Editorial

3 pages, 175 KiB

Open AccessEditorial

by Jan Awrejcewicz and José A. Tenreiro Machado

Entropy 2019, 21(9), 896; https://0-doi-org.brum.beds.ac.uk/10.3390/e21090896 - 16 Sep 2019

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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)

Research

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10 pages, 2908 KiB

Open AccessArticle

An Analysis of Deterministic Chaos as an Entropy Source for Random Number Generators

by Kaya Demir and Salih Ergün

Entropy 2018, 20(12), 957; https://0-doi-org.brum.beds.ac.uk/10.3390/e20120957 - 11 Dec 2018

Cited by 24 | Viewed by 3296

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)

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22 pages, 1349 KiB

Open AccessArticle

Information Transfer Among the Components in Multi-Dimensional Complex Dynamical Systems

by Yimin Yin and Xiaojun Duan

Entropy 2018, 20(10), 774; https://0-doi-org.brum.beds.ac.uk/10.3390/e20100774 - 09 Oct 2018

Cited by 13 | Viewed by 2717

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)

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14 pages, 3238 KiB

Open AccessArticle

Fractional Form of a Chaotic Map without Fixed Points: Chaos, Entropy and Control

by Adel Ouannas, Xiong Wang, Amina-Aicha Khennaoui, Samir Bendoukha, Viet-Thanh Pham and Fawaz E. Alsaadi

Entropy 2018, 20(10), 720; https://0-doi-org.brum.beds.ac.uk/10.3390/e20100720 - 20 Sep 2018

Cited by 23 | Viewed by 3267

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)

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19 pages, 1075 KiB

Open AccessArticle

Tsallis Entropy of Product MV-Algebra Dynamical Systems

by Dagmar Markechová and Beloslav Riečan

Entropy 2018, 20(8), 589; https://0-doi-org.brum.beds.ac.uk/10.3390/e20080589 - 09 Aug 2018

Cited by 3 | Viewed by 2566

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 \in (0, 1) \cup (1, \infty),

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 \in (0, 1) \cup (1, \infty),

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)

13 pages, 3491 KiB

Open AccessArticle

A Novel Image Encryption Scheme Based on Self-Synchronous Chaotic Stream Cipher and Wavelet Transform

by Chunlei Fan and Qun Ding

Entropy 2018, 20(6), 445; https://0-doi-org.brum.beds.ac.uk/10.3390/e20060445 - 06 Jun 2018

Cited by 17 | Viewed by 3236

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)

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14 pages, 298 KiB

Open AccessArticle

The General Solution of Singular Fractional-Order Linear Time-Invariant Continuous Systems with Regular Pencils

by Iqbal M. Batiha, Reyad El-Khazali, Ahmed AlSaedi and Shaher Momani

Entropy 2018, 20(6), 400; https://0-doi-org.brum.beds.ac.uk/10.3390/e20060400 - 23 May 2018

Cited by 28 | Viewed by 3414

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)

28 pages, 14466 KiB

Open AccessArticle

Quantifying Chaos by Various Computational Methods. Part 1: Simple Systems

by Jan Awrejcewicz, Anton V. Krysko, Nikolay P. Erofeev, Vitalyj Dobriyan, Marina A. Barulina and Vadim A. Krysko

Entropy 2018, 20(3), 175; https://0-doi-org.brum.beds.ac.uk/10.3390/e20030175 - 06 Mar 2018

Cited by 42 | Viewed by 5182

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)

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13 pages, 35014 KiB

Open AccessFeature PaperArticle

Quantifying Chaos by Various Computational Methods. Part 2: Vibrations of the Bernoulli–Euler Beam Subjected to Periodic and Colored Noise

by Jan Awrejcewicz, Anton V. Krysko, Nikolay P. Erofeev, Vitalyi Dobriyan, Marina A. Barulina and Vadim A. Krysko

Entropy 2018, 20(3), 170; https://0-doi-org.brum.beds.ac.uk/10.3390/e20030170 - 05 Mar 2018

Cited by 14 | Viewed by 3821

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)

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11 pages, 296 KiB

Open AccessArticle

On Points Focusing Entropy

by Ewa Korczak-Kubiak, Anna Loranty and Ryszard J. Pawlak

Entropy 2018, 20(2), 128; https://0-doi-org.brum.beds.ac.uk/10.3390/e20020128 - 16 Feb 2018

Cited by 5 | Viewed by 3281

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)

15 pages, 276 KiB

Open AccessArticle

Analytical Solutions for Multi-Time Scale Fractional Stochastic Differential Equations Driven by Fractional Brownian Motion and Their Applications

by Xiao-Li Ding and Juan J. Nieto

Entropy 2018, 20(1), 63; https://0-doi-org.brum.beds.ac.uk/10.3390/e20010063 - 16 Jan 2018

Cited by 18 | Viewed by 4252

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)

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