Special Issue "Entropy in Dynamic Systems"

A special issue of Entropy (ISSN 1099-4300).

Deadline for manuscript submissions: closed (30 September 2018).

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A printed edition of this Special Issue is available here.

Special Issue Editors

Prof. Dr. Jan Awrejcewicz
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Guest Editor
Department of Automation, Biomechanics and Mechatronics, The Lodz University of Technology, 1/15 Stefanowski St., 90-924 Łódź, Poland
Interests: linear/nonlinear vibrations and control; lumped and continuous parameter systems; PDEs and ODEs; analytical/numerical methods; non-smooth dynamical systems (impacts and friction)
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Prof. José A. Tenreiro Machado
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Guest Editor
Department of Electrical Engineering, Institute of Engineering, Polytechnic Institute of Porto, 4249-015 Porto, Portugal
Interests: nonlinear dynamics; fractional calculus; modeling; control; evolutionary computing; genomics; robotics; and intelligent transportation systems
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor and control complicated chaotic and stochastic processes governed by difference and differential (ordinary and partial) equations, algebraic-differential equations, integral-differential equations and equations with time delay. Though the roots of dynamical entropy are associated with the names of influential mathematicians, such as Kolmogrov, Sinai, Shannon, Krieger, Orstein, and Dinsburg, nowadays, it wandered to different branches of pure and applied sciences, and it possesses various meanings.

Multiple meanings of entropy is exhibited in physics and engineering, where it refers to the first and second thermodynamics laws and the Shannon and Boltzman entropies (in probabilistic theory and statistical mechanics), dynamical entropy (mathematics and applied mathematics, physics, economy, history, biology, social sciences, immunological systems), topological entropy (including information about the system evolutions), symbolic extension entropy (it allows for controlling the data compression based on entropy structure), digitalization entropy, etc.

Though the term of entropy came from Greek and emphasizes its analogy to energy, its multiple meanings in numerous branches of sciences are understood in a rather rough way, with an emphasis on transition from regular to chaotic states, stochastic and deterministic disorder, uniform and non-uniform distribution or decay of diversity.

This Special Issue addresses the notion of entropy in its broader sense and hence the manuscripts from different branches of mathematical/physical sciences, natural/social sciences and engineering oriented sciences are invited putting emphasis on complexity of dynamical systems including the features like timing chaos and spatio-temporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel non-linear phenomena, resonances, and beyond.

This Special Issue aims at covering diverse research from qualitatively different sciences, linked by dynamical entropy phenomena, understood in a broad manner. In particular, the following topics are of interest:

  • Complex analysis of difference and differential equations;
  • Dynamics and control of complex engineering systems;
  • Advances in fractional calculus;
  • Mathematical modelling of entropy in classical and generalized dynamical systems;
  • Entropy in physics, applied mathematics and information theory;
  • Entropy-based approaches to study transportation, social, financial and economical networks;
  • Deterministic chaotic versus stochastic processes;
  • Vibration signal processing and complex dynamics;
  • Entropy, Lyapunov exponents, Fourier and wavelet transforms and dimension;
  • Local, metric, topological, symbolic extension and smooth/non-smooth dynamical entropy.
Prof. Dr. Jan Awrejcewicz
Prof. Dr. Jose A.T. Machado
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Published Papers (11 papers)

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Editorial

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Open AccessEditorial
Entropy in Dynamic Systems
Entropy 2019, 21(9), 896; https://doi.org/10.3390/e21090896 - 16 Sep 2019
Abstract
In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor and control complicated chaotic and stochastic processes [...] Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems) Printed Edition available

