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Special Issue "Recent Advances in Chaos Theory and Complex Networks"

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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 (30 April 2015)

Special Issue Editors

Guest Editor
Prof. Dr. Guanrong Chen

Department of Electronic Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong SAR, China
Website | E-Mail
Interests: chaos; complex networks; dynamical systems; bifurcation; synchronization
Guest Editor
Prof. Dr. C.K. Michael Tse

Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Kowloon, Hong Kong SAR, China
Website | E-Mail
Interests: chaos; power electronics; communications; complex networks applications
Guest Editor
Prof. Dr. Mustak E. Yalcin

Department of Electronics and Communication Engineering, Faculty of Electrical and Electronic Engineering, Istanbul Technical University, Maslak, 34469, Istanbul, Turkey
Website | E-Mail
Interests: nonlinear circuits and systems; neural networks; complex systems; nonlinear dynamics; complex networks
Guest Editor
Dr. Hai Yu

Software College, Northeastern University, Shenyang, Liaoning, P. R. China
Website | E-Mail
Interests: chaos; complex networks; communications; video coding; digital chaotic cipher
Guest Editor
Dr. Mattia Frasca

Department of Electrical, Electronic and Computer Science Engineering, University of Catania, Italy
Website | E-Mail
Interests: complex networks; chaotic circuits; memristors

Special Issue Information

Dear Colleagues,

As one of the greatest natural science discoveries in the 20th century, chaos theory has been extensively investigated, and significant progress has been made in the past several decades. It has become an exciting emerging interdisciplinary area in which a broad spectrum of methodologies and technologies are being studied to deal with large, complex, and dynamical problems. In the last fifteen years, the notion of complex networks as part of the complexity studies is actively and intensively investigated in the broad areas of computer and communication networks, physical and biological networks, as well as social and financial networks. In particular, chaotic networks, i.e., networks of chaotic maps, have emerged as a new trend of proactive research, extending the classical subject of coupled map lattices regarding their dynamical properties such as spatiotemporal chaos, bifurcations and entropy.

The main focus of this Special Issue will be on the state-of-the-art advancements in chaos theory and complex networks, as well as their applications to dynamical systems, nonlinear circuits, information processing, communications, cryptography, systems biology, and so on.

Prof. Dr. Guanrong Chen
Prof. Dr. C.K. Michael Tse
Prof. Dr. Mustak E. Yalcin
Dr. Hai Yu
Dr. Mattia Frasca
Guest Editors

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).

Keywords

  • chaos
  • complex network
  • complexity
  • entropy
  • fractional calculus
  • information theory
  • nonlinearity
  • nonlinear dynamics
  • systems biology

Published Papers (22 papers)

