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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 (31 August 2013)

Special Issue Editor

Guest Editor
Prof. Dr. J. A. Tenreiro Machado (Website)

Department of Electrical Engineering, Institute of Engineering, Polytechnic Institute of Porto, Porto, Portugal
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 area 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 behaviour of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences.
The special issue focus 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.

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
Prof. J. A. Tenreiro Machado
Guest Editor


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Published Papers (34 papers)

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Research

Open AccessArticle Finite-Time Chaos Suppression of Permanent Magnet Synchronous Motor Systems
Entropy 2014, 16(4), 2234-2243; doi:10.3390/e16042234
Received: 12 September 2013 / Revised: 11 April 2014 / Accepted: 11 April 2014 / Published: 21 April 2014
Cited by 3 | PDF Full-text (303 KB) | HTML Full-text | XML Full-text
Abstract
This paper considers the problem of the chaos suppression for the Permanent Magnet Synchronous Motor (PMSM) system via the finite-time control. Based on Lyapunov stability theory and the finite-time controller are developed such that the chaos behaviors of PMSM system can be [...] Read more.
This paper considers the problem of the chaos suppression for the Permanent Magnet Synchronous Motor (PMSM) system via the finite-time control. Based on Lyapunov stability theory and the finite-time controller are developed such that the chaos behaviors of PMSM system can be suppressed. The effectiveness and accuracy of the proposed methods are shown in numerical simulations. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations
Entropy 2014, 16(4), 1949-1968; doi:10.3390/e16041949
Received: 2 September 2013 / Revised: 20 March 2014 / Accepted: 20 March 2014 / Published: 1 April 2014
Cited by 1 | PDF Full-text (1553 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and Discontinuous PWS [...] Read more.
In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and Discontinuous PWS (or Filippov-type PWS) systems. The proposed qualitative approach explicitly includes two main aspects: multiple discontinuity boundaries (DBs) in the phase space and multiple intersections between DBs (or corner manifolds—CMs). Previous classifications of DIBs of limit cycles have been restricted to generic cases with a single DB or a single CM. We use the definition of piecewise topological equivalence in order to synthesize all possibilities of nonsmooth limit cycles. Families, groups and subgroups of cycles are defined depending on smoothness zones and discontinuity boundaries (DB) involved. The synthesized cycles are used to define bifurcation patterns when the system is perturbed with parametric changes. Four families of DIBs of limit cycles are defined depending on the properties of the cycles involved. Well-known and novel bifurcations can be classified using this approach. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Figures

Open AccessArticle Analysis of Solar Neutrino Data from Super-Kamiokande I and II
Entropy 2014, 16(3), 1414-1425; doi:10.3390/e16031414
Received: 5 December 2013 / Revised: 16 December 2013 / Accepted: 27 February 2014 / Published: 10 March 2014
Cited by 7 | PDF Full-text (550 KB) | HTML Full-text | XML Full-text
Abstract
We are going back to the roots of the original solar neutrino problem: the analysis of data from solar neutrino experiments. The application of standard deviation analysis (SDA) and diffusion entropy analysis (DEA) to the Super-Kamiokande I and II data reveals that [...] Read more.
We are going back to the roots of the original solar neutrino problem: the analysis of data from solar neutrino experiments. The application of standard deviation analysis (SDA) and diffusion entropy analysis (DEA) to the Super-Kamiokande I and II data reveals that they represent a non-Gaussian signal. The Hurst exponent is different from the scaling exponent of the probability density function, and both the Hurst exponent and scaling exponent of the probability density function of the Super-Kamiokande data deviate considerably from the value of 0.5, which indicates that the statistics of the underlying phenomenon is anomalous. To develop a road to the possible interpretation of this finding, we utilize Mathai’s pathway model and consider fractional reaction and fractional diffusion as possible explanations of the non-Gaussian content of the Super-Kamiokande data. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Robust Control of a Class of Uncertain Fractional-Order Chaotic Systems with Input Nonlinearity via an Adaptive Sliding Mode Technique
Entropy 2014, 16(2), 729-746; doi:10.3390/e16020729
Received: 10 December 2013 / Revised: 23 January 2014 / Accepted: 26 January 2014 / Published: 7 February 2014
Cited by 9 | PDF Full-text (399 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the problem of stabilizing a class of fractional-order chaotic systems with sector and dead-zone nonlinear inputs is investigated. The effects of model uncertainties and external disturbances are fully taken into account. Moreover, the bounds of both model uncertainties and [...] Read more.
In this paper, the problem of stabilizing a class of fractional-order chaotic systems with sector and dead-zone nonlinear inputs is investigated. The effects of model uncertainties and external disturbances are fully taken into account. Moreover, the bounds of both model uncertainties and external disturbances are assumed to be unknown in advance. To deal with the system’s nonlinear items and unknown bounded uncertainties, an adaptive fractional-order sliding mode (AFSM) controller is designed. Then, Lyapunov’s stability theory is used to prove the stability of the designed control scheme. Finally, two simulation examples are given to verify the effectiveness and robustness of the proposed control approach. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Dynamical Stability and Predictability of Football Players: The Study of One Match
Entropy 2014, 16(2), 645-674; doi:10.3390/e16020645
Received: 2 August 2013 / Revised: 11 January 2014 / Accepted: 14 January 2014 / Published: 23 January 2014
Cited by 8 | PDF Full-text (2141 KB) | HTML Full-text | XML Full-text
Abstract
The game of football demands new computational approaches to measure individual and collective performance. Understanding the phenomena involved in the game may foster the identification of strengths and weaknesses, not only of each player, but also of the whole team. The development [...] Read more.