Research

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Open AccessArticle
An Analysis of Deterministic Chaos as an Entropy Source for Random Number Generators
Entropy 2018, 20(12), 957; https://doi.org/10.3390/e20120957 - 11 Dec 2018
Cited by 6
Abstract
This paper presents an analytical study on the use of deterministic chaos as an entropy source for the generation of random numbers. The chaotic signal generated by a phase-locked loop (PLL) device is investigated using numerical simulations. Depending on the system parameters, the [...] Read more.
This paper presents an analytical study on the use of deterministic chaos as an entropy source for the generation of random numbers. The chaotic signal generated by a phase-locked loop (PLL) device is investigated using numerical simulations. Depending on the system parameters, the chaos originating from the PLL device can be either bounded or unbounded in the phase direction. Bounded and unbounded chaos differs in terms of the flatness of the power spectrum associated with the chaotic signal. Random bits are generated by regular sampling of the signal from bounded and unbounded chaos. A white Gaussian noise source is also sampled regularly to generate random bits. By varying the sampling frequency, and based on the autocorrelation and the approximate entropy analysis of the resulting bit sequences, a comparison is made between bounded chaos, unbounded chaos and Gaussian white noise as an entropy source for random number generators. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems) Printed Edition available
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Open AccessArticle
Information Transfer Among the Components in Multi-Dimensional Complex Dynamical Systems
Entropy 2018, 20(10), 774; https://doi.org/10.3390/e20100774 - 09 Oct 2018
Cited by 7
Abstract
In this paper, a rigorous formalism of information transfer within a multi-dimensional deterministic dynamic system is established for both continuous flows and discrete mappings. The underlying mechanism is derived from entropy change and transfer during the evolutions of multiple components. While this work [...] Read more.
In this paper, a rigorous formalism of information transfer within a multi-dimensional deterministic dynamic system is established for both continuous flows and discrete mappings. The underlying mechanism is derived from entropy change and transfer during the evolutions of multiple components. While this work is mainly focused on three-dimensional systems, the analysis of information transfer among state variables can be generalized to high-dimensional systems. Explicit formulas are given and verified in the classical Lorenz and Chua’s systems. The uncertainty of information transfer is quantified for all variables, with which a dynamic sensitivity analysis could be performed statistically as an additional benefit. The generalized formalisms can be applied to study dynamical behaviors as well as asymptotic dynamics of the system. The simulation results can help to reveal some underlying information for understanding the system better, which can be used for prediction and control in many diverse fields. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems) Printed Edition available
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Open AccessArticle
Fractional Form of a Chaotic Map without Fixed Points: Chaos, Entropy and Control
Entropy 2018, 20(10), 720; https://doi.org/10.3390/e20100720 - 20 Sep 2018
Cited by 10
Abstract
In this paper, we investigate the dynamics of a fractional order chaotic map corresponding to a recently developed standard map that exhibits a chaotic behavior with no fixed point. This is the first study to explore a fractional chaotic map without a fixed [...] Read more.
In this paper, we investigate the dynamics of a fractional order chaotic map corresponding to a recently developed standard map that exhibits a chaotic behavior with no fixed point. This is the first study to explore a fractional chaotic map without a fixed point. In our investigation, we use phase plots and bifurcation diagrams to examine the dynamics of the fractional map and assess the effect of varying the fractional order. We also use the approximate entropy measure to quantify the level of chaos in the fractional map. In addition, we propose a one-dimensional stabilization controller and establish its asymptotic convergence by means of the linearization method. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems) Printed Edition available
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Open AccessArticle
Tsallis Entropy of Product MV-Algebra Dynamical Systems
Entropy 2018, 20(8), 589; https://doi.org/10.3390/e20080589 - 09 Aug 2018
Cited by 3
Abstract
This paper is concerned with the mathematical modelling of Tsallis entropy in product MV-algebra dynamical systems. We define the Tsallis entropy of order α, where α(0,1)(1,), of a partition [...] Read more.
This paper is concerned with the mathematical modelling of Tsallis entropy in product MV-algebra dynamical systems. We define the Tsallis entropy of order α , where α ( 0 , 1 ) ( 1 , ) , of a partition in a product MV-algebra and its conditional version and we examine their properties. Among other, it is shown that the Tsallis entropy of order α , where α > 1 , has the property of sub-additivity. This property allows us to define, for α > 1 , the Tsallis entropy of a product MV-algebra dynamical system. It is proven that the proposed entropy measure is invariant under isomorphism of product MV-algebra dynamical systems. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems) Printed Edition available
Open AccessArticle
A Novel Image Encryption Scheme Based on Self-Synchronous Chaotic Stream Cipher and Wavelet Transform
Entropy 2018, 20(6), 445; https://doi.org/10.3390/e20060445 - 06 Jun 2018
Cited by 7
Abstract
In this paper, a novel image encryption scheme is proposed for the secure transmission of image data. A self-synchronous chaotic stream cipher is designed with the purpose of resisting active attack and ensures the limited error propagation of image data. Two-dimensional discrete wavelet [...] Read more.
In this paper, a novel image encryption scheme is proposed for the secure transmission of image data. A self-synchronous chaotic stream cipher is designed with the purpose of resisting active attack and ensures the limited error propagation of image data. Two-dimensional discrete wavelet transform and Arnold mapping are used to scramble the pixel value of the original image. A four-dimensional hyperchaotic system with four positive Lyapunov exponents serve as the chaotic sequence generator of the self-synchronous stream cipher in order to enhance the security and complexity of the image encryption system. Finally, the simulation experiment results show that this image encryption scheme is both reliable and secure. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems) Printed Edition available
