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Special Issue "Complex Systems and Fractional Dynamics"

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

Deadline for manuscript submissions: closed (31 May 2017)

Special Issue Editors

Guest Editor
Prof. Dr. António M. Lopes

UISPA – LAETA/INEGI, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, Porto, Portugal
E-Mail
Phone: +351 22 5081758
Fax: +351 22 5081445
Interests: complex systems; nonlinear dynamics; robotics; control; fractional calculus; simulation
Guest Editor
Prof. Dr. J. A. Tenreiro Machado

Institute of Engineering, Department of Electrical Engineering, Polytechnic Institute of Porto, R. Dr. Roberto Frias, 4200-465 Porto, Portugal
Website | E-Mail
Phone: +351 22 8340500
Fax: +351 22 8321159
Interests: complex systems; nonlinear dynamics; fractional calculus; modeling; entropy; control; evolutionary computing; genomics

Special Issue Information

Dear Colleagues,

Complex systems are pervasive in many areas of science and we find them everyday and everywhere. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of large number of interconnected and interacting entities exhibiting much richer global scale dynamics than they could be inferred from the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences.

This Special Issue focuses on original and new research results on systems dynamics in science and engineering. Manuscripts in complex dynamical systems, nonlinearity, chaos and fractional dynamics in the thermodynamics or information processing perspectives are solicited. We welcome submissions addressing novel issues, as well as those on more specific topics illustrating the broad impact of entropy-based techniques in complexity, nonlinearity and fractionality.

Papers should fit the scope of the journal Entropy and topics of interest include (but are not limited to):
- Complex dynamics
- Nonlinear dynamical systems
- Advanced control systems
- Fractional calculus and its applications
- Evolutionary computing
- Finance and economy dynamics
- Fractals and chaos
- Biological systems and bioinformatics
- Nonlinear waves and acoustics
- Image and signal processing
- Transportation systems
- Geosciences
- Astronomy and cosmology
- Nuclear physics

Kindly note that submissions need to include content about entropy or information theory so that they fit within the scope of this journal.


Prof. Dr. J. A. Tenreiro Machado
Prof. Dr. António M. Lopes
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 1500 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.

Keywords

  • Dynamics
  • Complex systems
  • Fractional calculus
  • Entropy
  • Information

Published Papers (13 papers)

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Editorial

Jump to: Research

Open AccessEditorial Complex Systems and Fractional Dynamics
Entropy 2018, 20(7), 507; https://doi.org/10.3390/e20070507
Received: 28 June 2018 / Accepted: 2 July 2018 / Published: 3 July 2018
PDF Full-text (169 KB) | HTML Full-text | XML Full-text
Abstract
Complex systems (CS) are pervasive in many areas of science and technology, namely in financial
markets, transportation, telecommunication and social networks, world and country economies,
immunological systems, living organisms, computational systems, and electrical and mechanical
structures [...] Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)

