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Special Issue "Complex Systems and Nonlinear Dynamics"

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

Deadline for manuscript submissions: closed (15 October 2014)

Special Issue Editor

Guest Editor
Prof. Dr. J. A. Tenreiro Machado

Department of Electrical Engineering, Institute of Engineering, Polytechnic Institute of Porto, Porto, Portugal
Website | E-Mail
Phone: +351 22 8340500
Fax: +351 22 8321159
Interests: nonlinear dynamics; fractional calculus; modeling; control; evolutionary computing; genomics; robotics; and intelligent transportation systems

Special Issue Information

Dear Colleagues,

Complex systems are pervasive in many areas of science and we find them every day and everywhere. Examples include financial markets, highway transportation networks, telecommunication networks, world and national economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of large numbers of interconnected and interacting entities that exhibit much richer global scale dynamics than can be inferred from the properties and behaviors of individual entities. Complex systems are studied in many areas of the natural sciences, social sciences, and engineering and mathematical sciences.

This Special Issue focuses on original and new research results concerning systems dynamics in science and engineering. Manuscripts discussing 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 addressing more specific topics that illustrate the broad impact of entropy-based techniques in complexity, nonlinearity, and fractionality.

Specific topics of interest include (but are not limited to):

Complex dynamics
Nonlinear dynamical systems
Advanced control systems
Fractional calculus and its applications
Chemical dynamics
Economical dynamics and predictions
Dynamical systems synchronization
Biological systems and bioinformatics
Nonlinear waves and acoustics
Image and signal processing
Transportation system
Biomedical engineering

Prof. J. A. Tenreiro Machado
Guest Editor

Submission

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. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as 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 refereed through a 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 1400 CHF (Swiss Francs).


Published Papers (15 papers)