The game of football demands new computational approaches to measure individual and collective performance. Understanding the phenomena involved in the game may foster the identification of strengths and weaknesses, not only of each player, but also of the whole team. The development of assertive quantitative methodologies constitutes a key element in sports training. In football, the predictability and stability inherent in the motion of a given player may be seen as one of the most important concepts to fully characterise the variability of the whole team. This paper characterises the predictability and stability levels of players during an official football match. A Fractional Calculus (FC) approach to define a player’s trajectory. By applying FC, one can benefit from newly considered modeling perspectives, such as the fractional coefficient, to estimate a player’s predictability and stability. This paper also formulates the concept of attraction domain, related to the tactical region of each player, inspired by stability theory principles. To compare the variability inherent in the player’s process variables (e.g., distance covered) and to assess his predictability and stability, entropy measures are considered. Experimental results suggest that the most predictable player is the goalkeeper while, conversely, the most unpredictable players are the midfielders. We also conclude that, despite his predictability, the goalkeeper is the most unstable player, while lateral defenders are the most stable during the match. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Figures

Open AccessArticle Multiple Solutions of Nonlinear Boundary Value Problems of Fractional Order: A New Analytic Iterative Technique
Entropy 2014, 16(1), 471-493; doi:10.3390/e16010471
Received: 10 November 2013 / Revised: 11 December 2013 / Accepted: 11 December 2013 / Published: 9 January 2014
Cited by 12 | PDF Full-text (402 KB) | HTML Full-text | XML Full-text
Abstract
The purpose of this paper is to present a new kind of analytical method, the so-called residual power series, to predict and represent the multiplicity of solutions to nonlinear boundary value problems of fractional order. The present method is capable of calculating [...] Read more.
The purpose of this paper is to present a new kind of analytical method, the so-called residual power series, to predict and represent the multiplicity of solutions to nonlinear boundary value problems of fractional order. The present method is capable of calculating all branches of solutions simultaneously, even if these multiple solutions are very close and thus rather difficult to distinguish even by numerical techniques. To verify the computational efficiency of the designed proposed technique, two nonlinear models are performed, one of them arises in mixed convection flows and the other one arises in heat transfer, which both admit multiple solutions. The results reveal that the method is very effective, straightforward, and powerful for formulating these multiple solutions. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Adaptive Switched Generalized Function Projective Synchronization between Two Hyperchaotic Systems with Unknown Parameters
Entropy 2014, 16(1), 377-388; doi:10.3390/e16010377
Received: 17 October 2013 / Revised: 15 December 2013 / Accepted: 16 December 2013 / Published: 31 December 2013
Cited by 3 | PDF Full-text (1113 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we investigate adaptive switched generalized function projective synchronization between two new different hyperchaotic systems with unknown parameters, which is an extension of the switched modified function projective synchronization scheme. Based on the Lyapunov stability theory, corresponding adaptive controllers with [...] Read more.
In this paper, we investigate adaptive switched generalized function projective synchronization between two new different hyperchaotic systems with unknown parameters, which is an extension of the switched modified function projective synchronization scheme. Based on the Lyapunov stability theory, corresponding adaptive controllers with appropriate parameter update laws are constructed to achieve adaptive switched generalized function projective synchronization between two different hyperchaotic systems. A numerical simulation is conducted to illustrate the validity and feasibility of the proposed synchronization scheme. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Entropy Diversity in Multi-Objective Particle Swarm Optimization
Entropy 2013, 15(12), 5475-5491; doi:10.3390/e15125475
Received: 30 August 2013 / Revised: 30 November 2013 / Accepted: 3 December 2013 / Published: 10 December 2013
Cited by 4 | PDF Full-text (428 KB) | HTML Full-text | XML Full-text
Abstract
Multi-objective particle swarm optimization (MOPSO) is a search algorithm based on social behavior. Most of the existing multi-objective particle swarm optimization schemes are based on Pareto optimality and aim to obtain a representative non-dominated Pareto front for a given problem. Several approaches [...] Read more.
Multi-objective particle swarm optimization (MOPSO) is a search algorithm based on social behavior. Most of the existing multi-objective particle swarm optimization schemes are based on Pareto optimality and aim to obtain a representative non-dominated Pareto front for a given problem. Several approaches have been proposed to study the convergence and performance of the algorithm, particularly by accessing the final results. In the present paper, a different approach is proposed, by using Shannon entropy to analyze the MOPSO dynamics along the algorithm execution. The results indicate that Shannon entropy can be used as an indicator of diversity and convergence for MOPSO problems. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle New Results on Fractional Power Series: Theories and Applications
Entropy 2013, 15(12), 5305-5323; doi:10.3390/e15125305
Received: 12 September 2013 / Revised: 9 October 2013 / Accepted: 9 October 2013 / Published: 2 December 2013
Cited by 12 | PDF Full-text (1156 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, some theorems of the classical power series are generalized for the fractional power series. Some of these theorems are constructed by using Caputo fractional derivatives. Under some constraints, we proved that the Caputo fractional derivative can be expressed in [...] Read more.