Research

Jump to: Editorial

Open AccessArticle Fractional Derivative Phenomenology of Percolative Phonon-Assisted Hopping in Two-Dimensional Disordered Systems
Entropy 2017, 19(9), 463; https://doi.org/10.3390/e19090463
Received: 22 June 2017 / Revised: 28 August 2017 / Accepted: 28 August 2017 / Published: 1 September 2017
Cited by 1 | PDF Full-text (2982 KB) | HTML Full-text | XML Full-text
Abstract
Anomalous advection-diffusion in two-dimensional semiconductor systems with coexisting energetic and structural disorder is described in the framework of a generalized model of multiple trapping on a comb-like structure. The basic equations of the model contain fractional-order derivatives. To validate the model, we compare
[...] Read more.
Anomalous advection-diffusion in two-dimensional semiconductor systems with coexisting energetic and structural disorder is described in the framework of a generalized model of multiple trapping on a comb-like structure. The basic equations of the model contain fractional-order derivatives. To validate the model, we compare analytical solutions with results of a Monte Carlo simulation of phonon-assisted tunneling in two-dimensional patterns of a porous nanoparticle agglomerate and a phase-separated bulk heterojunction. To elucidate the role of directed percolation, we calculate transient current curves of the time-of-flight experiment and the evolution of the mean squared displacement averaged over medium realizations. The variations of the anomalous advection-diffusion parameters as functions of electric field intensity, levels of energetic, and structural disorder are presented. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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Open AccessArticle On the Modelling and Control of a Laboratory Prototype of a Hydraulic Canal Based on a TITO Fractional-Order Model
Entropy 2017, 19(8), 401; https://doi.org/10.3390/e19080401
Received: 30 June 2017 / Revised: 28 July 2017 / Accepted: 1 August 2017 / Published: 3 August 2017
Cited by 2 | PDF Full-text (6305 KB) | HTML Full-text | XML Full-text
Abstract
In this paper a two-input, two-output (TITO) fractional order mathematical model of a laboratory prototype of a hydraulic canal is proposed. This canal is made up of two pools that have a strong interaction between them. The inputs of the TITO model are
[...] Read more.
In this paper a two-input, two-output (TITO) fractional order mathematical model of a laboratory prototype of a hydraulic canal is proposed. This canal is made up of two pools that have a strong interaction between them. The inputs of the TITO model are the pump flow and the opening of an intermediate gate, and the two outputs are the water levels in the two pools. Based on the experiments developed in a laboratory prototype the parameters of the mathematical models have been identified. Then, considering the TITO model, a first control loop of the pump is closed to reproduce real-world conditions in which the water level of the first pool is not dependent on the opening of the upstream gate, thus leading to an equivalent single input, single output (SISO) system. The comparison of the resulting system with the classical first order systems typically utilized to model hydraulic canals shows that the proposed model has significantly lower error: about 50%, and, therefore, higher accuracy in capturing the canal dynamics. This model has also been utilized to optimize the design of the controller of the pump of the canal, thus achieving a faster response to step commands and thus minimizing the interaction between the two pools of the experimental platform. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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Open AccessArticle Toward a Theory of Industrial Supply Networks: A Multi-Level Perspective via Network Analysis
Entropy 2017, 19(8), 382; https://doi.org/10.3390/e19080382
Received: 30 May 2017 / Revised: 21 July 2017 / Accepted: 21 July 2017 / Published: 25 July 2017
Cited by 1 | PDF Full-text (22422 KB) | HTML Full-text | XML Full-text
Abstract
In most supply chains (SCs), transaction relationships between suppliers and customers are commonly considered to be an extrapolation from a linear perspective. However, this traditional linear concept of an SC is egotistic and oversimplified and does not sufficiently reflect the complex and cyclical
[...] Read more.
In most supply chains (SCs), transaction relationships between suppliers and customers are commonly considered to be an extrapolation from a linear perspective. However, this traditional linear concept of an SC is egotistic and oversimplified and does not sufficiently reflect the complex and cyclical structure of supplier-customer relationships in current economic and industrial situations. The interactional relationships and topological characteristics between suppliers and customers should be analyzed using supply networks (SNs) rather than traditional linear SCs. Therefore, this paper reconceptualizes SCs as SNs in complex adaptive systems (CAS), and presents three main contributions. First, we propose an integrated framework of CAS network by synthesizing multi-level network analysis from the network-, community- and vertex-perspective. The CAS perspective enables us to understand the advances of SN properties. Second, in order to emphasize the CAS properties of SNs, we conducted a real-world SN based on the Japanese industry and describe an advanced investigation of SN theory. The CAS properties help in enriching the SN theory, which can benefit SN management, community economics and industrial resilience. Third, we propose a quantitative metric of entropy to measure the complexity and robustness of SNs. The results not only support a specific understanding of the structural outcomes relevant to SNs, but also deliver efficient and effective support to the management and design of SNs. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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Open AccessArticle A Novel Numerical Approach for a Nonlinear Fractional Dynamical Model of Interpersonal and Romantic Relationships
Entropy 2017, 19(7), 375; https://doi.org/10.3390/e19070375
Received: 28 May 2017 / Revised: 4 July 2017 / Accepted: 19 July 2017 / Published: 22 July 2017
Cited by 20 | PDF Full-text (1978 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we propose a new numerical algorithm, namely q-homotopy analysis Sumudu transform method (q-HASTM), to obtain the approximate solution for the nonlinear fractional dynamical model of interpersonal and romantic relationships. The suggested algorithm examines the dynamics of love
[...] Read more.
In this paper, we propose a new numerical algorithm, namely q-homotopy analysis Sumudu transform method (q-HASTM), to obtain the approximate solution for the nonlinear fractional dynamical model of interpersonal and romantic relationships. The suggested algorithm examines the dynamics of love affairs between couples. The q-HASTM is a creative combination of Sumudu transform technique, q-homotopy analysis method and homotopy polynomials that makes the calculation very easy. To compare the results obtained by using q-HASTM, we solve the same nonlinear problem by Adomian’s decomposition method (ADM). The convergence of the q-HASTM series solution for the model is adapted and controlled by auxiliary parameter ℏ and asymptotic parameter n. The numerical results are demonstrated graphically and in tabular form. The result obtained by employing the proposed scheme reveals that the approach is very accurate, effective, flexible, simple to apply and computationally very nice. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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Open AccessArticle Two Approaches to Obtaining the Space-Time Fractional Advection-Diffusion Equation
Entropy 2017, 19(7), 297; https://doi.org/10.3390/e19070297
Received: 3 May 2017 / Revised: 13 June 2017 / Accepted: 21 June 2017 / Published: 23 June 2017
Cited by 3 | PDF Full-text (1044 KB) | HTML Full-text | XML Full-text
Abstract
Two approaches resulting in two different generalizations of the space-time-fractional advection-diffusion equation are discussed. The Caputo time-fractional derivative and Riesz fractional Laplacian are used. The fundamental solutions to the corresponding Cauchy and source problems in the case of one spatial variable are studied
[...] Read more.
Two approaches resulting in two different generalizations of the space-time-fractional advection-diffusion equation are discussed. The Caputo time-fractional derivative and Riesz fractional Laplacian are used. The fundamental solutions to the corresponding Cauchy and source problems in the case of one spatial variable are studied using the Laplace transform with respect to time and the Fourier transform with respect to the spatial coordinate. The numerical results are illustrated graphically. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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Open AccessArticle Analytical Approximate Solutions of (n + 1)-Dimensional Fractal Heat-Like and Wave-Like Equations
Entropy 2017, 19(7), 296; https://doi.org/10.3390/e19070296
Received: 27 May 2017 / Revised: 15 June 2017 / Accepted: 20 June 2017 / Published: 22 June 2017
Cited by 1 | PDF Full-text (1634 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we propose a new type (n + 1)-dimensional reduced differential transform method (RDTM) based on a local fractional derivative (LFD) to solve (n + 1)-dimensional local fractional partial differential equations (PDEs) in Cantor sets. The presented method is
[...] Read more.
In this paper, we propose a new type (n + 1)-dimensional reduced differential transform method (RDTM) based on a local fractional derivative (LFD) to solve (n + 1)-dimensional local fractional partial differential equations (PDEs) in Cantor sets. The presented method is named the (n + 1)-dimensional local fractional reduced differential transform method (LFRDTM). First the theories, their proofs and also some basic properties of this procedure are given. To understand the introduced method clearly, we apply it on the (n + 1)-dimensional fractal heat-like equations (HLEs) and wave-like equations (WLEs). The applications show that this new technique is efficient, simply applicable and has powerful effects in (n + 1)-dimensional local fractional problems. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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Open AccessArticle Information Technology Project Portfolio Implementation Process Optimization Based on Complex Network Theory and Entropy