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Research

Open AccessArticle Robust H Finite-Time Control for Discrete Markovian Jump Systems with Disturbances of Probabilistic Distributions
Entropy 2015, 17(1), 346-367; doi:10.3390/e17010346
Received: 15 October 2014 / Accepted: 5 January 2015 / Published: 14 January 2015
Cited by 5 | PDF Full-text (148 KB) | HTML Full-text | XML Full-text
Abstract
This paper is concerned with the robust H finite-time control for discrete delayed nonlinear systems with Markovian jumps and external disturbances. It is usually assumed that the disturbance affects the system states and outputs with the same influence degree of 100%, which
[...] Read more.
This paper is concerned with the robust H finite-time control for discrete delayed nonlinear systems with Markovian jumps and external disturbances. It is usually assumed that the disturbance affects the system states and outputs with the same influence degree of 100%, which is not evident enough to reflect the situation where the disturbance affects these two parts by different influence degrees. To tackle this problem, a probabilistic distribution denoted by binomial sequences is introduced to describe the external disturbance. Throughout the paper, the definitions of the finite-time boundedness (FTB) and the H FTB are firstly given respectively. To extend the results further, a model which combines a linear dynamic system and a static nonlinear operator is referred to describe the system under discussion. Then by virtue of state feedback control method, some new sufficient criteria are derived which guarantee the FTB and H FTB performances for the considered system. Finally, an example is provided to demonstrate the effectiveness of the developed control laws. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Chaos Control and Synchronization of a Hyperchaotic Zhou System by Integral Sliding Mode control
Entropy 2014, 16(12), 6539-6552; doi:10.3390/e16126539
Received: 25 September 2014 / Revised: 25 November 2014 / Accepted: 3 December 2014 / Published: 12 December 2014
Cited by 2 | PDF Full-text (306 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, an adaptive integral sliding mode control scheme is proposed for synchronization of hyperchaotic Zhou systems. In the proposed scheme, an integral sliding mode control is designed to stabilize a hyperchaotic Zhou system with known parameters to its unstable equilibrium at
[...] Read more.
In this paper, an adaptive integral sliding mode control scheme is proposed for synchronization of hyperchaotic Zhou systems. In the proposed scheme, an integral sliding mode control is designed to stabilize a hyperchaotic Zhou system with known parameters to its unstable equilibrium at the origin. The control is then applied to the synchronization of two identical systems, i.e., a slave and a master hyperchaotic Zhou system with unknown parameters. The adaptive control mechanism introduced synchronizes the systems by estimating the unknown parameters. Simulation results have shown that the proposed method has an excellent convergence from both speed and accuracy points of view, and it outperforms Vaidyanathan’s scheme, which is a well-recognized scheme in this area. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Consensus of Discrete Multiagent System with Various Time Delays and Environmental Disturbances
Entropy 2014, 16(12), 6524-6538; doi:10.3390/e16126524
Received: 4 October 2014 / Revised: 9 December 2014 / Accepted: 9 December 2014 / Published: 11 December 2014
Cited by 2 | PDF Full-text (209 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the consensus problem of discrete multiagent systems with time varying sampling periods is studied. Firstly, with thorough analysis of various delays among agents, the control input of each agent is designed with consideration of sending delay and receiving delay. With
[...] Read more.
In this paper, the consensus problem of discrete multiagent systems with time varying sampling periods is studied. Firstly, with thorough analysis of various delays among agents, the control input of each agent is designed with consideration of sending delay and receiving delay. With construction of discrete dynamics of state error vector, it is proved by applying Halanay inequality that consensus of the system can be reached. Further, the definition of bounded consensus is proposed in the situation where environmental disturbances exist. In order to handle this problem, the Halanay inequality is extended into a more general one with boundedness property. Based on the new Halanay inequality obtained, the boundedness of consensus error is guaranteed. At last, simulation examples are presented to demonstrate the theoretical conclusions. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle A Memristive Diode Bridge-Based Canonical Chua’s Circuit