In this paper, some theorems of the classical power series are generalized for the fractional power series. Some of these theorems are constructed by using Caputo fractional derivatives. Under some constraints, we proved that the Caputo fractional derivative can be expressed in terms of the ordinary derivative. A new construction of the generalized Taylor’s power series is obtained. Some applications including approximation of fractional derivatives and integrals of functions and solutions of linear and nonlinear fractional differential equations are also given. In the nonlinear case, the new and simple technique is used to find out the recurrence relation that determines the coefficients of the fractional power series. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Synchronization of a Novel Hyperchaotic Complex-Variable System Based on Finite-Time Stability Theory
Entropy 2013, 15(10), 4334-4344; doi:10.3390/e15104334
Received: 31 August 2013 / Revised: 6 October 2013 / Accepted: 11 October 2013 / Published: 16 October 2013
Cited by 3 | PDF Full-text (278 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we investigate the finite-time synchronization problem of a novel hyperchaotic complex-variable system which generates 2-, 3- and 4-scroll attractors. Based on the finite-time stability theory, two control strategies are proposed to realize synchronization of the novel hyperchaotic complex-variable system [...] Read more.
In this paper, we investigate the finite-time synchronization problem of a novel hyperchaotic complex-variable system which generates 2-, 3- and 4-scroll attractors. Based on the finite-time stability theory, two control strategies are proposed to realize synchronization of the novel hyperchaotic complex-variable system in finite time. Finally, two numerical examples have been provided to illustrate the effectiveness of the theoretical analysis. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Diffusion in Relatively Homogeneous Sand Columns: A Scale-Dependent or Scale-Independent Process?
Entropy 2013, 15(10), 4376-4391; doi:10.3390/e15104376
Received: 30 August 2013 / Revised: 23 September 2013 / Accepted: 10 October 2013 / Published: 16 October 2013
PDF Full-text (428 KB) | HTML Full-text | XML Full-text
Abstract
Solute transport through homogeneous media has long been assumed to be scale-independent and can be quantified by the second-order advection-dispersion equation (ADE). This study, however, observed the opposite in the laboratory, where transport of CuSO4 through relatively homogeneous silica-sand columns exhibits sub-diffusion [...] Read more.
Solute transport through homogeneous media has long been assumed to be scale-independent and can be quantified by the second-order advection-dispersion equation (ADE). This study, however, observed the opposite in the laboratory, where transport of CuSO4 through relatively homogeneous silica-sand columns exhibits sub-diffusion growing with the spatial scale. Only at a very small travel distance (approximately 10 cm) and a relatively short temporal scale can the transport be approximated by normal diffusion. This is also the only spatiotemporal scale where the fundamental concept of the “representative element volume” (which defines the scale of homogeneous cells used by the ADE-based hydrologic models) is valid. The failure of the standard ADE motivated us to apply a tempered-stable, fractional advection-dispersion equation (TS-FADE) to capture the transient anomalous dispersion with exponentially truncated power-law late-time tails in CuSO4 breakthrough curves. Results show that the tempering parameter in the TS-FADE model generally decreases with an increase of the column length (probably due to the higher probability of long retention processes), while the time index (which is a non-local parameter) remains stable for the uniformly packed columns. Transport in sand columns filled with relatively homogeneous silica sand, therefore, is scale-dependent, and the resultant transient sub-diffusion can be quantified by the TS-FADE model. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Relaxation to Fixed Points in the Logistic and Cubic Maps: Analytical and Numerical Investigation
Entropy 2013, 15(10), 4310-4318; doi:10.3390/e15104310
Received: 14 August 2013 / Revised: 25 September 2013 / Accepted: 1 October 2013 / Published: 14 October 2013
Cited by 3 | PDF Full-text (133 KB) | HTML Full-text | XML Full-text
Abstract
Convergence to a period one fixed point is investigated for both logistic and cubic maps. For the logistic map the relaxation to the fixed point is considered near a transcritical bifurcation while for the cubic map it is near a pitchfork bifurcation. [...] Read more.
Convergence to a period one fixed point is investigated for both logistic and cubic maps. For the logistic map the relaxation to the fixed point is considered near a transcritical bifurcation while for the cubic map it is near a pitchfork bifurcation. We confirmed that the convergence to the fixed point in both logistic and cubic maps for a region close to the fixed point goes exponentially fast to the fixed point and with a relaxation time described by a power law of exponent -1. At the bifurcation point, the exponent is not universal and depends on the type of the bifurcation as well as on the nonlinearity of the map. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Development of Metrics and a Complexity Scale for the Topology of Assembly Supply Chains
Entropy 2013, 15(10), 4285-4299; doi:10.3390/e15104285
Received: 29 August 2013 / Revised: 30 September 2013 / Accepted: 30 September 2013 / Published: 10 October 2013
Cited by 2 | PDF Full-text (452 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we present a methodological framework for conceptual modeling of assembly supply chain (ASC) networks. Models of such ASC networks are divided into classes on the basis of the numbers of initial suppliers. We provide a brief overview of select [...] Read more.