Entropy 2017, 19(6), 287; https://doi.org/10.3390/e19060287
Received: 4 April 2017 / Revised: 12 June 2017 / Accepted: 13 June 2017 / Published: 19 June 2017
Cited by 2 | PDF Full-text (6965 KB) | HTML Full-text | XML Full-text
Abstract
In traditional information technology project portfolio management (ITPPM), managers often pay more attention to the optimization of portfolio selection in the initial stage. In fact, during the portfolio implementation process, there are still issues to be optimized. Organizing cooperation will enhance the efficiency,
[...] Read more.
In traditional information technology project portfolio management (ITPPM), managers often pay more attention to the optimization of portfolio selection in the initial stage. In fact, during the portfolio implementation process, there are still issues to be optimized. Organizing cooperation will enhance the efficiency, although it brings more immediate risk due to the complex variety of links between projects. In order to balance the efficiency and risk, an optimization method is presented based on the complex network theory and entropy, which will assist portfolio managers in recognizing the structure of the portfolio and determine the cooperation range. Firstly, a complex network model for an IT project portfolio is constructed, in which the project is simulated as an artificial life agent. At the same time, the portfolio is viewed as a small scale of society. Following this, social network analysis is used to detect and divide communities in order to estimate the roles of projects between different portfolios. Based on these, the efficiency and the risk are measured using entropy and are balanced through searching for adequate hierarchy community divisions. Thus, the activities of cooperation in organizations, risk management, and so on—which are usually viewed as an important art—can be discussed and conducted based on quantity calculations. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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Open AccessArticle Minimum Sample Size for Reliable Causal Inference Using Transfer Entropy
Entropy 2017, 19(4), 150; https://doi.org/10.3390/e19040150
Received: 20 February 2017 / Revised: 29 March 2017 / Accepted: 29 March 2017 / Published: 31 March 2017
Cited by 1 | PDF Full-text (698 KB) | HTML Full-text | XML Full-text
Abstract
Transfer Entropy has been applied to experimental datasets to unveil causality between variables. In particular, its application to non-stationary systems has posed a great challenge due to restrictions on the sample size. Here, we have investigated the minimum sample size that produces a
[...] Read more.
Transfer Entropy has been applied to experimental datasets to unveil causality between variables. In particular, its application to non-stationary systems has posed a great challenge due to restrictions on the sample size. Here, we have investigated the minimum sample size that produces a reliable causal inference. The methodology has been applied to two prototypical models: the linear model autoregressive-moving average and the non-linear logistic map. The relationship between the Transfer Entropy value and the sample size has been systematically examined. Additionally, we have shown the dependence of the reliable sample size and the strength of coupling between the variables. Our methodology offers a realistic lower bound for the sample size to produce a reliable outcome. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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Open AccessArticle Fractional Jensen–Shannon Analysis of the Scientific Output of Researchers in Fractional Calculus
Entropy 2017, 19(3), 127; https://doi.org/10.3390/e19030127
Received: 9 March 2017 / Revised: 14 March 2017 / Accepted: 15 March 2017 / Published: 17 March 2017
Cited by 4 | PDF Full-text (359 KB) | HTML Full-text | XML Full-text
Abstract
This paper analyses the citation profiles of researchers in fractional calculus. Different metrics are used to quantify the dissimilarities between the data, namely the Canberra distance, and the classical and the generalized (fractional) Jensen–Shannon divergence. The information is then visualized by means of
[...] Read more.
This paper analyses the citation profiles of researchers in fractional calculus. Different metrics are used to quantify the dissimilarities between the data, namely the Canberra distance, and the classical and the generalized (fractional) Jensen–Shannon divergence. The information is then visualized by means of multidimensional scaling and hierarchical clustering. The mathematical tools and metrics allow for direct comparison and visualization of researchers based on their relative positioning and on patterns displayed in two- or three-dimensional maps. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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Open AccessArticle Multiplicity of Homoclinic Solutions for Fractional Hamiltonian Systems with Subquadratic Potential