Entropy 2014, 16(12), 6464-6476; doi:10.3390/e16126464
Received: 30 September 2014 / Revised: 19 November 2014 / Accepted: 2 December 2014 / Published: 8 December 2014
Cited by 8 | PDF Full-text (1505 KB) | HTML Full-text | XML Full-text
Abstract
A novel memristor circuit is presented, which is generated from the canonical Chua’s circuit by replacing the Chua’s diode with a first order memristive diode bridge. The circuit dynamical characteristics with the variations of circuit parameters are investigated both theoretically and numerically. It
[...] Read more.
A novel memristor circuit is presented, which is generated from the canonical Chua’s circuit by replacing the Chua’s diode with a first order memristive diode bridge. The circuit dynamical characteristics with the variations of circuit parameters are investigated both theoretically and numerically. It can be found that the circuit has three determined equilibrium points, including a zero saddle point and two nonzero saddle-foci with index 2. Specially, the circuit is non-dissipative in the neighborhood of the zero saddle point, and there exists complex nonlinear phenomena of coexisting bifurcation modes and coexisting chaotic attractors. Experimental observations are performed to verify the numerical simulation results. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Detecting Chaos from Agricultural Product Price Time Series
Entropy 2014, 16(12), 6415-6433; doi:10.3390/e16126415
Received: 13 July 2014 / Revised: 28 September 2014 / Accepted: 20 November 2014 / Published: 5 December 2014
Cited by 1 | PDF Full-text (1714 KB) | HTML Full-text | XML Full-text
Abstract
Analysis of the characteristics of agricultural product price volatility and trend forecasting are necessary to formulate and implement agricultural price control policies. Taking wholesale cabbage prices as an example, a multiple test methodology has been adopted to identify the nonlinearity, fractality, and chaos
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Analysis of the characteristics of agricultural product price volatility and trend forecasting are necessary to formulate and implement agricultural price control policies. Taking wholesale cabbage prices as an example, a multiple test methodology has been adopted to identify the nonlinearity, fractality, and chaos of the data. The approaches used include the R/S analysis, the BDS test, the power spectra, the recurrence plot, the largest Lyapunov exponent, the Kolmogorov entropy, and the correlation dimension. The results show that there is chaos in agricultural wholesale price data, which provides a good theoretical basis for selecting reasonable forecasting models as prediction techniques based on chaos theory can be applied to forecasting agricultural prices. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Adaptive Synchronization of Fractional Neural Networks with Unknown Parameters and Time Delays
Entropy 2014, 16(12), 6286-6299; doi:10.3390/e16126286
Received: 11 October 2014 / Revised: 23 November 2014 / Accepted: 26 November 2014 / Published: 1 December 2014
Cited by 9 | PDF Full-text (1391 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain
[...] Read more.
In this paper, the parameters identification and synchronization problem of fractional-order neural networks with time delays are investigated. Based on some analytical techniques and an adaptive control method, a simple adaptive synchronization controller and parameter update laws are designed to synchronize two uncertain complex networks with time delays. Besides, the system parameters in the uncertain network can be identified in the process of synchronization. To demonstrate the validity of the proposed method, several illustrative examples are presented. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Complex Modified Hybrid Projective Synchronization of Different Dimensional Fractional-Order Complex Chaos and Real Hyper-Chaos
Entropy 2014, 16(12), 6195-6211; doi:10.3390/e16126195
Received: 27 August 2014 / Revised: 16 November 2014 / Accepted: 17 November 2014 / Published: 27 November 2014
Cited by 8 | PDF Full-text (2785 KB) | HTML Full-text | XML Full-text
Abstract
This paper introduces a type of modified hybrid projective synchronization with complex transformationmatrix (CMHPS) for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformationmatrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based
[...] Read more.
This paper introduces a type of modified hybrid projective synchronization with complex transformationmatrix (CMHPS) for different dimensional fractional-order complex chaos and fractional-order real hyper-chaos. The transformationmatrix in this type of chaotic synchronization is a non-square matrix, and its elements are complex numbers. Based on the stability theory of fractional-order systems, by employing the feedback control technique, necessary and sufficient criteria on CMHPS are derived. Furthermore, CMHPS between fractional-order real hyper-chaotic Rössler system and other two different dimensional fractional-order complex Lorenz-like chaotic systems is provided as two examples to discuss reduced order and increased order synchronization, respectively. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis
Entropy 2014, 16(11), 5838-5852; doi:10.3390/e16115838
Received: 17 October 2014 / Accepted: 31 October 2014 / Published: 6 November 2014
Cited by 10 | PDF Full-text (503 KB) | HTML Full-text | XML Full-text
Abstract
Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of
[...] Read more.
Fractional order derivative operators offer a concise description to model multi-scale, heterogeneous and non-local systems. Specifically, in magnetic resonance imaging, there has been recent work to apply fractional order derivatives to model the non-Gaussian diffusion signal, which is ubiquitous in the movement of water protons within biological tissue. To provide a new perspective for establishing the utility of fractional order models, we apply entropy for the case of anomalous diffusion governed by a fractional order diffusion equation generalized in space and in time. This fractional order representation, in the form of the Mittag–Leffler function, gives an entropy minimum for the integer case of Gaussian diffusion and greater values of spectral entropy for non-integer values of the space and time derivatives. Furthermore, we consider kurtosis, defined as the normalized fourth moment, as another probabilistic description of the fractional time derivative. Finally, we demonstrate the implementation of anomalous diffusion, entropy and kurtosis measurements in diffusion weighted magnetic resonance imaging in the brain of a chronic ischemic stroke patient. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle A Note on Distance-based Graph Entropies
Entropy 2014, 16(10), 5416-5427; doi:10.3390/e16105416
Received: 26 July 2014 / Revised: 18 September 2014 / Accepted: 15 October 2014 / Published: 20 October 2014
Cited by 17 | PDF Full-text (161 KB) | HTML Full-text | XML Full-text
Abstract
A variety of problems in, e.g., discrete mathematics, computer science, information theory, statistics, chemistry, biology, etc., deal with inferring and characterizing relational structures by using graph measures. In this sense, it has been proven that information-theoretic quantities representing graph entropies possess useful properties
[...] Read more.
A variety of problems in, e.g., discrete mathematics, computer science, information theory, statistics, chemistry, biology, etc., deal with inferring and characterizing relational structures by using graph measures. In this sense, it has been proven that information-theoretic quantities representing graph entropies possess useful properties such as a meaningful structural interpretation and uniqueness. As classical work, many distance-based graph entropies, e.g., the ones due to Bonchev et al. and related quantities have been proposed and studied. Our contribution is to explore graph entropies that are based on a novel information functional, which is the number of vertices with distance \(k\) to a given vertex. In particular, we investigate some properties thereof leading to a better understanding of this new information-theoretic quantity. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Adaptive Leader-Following Consensus of Multi-Agent Systems with Unknown Nonlinear Dynamics
Entropy 2014, 16(9), 5020-5031; doi:10.3390/e16095020
Received: 10 August 2014 / Revised: 13 September 2014 / Accepted: 15 September 2014 / Published: 22 September 2014
Cited by 6 | PDF Full-text (556 KB) | HTML Full-text | XML Full-text
Abstract
This paper deals with the leader-following consensus of multi-agent systems with matched nonlinear dynamics. Compared with previous works, the major difficulty here is caused by the simultaneous existence of nonidentical agent dynamics and unknown system parameters, which are more practical in real-world applications.
[...] Read more.
This paper deals with the leader-following consensus of multi-agent systems with matched nonlinear dynamics. Compared with previous works, the major difficulty here is caused by the simultaneous existence of nonidentical agent dynamics and unknown system parameters, which are more practical in real-world applications. To tackle this difficulty, a distributed adaptive control law for each follower is proposed based on algebraic graph theory and algebraic Riccati equation. By a Lyapunov function method, we show that the designed control law guarantees that each follower asymptotically converges to the leader under connected communication graphs. A simulation example demonstrates the effectiveness of the proposed scheme. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Figures