In this paper, we present a methodological framework for conceptual modeling of assembly supply chain (ASC) networks. Models of such ASC networks are divided into classes on the basis of the numbers of initial suppliers. We provide a brief overview of select literature on the topic of structural complexity in assembly systems. Subsequently, the so called Vertex degree index for measuring a structural complexity of ASC networks is applied. This measure, which is based on the Shannon entropy, is well suited for the given purpose. Finally, we outline a generic model of quantitative complexity scale for ASC Networks. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Analogue Realization of Fractional-Order Dynamical Systems
Entropy 2013, 15(10), 4199-4214; doi:10.3390/e15104199
Received: 27 August 2013 / Revised: 24 September 2013 / Accepted: 25 September 2013 / Published: 7 October 2013
Cited by 10 | PDF Full-text (844 KB) | HTML Full-text | XML Full-text
Abstract
As it results from many research works, the majority of real dynamical objects are fractional-order systems, although in some types of systems the order is very close to integer order. Application of fractional-order models is more adequate for the description and analysis [...] Read more.
As it results from many research works, the majority of real dynamical objects are fractional-order systems, although in some types of systems the order is very close to integer order. Application of fractional-order models is more adequate for the description and analysis of real dynamical systems than integer-order models, because their total entropy is greater than in integer-order models with the same number of parameters. A great deal of modern methods for investigation, monitoring and control of the dynamical processes in different areas utilize approaches based upon modeling of these processes using not only mathematical models, but also physical models. This paper is devoted to the design and analogue electronic realization of the fractional-order model of a fractional-order system, e.g., of the controlled object and/or controller, whose mathematical model is a fractional-order differential equation. The electronic realization is based on fractional-order differentiator and integrator where operational amplifiers are connected with appropriate impedance, with so called Fractional Order Element or Constant Phase Element. Presented network model approximates quite well the properties of the ideal fractional-order system compared with e.g., domino ladder networks. Along with the mathematical description, circuit diagrams and design procedure, simulation and measured results are also presented. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle The Fractional Differential Polynomial Neural Network for Approximation of Functions
Entropy 2013, 15(10), 4188-4198; doi:10.3390/e15104188
Received: 26 August 2013 / Revised: 5 September 2013 / Accepted: 24 September 2013 / Published: 30 September 2013
Cited by 7 | PDF Full-text (296 KB) | HTML Full-text | XML Full-text
Abstract
In this work, we introduce a generalization of the differential polynomial neural network utilizing fractional calculus. Fractional calculus is taken in the sense of the Caputo differential operator. It approximates a multi-parametric function with particular polynomials characterizing its functional output as a [...] Read more.
In this work, we introduce a generalization of the differential polynomial neural network utilizing fractional calculus. Fractional calculus is taken in the sense of the Caputo differential operator. It approximates a multi-parametric function with particular polynomials characterizing its functional output as a generalization of input patterns. This method can be employed on data to describe modelling of complex systems. Furthermore, the total information is calculated by using the fractional Poisson process. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Fractional Heat Conduction in an Infinite Medium with a Spherical Inclusion
Entropy 2013, 15(10), 4122-4133; doi:10.3390/e15104122
Received: 27 August 2013 / Revised: 22 September 2013 / Accepted: 22 September 2013 / Published: 27 September 2013
Cited by 9 | PDF Full-text (230 KB) | HTML Full-text | XML Full-text
Abstract
The problem of fractional heat conduction in a composite medium consisting of a spherical inclusion (0< r < R)  and a matrix (R <  r < ∞)  being in perfect thermal contact at r = R  is considered. The heat conduction in each [...] Read more.
The problem of fractional heat conduction in a composite medium consisting of a spherical inclusion (0< r < R)  and a matrix (R <  r < ∞)  being in perfect thermal contact at r = R  is considered. The heat conduction in each region is described by the time-fractional heat conduction equation with the Caputo derivative of fractional order 0 < a ≤ 2 and 0 < β ≤ 2,  respectively. The Laplace transform with respect to time is used. The approximate solution valid for small values of time is obtained in terms of the Mittag-Leffler, Wright, and Mainardi functions. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Learning Entropy: Multiscale Measure for Incremental Learning
Entropy 2013, 15(10), 4159-4187; doi:10.3390/e15104159
Received: 26 July 2013 / Revised: 17 September 2013 / Accepted: 22 September 2013 / Published: 27 September 2013
Cited by 5 | PDF Full-text (729 KB) | HTML Full-text | XML Full-text
Abstract
First, this paper recalls a recently introduced method of adaptive monitoring of dynamical systems and presents the most recent extension with a multiscale-enhanced approach. Then, it is shown that this concept of real-time data monitoring establishes a novel non-Shannon and non-probabilistic concept [...] Read more.