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Open AccessArticle
The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils
Entropy 2018, 20(6), 400; https://doi.org/10.3390/e20060400 - 23 May 2018
Cited by 6
Abstract
This paper introduces a general solution of singular fractional-order linear-time invariant (FoLTI) continuous systems using the Adomian Decomposition Method (ADM) based on the Caputo's definition of the fractional-order derivative. The complexity of their entropy lies in defining the complete solution of such systems, [...] Read more.
This paper introduces a general solution of singular fractional-order linear-time invariant (FoLTI) continuous systems using the Adomian Decomposition Method (ADM) based on the Caputo's definition of the fractional-order derivative. The complexity of their entropy lies in defining the complete solution of such systems, which depends on introducing a method of decomposing their dynamic states from their static states. The solution is formulated by converting the singular system of regular pencils into a recursive form using the sequence of transformations, which separates the dynamic variables from the algebraic variables. The main idea of this work is demonstrated via numerical examples. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems) Printed Edition available
Open AccessArticle
Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems
Entropy 2018, 20(3), 175; https://doi.org/10.3390/e20030175 - 06 Mar 2018
Cited by 10
Abstract
The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the [...] Read more.
The aim of the paper was to analyze the given nonlinear problem by different methods of computation of the Lyapunov exponents (Wolf method, Rosenstein method, Kantz method, the method based on the modification of a neural network, and the synchronization method) for the classical problems governed by difference and differential equations (Hénon map, hyperchaotic Hénon map, logistic map, Rössler attractor, Lorenz attractor) and with the use of both Fourier spectra and Gauss wavelets. It has been shown that a modification of the neural network method makes it possible to compute a spectrum of Lyapunov exponents, and then to detect a transition of the system regular dynamics into chaos, hyperchaos, and others. The aim of the comparison was to evaluate the considered algorithms, study their convergence, and also identify the most suitable algorithms for specific system types and objectives. Moreover, an algorithm of calculation of the spectrum of Lyapunov exponents based on a trained neural network has been proposed. It has been proven that the developed method yields good results for different types of systems and does not require a priori knowledge of the system equations. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems) Printed Edition available
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Open AccessFeature PaperArticle
Quantifying Chaos by Various Computational Methods. Part 2: Vibrations of the Bernoulli–Euler Beam Subjected to Periodic and Colored Noise
Entropy 2018, 20(3), 170; https://doi.org/10.3390/e20030170 - 05 Mar 2018
Cited by 5
Abstract
In this part of the paper, the theory of nonlinear dynamics of flexible Euler–Bernoulli beams (the kinematic model of the first-order approximation) under transverse harmonic load and colored noise has been proposed. It has been shown that the introduced concept of phase transition [...] Read more.
In this part of the paper, the theory of nonlinear dynamics of flexible Euler–Bernoulli beams (the kinematic model of the first-order approximation) under transverse harmonic load and colored noise has been proposed. It has been shown that the introduced concept of phase transition allows for further generalization of the problem. The concept has been extended to a so-called noise-induced transition, which is a novel transition type exhibited by nonequilibrium systems embedded in a stochastic fluctuated medium, the properties of which depend on time and are influenced by external noise. Colored noise excitation of a structural system treated as a system with an infinite number of degrees of freedom has been studied. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems) Printed Edition available
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Open AccessArticle
On Points Focusing Entropy
Entropy 2018, 20(2), 128; https://doi.org/10.3390/e20020128 - 16 Feb 2018
Cited by 2
Abstract
In the paper, we consider local aspects of the entropy of nonautonomous dynamical systems. For this purpose, we introduce the notion of a (asymptotical) focal entropy point. The notion of entropy appeared as a result of practical needs concerning thermodynamics and the problem [...] Read more.
In the paper, we consider local aspects of the entropy of nonautonomous dynamical systems. For this purpose, we introduce the notion of a (asymptotical) focal entropy point. The notion of entropy appeared as a result of practical needs concerning thermodynamics and the problem of information flow, and it is connected with the complexity of a system. The definition adopted in the paper specifies the notions that express the complexity of a system around certain points (the complexity of the system is the same as its complexity around these points), and moreover, the complexity of a system around such points does not depend on the behavior of the system in other parts of its domain. Any periodic system “acting” in the closed unit interval has an asymptotical focal entropy point, which justifies wide interest in these issues. In the paper, we examine the problems of the distortions of a system and the approximation of an autonomous system by a nonautonomous one, in the context of having a (asymptotical) focal entropy point. It is shown that even a slight modification of a system may lead to the arising of the respective focal entropy points. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems) Printed Edition available
Open AccessArticle
Analytical Solutions for Multi-Time Scale Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion and Their Applications
Entropy 2018, 20(1), 63; https://doi.org/10.3390/e20010063 - 16 Jan 2018
Cited by 8
Abstract
In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. [...] Read more.
In this paper, we investigate analytical solutions of multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. We firstly decompose homogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions into independent differential subequations, and give their analytical solutions. Then, we use the variation of constant parameters to obtain the solutions of nonhomogeneous multi-time scale fractional stochastic differential equations driven by fractional Brownian motions. Finally, we give three examples to demonstrate the applicability of our obtained results. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems) Printed Edition available
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