensional ◽

Piecewise Linear Map ◽

Piecewise Monotone ◽

Positive Topological Entropy ◽

Interval Map

Let f be a piecewise monotone interval map with positive topological entropy h(f)= log (s). Milnor and Thurston showed that f is topological semiconjugated to a piecewise linear map having slope s. Here we prove that these semiconjugacies are the eigenvectors of a certain linear endomorphism associated to f. Using this characterization, we prove a conjecture presented by those authors.

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COMPUTING THE TOPOLOGICAL ENTROPY OF UNIMODAL MAPS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412501520 ◽

2012 ◽

Vol 22 (06) ◽

pp. 1250152 ◽

Cited By ~ 8

Author(s):

RUI DILÃO ◽

JOSÉ AMIGÓ

Keyword(s):

Critical Point ◽

Topological Entropy ◽

End Points ◽

Unimodal Maps ◽

Boundary Points ◽

Interval Map ◽

Kneading Sequence ◽

Sign Analysis

We derive an algorithm to determine recursively the lap number (minimal number of monotone pieces) of the iterates of unimodal maps of an interval with free end-points. For this family of maps, the kneading sequence does not determine the lap numbers. The algorithm is obtained by the sign analysis of the itineraries of the critical point and of the boundary points of the interval map. We apply this algorithm to the estimation of the growth number and the topological entropy of maps with direct and reverse bifurcations.

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Embedding the symbolic dynamics of Lorenz maps

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004114000061 ◽

2014 ◽

Vol 156 (3) ◽

pp. 505-519 ◽

Cited By ~ 1

Author(s):

TONY SAMUEL ◽

NINA SNIGIREVA ◽

ANDREW VINCE

Keyword(s):

Topological Entropy ◽

Symbolic Dynamics ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Interval Map ◽

Piecewise Continuous ◽

Necessary And Sufficient ◽

Lorenz Maps

AbstractNecessary and sufficient conditions for the symbolic dynamics of a given Lorenz map to be fully embedded in the symbolic dynamics of a piecewise continuous interval map are given. As an application of this embedding result, we describe a new algorithm for calculating the topological entropy of a Lorenz map.

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Transmission rate conditions for distributed filtering in sensor networks against eavesdropper

Transactions of the Institute of Measurement and Control ◽

10.1177/01423312211005607 ◽

2021 ◽

pp. 014233122110056

Author(s):

Bingya Zhao ◽

Ya Zhang

Keyword(s):

Sensor Networks ◽

Topological Entropy ◽

Kalman Filtering ◽

Transmission Rate ◽

Necessary Condition ◽

Estimation Errors ◽

Positive Role ◽

Filtering Algorithm ◽

Wiretap Channels ◽

Kalman Filtering Algorithm

This paper studies the distributed secure estimation problem of sensor networks (SNs) in the presence of eavesdroppers. In an SN, sensors communicate with each other through digital communication channels, and the eavesdropper overhears the messages transmitted by the sensors over fading wiretap channels. The increasing transmission rate plays a positive role in the detectability of the network while playing a negative role in the secrecy. Two types of SNs under two cooperative filtering algorithms are considered. For networks with collectively observable nodes and the Kalman filtering algorithm, by studying the topological entropy of sensing measurements, a sufficient condition of distributed detectability and secrecy, under which there exists a code–decode strategy such that the sensors’ estimation errors are bounded while the eavesdropper’s error grows unbounded, is given. For collectively observable SNs under the consensus Kalman filtering algorithm, by studying the topological entropy of the sensors’ covariance matrices, a necessary condition of distributed detectability and secrecy is provided. A simulation example is given to illustrate the results.

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Entropy on noncompact sets

Topological Algebra and its Applications ◽

10.1515/taa-2019-0003 ◽

2019 ◽

Vol 7 (1) ◽

pp. 29-37

Author(s):

Jose S. Cánovas

Keyword(s):

Topological Entropy ◽

Topological Spaces ◽

Continuous Maps

AbstractIn this paper we review and explore the notion of topological entropy for continuous maps defined on non compact topological spaces which need not be metrizable. We survey the different notions, analyze their relationship and study their properties. Some questions remain open along the paper.

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