First, this paper recalls a recently introduced method of adaptive monitoring of dynamical systems and presents the most recent extension with a multiscale-enhanced approach. Then, it is shown that this concept of real-time data monitoring establishes a novel non-Shannon and non-probabilistic concept of novelty quantification, i.e., Entropy of Learning, or in short the Learning Entropy. This novel cognitive measure can be used for evaluation of each newly measured sample of data, or even of whole intervals. The Learning Entropy is quantified in respect to the inconsistency of data to the temporary governing law of system behavior that is incrementally learned by adaptive models such as linear or polynomial adaptive filters or neural networks. The paper presents this novel concept on the example of gradient descent learning technique with normalized learning rate. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Time Eigenstates for Potential Functions without Extremal Points
Entropy 2013, 15(10), 4105-4121; doi:10.3390/e15104105
Received: 18 August 2013 / Revised: 21 September 2013 / Accepted: 22 September 2013 / Published: 26 September 2013
Cited by 2 | PDF Full-text (379 KB)
Abstract
In a previous paper, we introduced a way to generate a time coordinate system for classical and quantum systems when the potential function has extremal points. In this paper, we deal with the case in which the potential function has no extremal [...] Read more.
In a previous paper, we introduced a way to generate a time coordinate system for classical and quantum systems when the potential function has extremal points. In this paper, we deal with the case in which the potential function has no extremal points at all, and we illustrate the method with the harmonic and linear potentials. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Figures

Open AccessArticle On a Generalized Entropy Measure Leading to the Pathway Model with a Preliminary Application to Solar Neutrino Data
Entropy 2013, 15(10), 4011-4025; doi:10.3390/e15104011
Received: 4 September 2013 / Accepted: 22 September 2013 / Published: 25 September 2013
Cited by 9 | PDF Full-text (257 KB) | HTML Full-text | XML Full-text
Abstract
An entropy for the scalar variable case, parallel to Havrda-Charvat entropy, was introduced by the first author, and the properties and its connection to Tsallis non-extensive statistical mechanics and the Mathai pathway model were examined by the authors in previous papers. In [...] Read more.
An entropy for the scalar variable case, parallel to Havrda-Charvat entropy, was introduced by the first author, and the properties and its connection to Tsallis non-extensive statistical mechanics and the Mathai pathway model were examined by the authors in previous papers. In the current paper, we extend the entropy to cover the scalar case, multivariable case, and matrix variate case. Then, this measure is optimized under different types of restrictions, and a number of models in the multivariable case and matrix variable case are obtained. Connections of these models to problems in statistical and physical sciences are pointed out. An application of the simplest case of the pathway model to the interpretation of solar neutrino data by applying standard deviation analysis and diffusion entropy analysis is provided. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Evaluating the Spectrum of Unlocked Injection Frequency Dividers in Pulling Mode
Entropy 2013, 15(10), 4026-4041; doi:10.3390/e15104026
Received: 12 August 2013 / Revised: 17 September 2013 / Accepted: 18 September 2013 / Published: 25 September 2013
Cited by 1 | PDF Full-text (290 KB) | HTML Full-text | XML Full-text
Abstract
We study the phenomenon of periodic pulling which occurs in certain integrated microcircuits of relevant interest in applications, namely the injection-locked frequency dividers (ILFDs). They are modelled as second-order driven oscillators working in the subharmonic (secondary) resonance regime, i.e., when the [...] Read more.
We study the phenomenon of periodic pulling which occurs in certain integrated microcircuits of relevant interest in applications, namely the injection-locked frequency dividers (ILFDs). They are modelled as second-order driven oscillators working in the subharmonic (secondary) resonance regime, i.e., when the self-oscillating frequency is close (resonant) to an integer submultiple n of the driving frequency. Under the assumption of weak injection, we find the spectrum of the system’s oscillatory response in the unlocked mode through closed-form expressions, showing that such spectrum is double-sided and asymmetric, unlike the single-sided spectrum of systems with primary resonance (n=1). An analytical expression for the amplitude modulation of the oscillatory response is also presented. Numerical results are presented to support theoretical relations derived. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Analysis and Visualization of Seismic Data Using Mutual Information
Entropy 2013, 15(9), 3892-3909; doi:10.3390/e15093892
Received: 26 July 2013 / Revised: 10 September 2013 / Accepted: 12 September 2013 / Published: 16 September 2013
Cited by 13 | PDF Full-text (843 KB) | HTML Full-text | XML Full-text
Abstract
Seismic data is difficult to analyze and classical mathematical tools reveal strong limitations in exposing hidden relationships between earthquakes. In this paper, we study earthquake phenomena in the perspective of complex systems. Global seismic data, covering the period from 1962 up to [...] Read more.
Seismic data is difficult to analyze and classical mathematical tools reveal strong limitations in exposing hidden relationships between earthquakes. In this paper, we study earthquake phenomena in the perspective of complex systems. Global seismic data, covering the period from 1962 up to 2011 is analyzed. The events, characterized by their magnitude, geographic location and time of occurrence, are divided into groups, either according to the Flinn-Engdahl (F-E) seismic regions of Earth or using a rectangular grid based in latitude and longitude coordinates. Two methods of analysis are considered and compared in this study. In a first method, the distributions of magnitudes are approximated by Gutenberg-Richter (G-R) distributions and the parameters used to reveal the relationships among regions. In the second method, the mutual information is calculated and adopted as a measure of similarity between regions. In both cases, using clustering analysis, visualization maps are generated, providing an intuitive and useful representation of the complex relationships that are present among seismic data. Such relationships might not be perceived on classical geographic maps. Therefore, the generated charts are a valid alternative to other visualization tools, for understanding the global behavior of earthquakes. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Combination Synchronization of Three Identical or Different Nonlinear Complex Hyperchaotic Systems
Entropy 2013, 15(9), 3746-3761; doi:10.3390/e15093746
Received: 5 August 2013 / Revised: 3 September 2013 / Accepted: 4 September 2013 / Published: 10 September 2013
Cited by 8 | PDF Full-text (280 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we investigate the combination synchronization of three nonlinear complex hyperchaotic systems: the complex hyperchaotic Lorenz system, the complex hyperchaotic Chen system and the complex hyperchaotic L¨u system. Based on the Lyapunov stability theory, corresponding controllers to achieve combination synchronization [...] Read more.