Open AccessArticle Existence of Entropy Solutions for Nonsymmetric Fractional Systems
Entropy 2014, 16(9), 4911-4922; doi:10.3390/e16094911
Received: 10 July 2014 / Revised: 24 August 2014 / Accepted: 10 September 2014 / Published: 12 September 2014
Cited by 6 | PDF Full-text (216 KB) | HTML Full-text | XML Full-text
Abstract
The present work focuses on entropy solutions for the fractional Cauchy problem of nonsymmetric systems. We impose sufficient conditions on the parameters to obtain bounded solutions of L . The solutions attained are unique and exclusive. Performance is established by utilizing the
[...] Read more.
The present work focuses on entropy solutions for the fractional Cauchy problem of nonsymmetric systems. We impose sufficient conditions on the parameters to obtain bounded solutions of L . The solutions attained are unique and exclusive. Performance is established by utilizing the maximum principle for certain generalized time and space-fractional diffusion equations. The fractional differential operator is inspected based on the interpretation of the Riemann–Liouville differential operator. Fractional entropy inequalities are imposed. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Information Entropy-Based Metrics for Measuring Emergences in Artificial Societies
Entropy 2014, 16(8), 4583-4602; doi:10.3390/e16084583
Received: 21 May 2014 / Revised: 30 June 2014 / Accepted: 12 August 2014 / Published: 15 August 2014
Cited by 2 | PDF Full-text (1661 KB) | HTML Full-text | XML Full-text
Abstract
Emergence is a common phenomenon, and it is also a general and important concept in complex dynamic systems like artificial societies. Usually, artificial societies are used for assisting in resolving several complex social issues (e.g., emergency management, intelligent transportation system) with the aid
[...] Read more.
Emergence is a common phenomenon, and it is also a general and important concept in complex dynamic systems like artificial societies. Usually, artificial societies are used for assisting in resolving several complex social issues (e.g., emergency management, intelligent transportation system) with the aid of computer science. The levels of an emergence may have an effect on decisions making, and the occurrence and degree of an emergence are generally perceived by human observers. However, due to the ambiguity and inaccuracy of human observers, to propose a quantitative method to measure emergences in artificial societies is a meaningful and challenging task. This article mainly concentrates upon three kinds of emergences in artificial societies, including emergence of attribution, emergence of behavior, and emergence of structure. Based on information entropy, three metrics have been proposed to measure emergences in a quantitative way. Meanwhile, the correctness of these metrics has been verified through three case studies (the spread of an infectious influenza, a dynamic microblog network, and a flock of birds) with several experimental simulations on the Netlogo platform. These experimental results confirm that these metrics increase with the rising degree of emergences. In addition, this article also has discussed the limitations and extended applications of these metrics. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Chaos Synchronization Error Technique-Based Defect Pattern Recognition for GIS through Partial Discharge Signal Analysis
Entropy 2014, 16(8), 4566-4582; doi:10.3390/e16084566
Received: 31 May 2014 / Revised: 30 July 2014 / Accepted: 31 July 2014 / Published: 13 August 2014
Cited by 3 | PDF Full-text (692 KB) | HTML Full-text | XML Full-text
Abstract
The work is aimed at using the chaos synchronization error dynamics (CSED) technique for defect pattern recognition in gas insulated switchgear (GIS). The radiated electromagnetic waves generated due to internal defects were measured by the self-made ultrahigh frequency (UHF) micro-strip antenna, so as
[...] Read more.
The work is aimed at using the chaos synchronization error dynamics (CSED) technique for defect pattern recognition in gas insulated switchgear (GIS). The radiated electromagnetic waves generated due to internal defects were measured by the self-made ultrahigh frequency (UHF) micro-strip antenna, so as to determine whether partial discharge will occur. Firstly, a data pretreatment is performed on the measured raw data for the purpose of computational burden reduction. A characteristic matrix is then constructed according to dynamic error trajectories in a chaos synchronization system, subsequent to which characteristics are extracted. A comparison with the existing Hilbert-Huang Transform (HHT) method reveals that the two characteristics extracted from the CSED results presented herein using the fractal theory were recognized at a higher rate pattern. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle Fractional Order Generalized Information
Entropy 2014, 16(4), 2350-2361; doi:10.3390/e16042350
Received: 3 April 2014 / Revised: 16 April 2014 / Accepted: 21 April 2014 / Published: 24 April 2014
Cited by 22 | PDF Full-text (4266 KB) | HTML Full-text | XML Full-text
Abstract
This paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow
[...] Read more.
This paper formulates a novel expression for entropy inspired in the properties of Fractional Calculus. The characteristics of the generalized fractional entropy are tested both in standard probability distributions and real world data series. The results reveal that tuning the fractional order allow an high sensitivity to the signal evolution, which is useful in describing the dynamics of complex systems. The concepts are also extended to relative distances and tested with several sets of data, confirming the goodness of the generalization. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)
Open AccessArticle A Fuzzy Parallel Processing Scheme for Enhancing the Effectiveness of a Dynamic Just-in-time Location-aware Service System
Entropy 2014, 16(4), 2001-2022; doi:10.3390/e16042001
Received: 26 February 2014 / Revised: 31 March 2014 / Accepted: 1 April 2014 / Published: 8 April 2014
Cited by 5 | PDF Full-text (542 KB) | HTML Full-text | XML Full-text
Abstract
Location-aware service systems are a hot topic in diverse research fields including mobile commerce, ambient intelligence, remote sensing and ubiquitous computing. However, the timeliness and efficiency of such systems are two issues that have rarely been emphasized. For this reason, this study tries
[...] Read more.
Location-aware service systems are a hot topic in diverse research fields including mobile commerce, ambient intelligence, remote sensing and ubiquitous computing. However, the timeliness and efficiency of such systems are two issues that have rarely been emphasized. For this reason, this study tries to establish a location-aware service system in which both the timeliness and efficiency of service provision are addressed. To this end, some innovative treatments have been used in the proposed methodology. First, the uncertainty of detecting a user’s location using the global positioning system is considered by modeling the location and speed of the user with fuzzy numbers. Subsequently, a fuzzy integer-nonlinear programming model is formulated to address the problem of finding the dynamic just-in-time service location and path for the user. To help solve the problem, the maximum entropy weighting function and the basic defuzzification distribution (BADD) method are applied to defuzzify the fuzzy variables. In addition, to enhance the efficiency of solving the problem, a fuzzy parallel processing scheme is also proposed for decomposing the problem into smaller pieces that can be handled by separate processing modules. An illustrative example is used to illustrate the proposed methodology. Finally, the effectiveness of the proposed methodology has been confirmed with an experiment. According to the results, using the proposed methodology the waiting time could be reduced by 60%. Full article
(This article belongs to the Special Issue Complex Systems and Nonlinear Dynamics)

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