Special Issue "Entropy in Dynamic Systems II"

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Complexity".

Deadline for manuscript submissions: 15 December 2022 | Viewed by 5404

Special Issue Editors

Prof. Dr. José A. Tenreiro Machado
grade Website
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, complex systems
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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. Jan Awrejcewicz
Prof. José A. Tenreiro 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 submissions that pass pre-check are 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 1800 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 (5 papers)

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Research

Article
Non-Linear Observer Design with Laguerre Polynomials
Entropy 2022, 24(7), 913; https://doi.org/10.3390/e24070913 - 30 Jun 2022
Viewed by 299
Abstract
In this paper, a methodology for a non-linear system state estimation is demonstrated, exploiting the input and parameter observability. For this purpose, the initial system is transformed into the canonical observability form, and the function that aggregates the non-linear dynamics of the system, [...] Read more.
In this paper, a methodology for a non-linear system state estimation is demonstrated, exploiting the input and parameter observability. For this purpose, the initial system is transformed into the canonical observability form, and the function that aggregates the non-linear dynamics of the system, which may be unknown or difficult to be computed, is approximated by a linear combination of Laguerre polynomials. Hence, the system identification translates into the estimation of the parameters involved in the linear combination in order for the system to be observable. For the validation of the elaborated observer, we consider a biological model from the literature, investigating whether it is practically possible to infer its states, taking into account the new coordinates to design the appropriate observer of the system states. Through simulations, we investigate the parameter settings under which the new observer can identify the state of the system. More specifically, as the parameter θ increases, the system converges more quickly to the steady-state, decreasing the respective distance from the system’s initial state. As for the first state, the estimation error is in the order of 102 for θ=15, and assuming c0={0,1},c1=1. Under the same conditions, the estimation error of the system’s second state is in the order of 101, setting a performance difference of 101 in relation to the first state. The outcomes show that the proposed observer’s performance can be further improved by selecting even higher values of θ. Hence, the system is observable through the measurement output. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems II)
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Article
Information Hiding Based on Statistical Features of Self-Organizing Patterns
Entropy 2022, 24(5), 684; https://doi.org/10.3390/e24050684 - 12 May 2022
Viewed by 473
Abstract
A computational technique for the determination of optimal hiding conditions of a digital image in a self-organizing pattern is presented in this paper. Three statistical features of the developing pattern (the Wada index based on the weighted and truncated Shannon entropy, the mean [...] Read more.
A computational technique for the determination of optimal hiding conditions of a digital image in a self-organizing pattern is presented in this paper. Three statistical features of the developing pattern (the Wada index based on the weighted and truncated Shannon entropy, the mean of the brightness of the pattern, and the p-value of the Kolmogorov-Smirnov criterion for the normality testing of the distribution function) are used for that purpose. The transition from the small-scale chaos of the initial conditions to the large-scale chaos of the developed pattern is observed during the evolution of the self-organizing system. Computational experiments are performed with the stripe-type patterns, spot-type patterns, and unstable patterns. It appears that optimal image hiding conditions are secured when the Wada index stabilizes after the initial decline, the mean of the brightness of the pattern remains stable before dropping down significantly below the average, and the p-value indicates that the distribution becomes Gaussian. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems II)
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Article
Quantifying Information without Entropy: Identifying Intermittent Disturbances in Dynamical Systems
Entropy 2020, 22(11), 1199; https://doi.org/10.3390/e22111199 - 23 Oct 2020
Cited by 2 | Viewed by 826
Abstract
A system’s response to disturbances in an internal or external driving signal can be characterized as performing an implicit computation, where the dynamics of the system are a manifestation of its new state holding some memory about those disturbances. Identifying small disturbances in [...] Read more.
A system’s response to disturbances in an internal or external driving signal can be characterized as performing an implicit computation, where the dynamics of the system are a manifestation of its new state holding some memory about those disturbances. Identifying small disturbances in the response signal requires detailed information about the dynamics of the inputs, which can be challenging. This paper presents a new method called the Information Impulse Function (IIF) for detecting and time-localizing small disturbances in system response data. The novelty of IIF is its ability to measure relative information content without using Boltzmann’s equation by modeling signal transmission as a series of dissipative steps. Since a detailed expression of the informational structure in the signal is achieved with IIF, it is ideal for detecting disturbances in the response signal, i.e., the system dynamics. Those findings are based on numerical studies of the topological structure of the dynamics of a nonlinear system due to perturbated driving signals. The IIF is compared to both the Permutation entropy and Shannon entropy to demonstrate its entropy-like relationship with system state and its degree of sensitivity to perturbations in a driving signal. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems II)
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Article
Optimization for Software Implementation of Fractional Calculus Numerical Methods in an Embedded System
Entropy 2020, 22(5), 566; https://doi.org/10.3390/e22050566 - 18 May 2020
Cited by 1 | Viewed by 1799
Abstract
In this article, some practical software optimization methods for implementations of fractional order backward difference, sum, and differintegral operator based on Grünwald–Letnikov definition are presented. These numerical algorithms are of great interest in the context of the evaluation of fractional-order differential equations in [...] Read more.
In this article, some practical software optimization methods for implementations of fractional order backward difference, sum, and differintegral operator based on Grünwald–Letnikov definition are presented. These numerical algorithms are of great interest in the context of the evaluation of fractional-order differential equations in embedded systems, due to their more convenient form compared to Caputo and Riemann–Liouville definitions or Laplace transforms, based on the discrete convolution operation. A well-known difficulty relates to the non-locality of the operator, implying continually increasing numbers of processed samples, which may reach the limits of available memory or lead to exceeding the desired computation time. In the study presented here, several promising software optimization techniques were analyzed and tested in the evaluation of the variable fractional-order backward difference and derivative on two different Arm® Cortex®-M architectures. Reductions in computation times of up to 75% and 87% were achieved compared to the initial implementation, depending on the type of Arm® core. Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems II)
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Article
Hausdorff Dimension and Topological Entropies of a Solenoid
Entropy 2020, 22(5), 506; https://doi.org/10.3390/e22050506 - 28 Apr 2020
Cited by 2 | Viewed by 1247 | Correction
Abstract
The purpose of this paper is to elucidate the interrelations between three essentially different concepts: solenoids, topological entropy, and Hausdorff dimension. For this purpose, we describe the dynamics of a solenoid by topological entropy-like quantities and investigate the relations between them. For L-Lipschitz [...] Read more.
The purpose of this paper is to elucidate the interrelations between three essentially different concepts: solenoids, topological entropy, and Hausdorff dimension. For this purpose, we describe the dynamics of a solenoid by topological entropy-like quantities and investigate the relations between them. For L-Lipschitz solenoids and locally λ expanding solenoids, we show that the topological entropy and fractal dimensions are closely related. For a locally λ expanding solenoid, we prove that its topological entropy is lower estimated by the Hausdorff dimension of X multiplied by the logarithm of λ . Full article
(This article belongs to the Special Issue Entropy in Dynamic Systems II)
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