In this paper, we investigate the combination synchronization of three nonlinear complex hyperchaotic systems: the complex hyperchaotic Lorenz system, the complex hyperchaotic Chen system and the complex hyperchaotic L¨u system. Based on the Lyapunov stability theory, corresponding controllers to achieve combination synchronization among three identical or different nonlinear complex hyperchaotic systems are derived, respectively. Numerical simulations are presented to demonstrate the validity and feasibility of the theoretical analysis. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle A Discrete Meta-Control Procedure for Approximating Solutions to Binary Programs
Entropy 2013, 15(9), 3592-3601; doi:10.3390/e15093592
Received: 15 June 2013 / Revised: 30 August 2013 / Accepted: 30 August 2013 / Published: 4 September 2013
PDF Full-text (231 KB) | HTML Full-text | XML Full-text
Abstract
Large-scale binary integer programs occur frequently in many real-world applications. For some binary integer problems, finding an optimal solution or even a feasible solution is computationally expensive. In this paper, we develop a discrete meta-control procedure to approximately solve large-scale binary integer [...] Read more.
Large-scale binary integer programs occur frequently in many real-world applications. For some binary integer problems, finding an optimal solution or even a feasible solution is computationally expensive. In this paper, we develop a discrete meta-control procedure to approximately solve large-scale binary integer programs efficiently. The key idea is to map the vector of n binary decision variables into a scalar function defined over a time interval [0; n] and construct a linear quadratic tracking (LQT) problem that can be solved efficiently. We prove that an LQT formulation has an optimal binary solution, analogous to a classical bang-bang control in continuous time. Our LQT approach can provide advantages in reducing computation while generating a good approximate solution. Numerical examples are presented to demonstrate the usefulness of the proposed method. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Information Entropy As a Basic Building Block of Complexity Theory
Entropy 2013, 15(9), 3396-3418; doi:10.3390/e15093396
Received: 27 June 2013 / Revised: 12 August 2013 / Accepted: 12 August 2013 / Published: 29 August 2013
Cited by 7 | PDF Full-text (913 KB) | HTML Full-text | XML Full-text
Abstract
What is information? What role does information entropy play in this information exploding age, especially in understanding emergent behaviors of complex systems? To answer these questions, we discuss the origin of information entropy, the difference between information entropy and thermodynamic entropy, the [...] Read more.
What is information? What role does information entropy play in this information exploding age, especially in understanding emergent behaviors of complex systems? To answer these questions, we discuss the origin of information entropy, the difference between information entropy and thermodynamic entropy, the role of information entropy in complexity theories, including chaos theory and fractal theory, and speculate new fields in which information entropy may play important roles. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Synchronization of a Class of Fractional-Order Chaotic Neural Networks
Entropy 2013, 15(8), 3265-3276; doi:10.3390/e15083355
Received: 5 June 2013 / Revised: 3 August 2013 / Accepted: 5 August 2013 / Published: 14 August 2013
Cited by 27 | PDF Full-text (421 KB)
Abstract
The synchronization problem is studied in this paper for a class of fractional-order chaotic neural networks. By using the Mittag-Leffler function, M-matrix and linear feedback control, a sufficient condition is developed ensuring the synchronization of such neural models with the Caputo fractional [...] Read more.
The synchronization problem is studied in this paper for a class of fractional-order chaotic neural networks. By using the Mittag-Leffler function, M-matrix and linear feedback control, a sufficient condition is developed ensuring the synchronization of such neural models with the Caputo fractional derivatives. The synchronization condition is easy to verify, implement and only relies on system structure. Furthermore, the theoretical results are applied to a typical fractional-order chaotic Hopfield neural network, and numerical simulation demonstrates the effectiveness and feasibility of the proposed method. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle A Maximum Entropy-Based Chaotic Time-Variant Fragile Watermarking Scheme for Image Tampering Detection
Entropy 2013, 15(8), 3170-3185; doi:10.3390/e15083260
Received: 16 April 2013 / Revised: 27 July 2013 / Accepted: 29 July 2013 / Published: 5 August 2013
Cited by 6 | PDF Full-text (1932 KB)
Abstract
The fragile watermarking technique is used to protect intellectual property rights while also providing security and rigorous protection. In order to protect the copyright of the creators, it can be implanted in some representative text or totem. Because all of the media [...] Read more.