Entropy 2017, 19(2), 50; https://doi.org/10.3390/e19020050
Received: 24 November 2016 / Revised: 11 January 2017 / Accepted: 19 January 2017 / Published: 24 January 2017
Cited by 1 | PDF Full-text (326 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we study the existence of homoclinic solutions for the fractional Hamiltonian systems with left and right Liouville–Weyl derivatives. We establish some new results concerning the existence and multiplicity of homoclinic solutions for the given system by using Clark’s theorem from
[...] Read more.
In this paper, we study the existence of homoclinic solutions for the fractional Hamiltonian systems with left and right Liouville–Weyl derivatives. We establish some new results concerning the existence and multiplicity of homoclinic solutions for the given system by using Clark’s theorem from critical point theory and fountain theorem. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
Open AccessArticle Research Entropy Complexity about the Nonlinear Dynamic Delay Game Model
Entropy 2017, 19(1), 22; https://doi.org/10.3390/e19010022
Received: 9 November 2016 / Revised: 15 December 2016 / Accepted: 22 December 2016 / Published: 9 January 2017
Cited by 2 | PDF Full-text (799 KB) | HTML Full-text | XML Full-text
Abstract
Based on the research of domestic and foreign scholars, this paper has improved and established a double oligopoly market model of renewable energy, and analyzed the complex dynamic characteristics of a system based on entropy theory and chaos theory, such as equilibrium point,
[...] Read more.
Based on the research of domestic and foreign scholars, this paper has improved and established a double oligopoly market model of renewable energy, and analyzed the complex dynamic characteristics of a system based on entropy theory and chaos theory, such as equilibrium point, stability, Hopf bifurcation conditions, etc. This paper also studied and simulated the effects of the natural growth rate of energy and the single delay decision on the renewable energy system by minimizing the entropy of the system and reducing the system instability to a minimum, so that the degree of disorder within the system was reduced. The results show that with the increase of the natural growth rate of energy, the stability of the system is not affected, but the market demand of the oligopoly 1 is gradually reducing and the market demand of the oligopoly 2 is gradually increasing. At the same time, a single oligopoly making the time delay decision will affect the stability of the two oligopolies. With the increase of delay, the time required to reach the stable state will grow, and the system will eventually enter the Hopf bifurcation, thus the system will have its entropy increased and fall into an unstable state. Therefore, in the actual market of renewable energy, oligopolies should pay attention to the natural growth rate of energy and time delay, ensuring the stability of the game process and the orderliness of the system. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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Open AccessArticle Fractional-Order Identification and Control of Heating Processes with Non-Continuous Materials
Entropy 2016, 18(11), 398; https://doi.org/10.3390/e18110398
Received: 26 September 2016 / Revised: 4 November 2016 / Accepted: 9 November 2016 / Published: 12 November 2016
Cited by 5 | PDF Full-text (1804 KB) | HTML Full-text | XML Full-text
Abstract
The paper presents a fractional order model of a heating process and a comparison of fractional and standard PI controllers in its closed loop system. Preliminarily, an enhanced fractional order model for the heating process on non-continuous materials has been identified through a
[...] Read more.
The paper presents a fractional order model of a heating process and a comparison of fractional and standard PI controllers in its closed loop system. Preliminarily, an enhanced fractional order model for the heating process on non-continuous materials has been identified through a fitting algorithm on experimental data. Experimentation has been carried out on a finite length beam filled with three non-continuous materials (air, styrofoam, metal buckshots) in order to identify a model in the frequency domain and to obtain a relationship between the fractional order of the heating process and the different materials’ properties. A comparison between the experimental model and the theoretical one has been performed, proving a significant enhancement of the fitting performances. Moreover the obtained modelling results confirm the fractional nature of the heating processes when diffusion occurs in non-continuous composite materials, and they show how the model’s fractional order can be used as a characteristic parameter for non-continuous materials with different composition and structure. Finally, three different kinds of controllers have been applied and compared in order to keep constant the beam temperature constant at a fixed length. Full article
(This article belongs to the Special Issue Complex Systems and Fractional Dynamics)
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