The fragile watermarking technique is used to protect intellectual property rights while also providing security and rigorous protection. In order to protect the copyright of the creators, it can be implanted in some representative text or totem. Because all of the media on the Internet are digital, protection has become a critical issue, and determining how to use digital watermarks to protect digital media is thus the topic of our research. This paper uses the Logistic map with parameter u = 4 to generate chaotic dynamic behavior with the maximum entropy 1. This approach increases the security and rigor of the protection. The main research target of information hiding is determining how to hide confidential data so that the naked eye cannot see the difference. Next, we introduce one method of information hiding. Generally speaking, if the image only goes through Arnold’s cat map and the Logistic map, it seems to lack sufficient security. Therefore, our emphasis is on controlling Arnold’s cat map and the initial value of the chaos system to undergo small changes and generate different chaos sequences. Thus, the current time is used to not only make encryption more stringent but also to enhance the security of the digital media. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle On the Topological Entropy of Some Skew-Product Maps
Entropy 2013, 15(8), 3100-3108; doi:10.3390/e15083100
Received: 5 July 2013 / Revised: 25 July 2013 / Accepted: 26 July 2013 / Published: 31 July 2013
Cited by 1 | PDF Full-text (294 KB) | HTML Full-text | XML Full-text
Abstract
The aim of this short note is to compute the topological entropy for a family of skew-product maps, whose base is a subshift of finite type, and the fiber maps are homeomorphisms defined in one dimensional spaces. We show that the skew-product [...] Read more.
The aim of this short note is to compute the topological entropy for a family of skew-product maps, whose base is a subshift of finite type, and the fiber maps are homeomorphisms defined in one dimensional spaces. We show that the skew-product map does not increase the topological entropy of the subshift. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle The Entropy of Co-Compact Open Covers
Entropy 2013, 15(7), 2464-2479; doi:10.3390/e15072464
Received: 3 April 2013 / Revised: 8 June 2013 / Accepted: 18 June 2013 / Published: 24 June 2013
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Abstract
Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability are not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy [...] Read more.
Co-compact entropy is introduced as an invariant of topological conjugation for perfect mappings defined on any Hausdorff space (compactness and metrizability are not necessarily required). This is achieved through the consideration of co-compact covers of the space. The advantages of co-compact entropy include: (1) it does not require the space to be compact and, thus, generalizes Adler, Konheim and McAndrew’s topological entropy of continuous mappings on compact dynamical systems; and (2) it is an invariant of topological conjugation, compared to Bowen’s entropy, which is metric-dependent. Other properties of co-compact entropy are investigated, e.g., the co-compact entropy of a subsystem does not exceed that of the whole system. For the linear system, (R; f), defined by f(x) = 2x, the co-compact entropy is zero, while Bowen’s entropy for this system is at least log 2. More generally, it is found that co-compact entropy is a lower bound of Bowen’s entropies, and the proof of this result also generates the Lebesgue Covering Theorem to co-compact open covers of non-compact metric spaces. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Effect of Prey Refuge on the Spatiotemporal Dynamics of a Modified Leslie-Gower Predator-Prey System with Holling Type III Schemes
Entropy 2013, 15(6), 2431-2447; doi:10.3390/e15062431
Received: 27 March 2013 / Revised: 5 June 2013 / Accepted: 6 June 2013 / Published: 19 June 2013
Cited by 3 | PDF Full-text (1087 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the spatiotemporal dynamics of a diffusive Leslie-Gower predator-prey model with prey refuge are investigated analytically and numerically. Mathematical theoretical works have considered the existence of global solutions, population permanence and the stability of equilibrium points, which depict the threshold [...] Read more.
In this paper, the spatiotemporal dynamics of a diffusive Leslie-Gower predator-prey model with prey refuge are investigated analytically and numerically. Mathematical theoretical works have considered the existence of global solutions, population permanence and the stability of equilibrium points, which depict the threshold expressions of some critical parameters. Numerical simulations are performed to explore the pattern formation of species. These results show that the prey refuge has a profound effect on predator-prey interactions and they have the potential to be useful for the study of the entropy theory of bioinformatics. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle A Method for Choosing an Initial Time Eigenstate in Classical and Quantum Systems
Entropy 2013, 15(6), 2415-2430; doi:10.3390/e15062415
Received: 23 April 2013 / Revised: 29 May 2013 / Accepted: 3 June 2013 / Published: 17 June 2013
Cited by 2 | PDF Full-text (1229 KB) | HTML Full-text | XML Full-text
Abstract
A subject of interest in classical and quantum mechanics is the development of the appropriate treatment of the time variable. In this paper we introduce a method of choosing the initial time eigensurface and how this method can be used to generate [...] Read more.
A subject of interest in classical and quantum mechanics is the development of the appropriate treatment of the time variable. In this paper we introduce a method of choosing the initial time eigensurface and how this method can be used to generate time-energy coordinates and, consequently, time-energy representations for classical and quantum systems. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Figures

Open AccessArticle Entropy Increase in Switching Systems
Entropy 2013, 15(6), 2363-2383; doi:10.3390/e15062363
Received: 10 May 2013 / Revised: 3 June 2013 / Accepted: 3 June 2013 / Published: 7 June 2013
Cited by 4 | PDF Full-text (156 KB) | HTML Full-text | XML Full-text
Abstract
The relation between the complexity of a time-switched dynamics and the complexity of its control sequence depends critically on the concept of a non-autonomous pullback attractor. For instance, the switched dynamics associated with scalar dissipative affine maps has a pullback attractor consisting [...] Read more.
The relation between the complexity of a time-switched dynamics and the complexity of its control sequence depends critically on the concept of a non-autonomous pullback attractor. For instance, the switched dynamics associated with scalar dissipative affine maps has a pullback attractor consisting of singleton component sets. This entails that the complexity of the control sequence and switched dynamics, as quantified by the topological entropy, coincide. In this paper we extend the previous framework to pullback attractors with nontrivial components sets in order to gain further insights in that relation. This calls, in particular, for distinguishing two distinct contributions to the complexity of the switched dynamics. One proceeds from trajectory segments connecting different component sets of the attractor; the other contribution proceeds from trajectory segments within the component sets. We call them “macroscopic” and “microscopic” complexity, respectively, because only the first one can be measured by our analytical tools. As a result of this picture, we obtain sufficient conditions for a switching system to be more complex than its unswitched subsystems, i.e., a complexity analogue of Parrondo’s paradox. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle A Unification between Dynamical System Theory and Thermodynamics Involving an Energy, Mass, and Entropy State Space Formalism
Entropy 2013, 15(5), 1821-1846; doi:10.3390/e15051821
Received: 29 March 2013 / Revised: 10 May 2013 / Accepted: 10 May 2013 / Published: 16 May 2013
Cited by 3 | PDF Full-text (177 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we combine the two universalisms of thermodynamics and dynamical systems theory to develop a dynamical system formalism for classical thermodynamics. Specifically, using a compartmental dynamical system energy flow model involving heat flow, work energy, and chemical reactions, we develop [...] Read more.
In this paper, we combine the two universalisms of thermodynamics and dynamical systems theory to develop a dynamical system formalism for classical thermodynamics. Specifically, using a compartmental dynamical system energy flow model involving heat flow, work energy, and chemical reactions, we develop a state-space dynamical system model that captures the key aspects of thermodynamics, including its fundamental laws. In addition, we show that our thermodynamically consistent dynamical system model is globally semistable with system states converging to a state of temperature equipartition. Furthermore, in the presence of chemical reactions, we use the law of mass-action and the notion of chemical potential to show that the dynamic system states converge to a state of temperature equipartition and zero affinity corresponding to a state of chemical equilibrium. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Genetic Algorithm-Based Identification of Fractional-Order Systems
Entropy 2013, 15(5), 1624-1642; doi:10.3390/e15051624
Received: 14 March 2013 / Revised: 23 April 2013 / Accepted: 25 April 2013 / Published: 6 May 2013
Cited by 16 | PDF Full-text (353 KB) | HTML Full-text | XML Full-text
Abstract
Fractional calculus has become an increasingly popular tool for modeling the complex behaviors of physical systems from diverse domains. One of the key issues to apply fractional calculus to engineering problems is to achieve the parameter identification of fractional-order systems. A time-domain [...] Read more.
Fractional calculus has become an increasingly popular tool for modeling the complex behaviors of physical systems from diverse domains. One of the key issues to apply fractional calculus to engineering problems is to achieve the parameter identification of fractional-order systems. A time-domain identification algorithm based on a genetic algorithm (GA) is proposed in this paper. The multi-variable parameter identification is converted into a parameter optimization by applying GA to the identification of fractional-order systems. To evaluate the identification accuracy and stability, the time-domain output error considering the condition variation is designed as the fitness function for parameter optimization. The identification process is established under various noise levels and excitation levels. The effects of external excitation and the noise level on the identification accuracy are analyzed in detail. The simulation results show that the proposed method could identify the parameters of both commensurate rate and non-commensurate rate fractional-order systems from the data with noise. It is also observed that excitation signal is an important factor influencing the identification accuracy of fractional-order systems. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available
Open AccessArticle Outer Synchronization between Fractional-Order Complex Networks: A Non-Fragile Observer-based Control Scheme
Entropy 2013, 15(4), 1357-1374; doi:10.3390/e15041357
Received: 25 February 2013 / Revised: 4 April 2013 / Accepted: 8 April 2013 / Published: 15 April 2013
Cited by 5 | PDF Full-text (625 KB) | HTML Full-text | XML Full-text
Abstract
This paper addresses the global outer synchronization problem between two fractional-order complex networks coupled in a drive-response configuration. In particular, for a given fractional-order complex network composed of Lur’e systems, an observer-type response network with non-fragile output feedback controllers is constructed. Both [...] Read more.
This paper addresses the global outer synchronization problem between two fractional-order complex networks coupled in a drive-response configuration. In particular, for a given fractional-order complex network composed of Lur’e systems, an observer-type response network with non-fragile output feedback controllers is constructed. Both additive and multiplicative uncertainties that perturb the control gain matrices are considered. Then, using the stability theory of fractional-order systems and eigenvalue distribution of the Kronecker sum of matrices, we establish some sufficient conditions for global outer synchronization. Interestingly, the developed results are cast in the format of linear matrix inequalities (LMIs), which can be efficiently solved via the MATLAB LMI Control Toolbox. Finally, numerical simulations on fractional-order networks with nearest-neighbor and small-world topologies are given to support the theoretical analysis. Full article
(This article belongs to the Special Issue Dynamical Systems) Print Edition available

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