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Research

Open AccessArticle Projective Exponential Synchronization for a Class of Complex PDDE Networks with Multiple Time Delays
Entropy 2015, 17(11), 7298-7309; doi:10.3390/e17117298
Received: 20 April 2015 / Revised: 18 October 2015 / Accepted: 19 October 2015 / Published: 28 October 2015
Cited by 1 | PDF Full-text (765 KB) | HTML Full-text | XML Full-text
Abstract
This paper addresses the problem of projective exponential synchronization for a class of complex spatiotemporal networks with multiple time delays satisfying the homogeneous Neumann boundary conditions, where the network is modeled by coupled partial differential-difference equations (PDDEs). A distributed proportional-spatial derivative (P-sD) controller
[...] Read more.
This paper addresses the problem of projective exponential synchronization for a class of complex spatiotemporal networks with multiple time delays satisfying the homogeneous Neumann boundary conditions, where the network is modeled by coupled partial differential-difference equations (PDDEs). A distributed proportional-spatial derivative (P-sD) controller is designed by employing Lyapunov’s direct method and Kronecker product. The controller ensures the projective exponential synchronization of the PDDE network. The main result of this paper is presented in terms of standard linear matrix inequality (LMI). A numerical example is provided to show the effectiveness of the proposed design method. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Dynamical Systems Induced on Networks Constructed from Time Series
Entropy 2015, 17(9), 6433-6446; doi:10.3390/e17096433
Received: 1 July 2015 / Revised: 15 September 2015 / Accepted: 16 September 2015 / Published: 18 September 2015
Cited by 1 | PDF Full-text (523 KB) | HTML Full-text | XML Full-text
Abstract
Several methods exist to construct complex networks from time series. In general, these methods claim to construct complex networks that preserve certain properties of the underlying dynamical system, and hence, they mark new ways of accessing quantitative indicators based on that dynamics. In
[...] Read more.
Several methods exist to construct complex networks from time series. In general, these methods claim to construct complex networks that preserve certain properties of the underlying dynamical system, and hence, they mark new ways of accessing quantitative indicators based on that dynamics. In this paper, we test this assertion by developing an algorithm to realize dynamical systems from these complex networks in such a way that trajectories of these dynamical systems produce time series that preserve certain statistical properties of the original time series (and hence, also the underlying true dynamical system). Trajectories from these networks are constructed from only the information in the network and are shown to be statistically equivalent to the original time series. In the context of this algorithm, we are able to demonstrate that the so-called adaptive k-nearest neighbour algorithm for generating networks out-performs methods based on ε-ball recurrence plots. For such networks, and with a suitable choice of parameter values, which we provide, the time series generated by this method function as a new kind of nonlinear surrogate generation algorithm. With this approach, we are able to test whether the simulation dynamics built from a complex network capture the underlying structure of the original system; whether the complex network is an adequate model of the dynamics. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Adaptive Fuzzy Control for Nonlinear Fractional-Order Uncertain Systems with Unknown Uncertainties and External Disturbance
Entropy 2015, 17(8), 5580-5592; doi:10.3390/e17085580
Received: 30 April 2015 / Revised: 19 July 2015 / Accepted: 30 July 2015 / Published: 3 August 2015
Cited by 5 | PDF Full-text (170 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, the problem of robust control of nonlinear fractional-order systems in the presence of uncertainties and external disturbance is investigated. Fuzzy logic systems are used for estimating the unknown nonlinear functions. Based on the fractional Lyapunov direct method and some proposed
[...] Read more.
In this paper, the problem of robust control of nonlinear fractional-order systems in the presence of uncertainties and external disturbance is investigated. Fuzzy logic systems are used for estimating the unknown nonlinear functions. Based on the fractional Lyapunov direct method and some proposed Lemmas, an adaptive fuzzy controller is designed. The proposed method can guarantee all the signals in the closed-loop systems remain bounded and the tracking errors converge to an arbitrary small region of the origin. Lastly, an illustrative example is given to demonstrate the effectiveness of the proposed results. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Entropy Minimization Design Approach of Supersonic Internal Passages
Entropy 2015, 17(8), 5593-5610; doi:10.3390/e17085593
Received: 25 May 2015 / Revised: 27 July 2015 / Accepted: 29 July 2015 / Published: 3 August 2015
Cited by 2 | PDF Full-text (2568 KB) | HTML Full-text | XML Full-text
Abstract
Fluid machinery operating in the supersonic regime unveil avenues towards more compact technology. However, internal supersonic flows are associated with high aerodynamic and thermal penalties, which usually prevent their practical implementation. Indeed, both shock losses and the limited operational range represent particular challenges
[...] Read more.
Fluid machinery operating in the supersonic regime unveil avenues towards more compact technology. However, internal supersonic flows are associated with high aerodynamic and thermal penalties, which usually prevent their practical implementation. Indeed, both shock losses and the limited operational range represent particular challenges to aerodynamic designers that should be taken into account at the initial phase of the design process. This paper presents a design methodology for supersonic passages based on direct evaluations of the velocity field using the method of characteristics and computation of entropy generation across shock waves. This meshless function evaluation tool is then coupled to an optimization scheme, based on evolutionary algorithms that minimize the entropy generation across the supersonic passage. Finally, we assessed the results with 3D Reynolds Averaged Navier Stokes calculations. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Generalized Combination Complex Synchronization for Fractional-Order Chaotic Complex Systems
Entropy 2015, 17(8), 5199-5217; doi:10.3390/e17085199
Received: 10 March 2015 / Revised: 29 June 2015 / Accepted: 17 July 2015 / Published: 24 July 2015
Cited by 4 | PDF Full-text (993 KB) | HTML Full-text | XML Full-text
Abstract
Based on two fractional-order chaotic complex drive systems and one fractional-order chaotic complex response system with different dimensions, we propose generalized combination complex synchronization. In this new synchronization scheme, there are two complex scaling matrices that are non-square matrices. On the basis of
[...] Read more.
Based on two fractional-order chaotic complex drive systems and one fractional-order chaotic complex response system with different dimensions, we propose generalized combination complex synchronization. In this new synchronization scheme, there are two complex scaling matrices that are non-square matrices. On the basis of the stability theory of fractional-order linear systems, we design a general controller via active control. Additionally, by virtue of two complex scaling matrices, generalized combination complex synchronization between fractional-order chaotic complex systems and real systems is investigated. Finally, three typical examples are given to demonstrate the effectiveness and feasibility of the schemes. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System
Entropy 2015, 17(7), 5101-5116; doi:10.3390/e17075101
Received: 26 May 2015 / Revised: 15 July 2015 / Accepted: 17 July 2015 / Published: 22 July 2015
Cited by 3 | PDF Full-text (3223 KB) | HTML Full-text | XML Full-text
Abstract
In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the Lyapunov dimension under
[...] Read more.
In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the Lyapunov dimension under diffeomorphism is demonstrated in the general case. This fact is used to obtain the analytic exact upper bound of the Lyapunov dimension of an attractor of the Shimizu–Morioka system. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle A Colour Image Encryption Scheme Using Permutation-Substitution Based on Chaos
Entropy 2015, 17(6), 3877-3897; doi:10.3390/e17063877
Received: 9 February 2015 / Revised: 1 April 2015 / Accepted: 8 April 2015 / Published: 9 June 2015
Cited by 2 | PDF Full-text (5792 KB) | HTML Full-text | XML Full-text
Abstract
An encryption scheme for colour images using a spatiotemporal chaotic system is proposed. Initially, we use the R, G and B components of a colour plain-image to form a matrix. Then the matrix is permutated by using zigzag path scrambling. The resultant matrix
[...] Read more.
An encryption scheme for colour images using a spatiotemporal chaotic system is proposed. Initially, we use the R, G and B components of a colour plain-image to form a matrix. Then the matrix is permutated by using zigzag path scrambling. The resultant matrix is then passed through a substitution process. Finally, the ciphered colour image is obtained from the confused matrix. Theoretical analysis and experimental results indicate that the proposed scheme is both secure and practical, which make it suitable for encrypting colour images of any size. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Exponential Outer Synchronization between Two Uncertain Time-Varying Complex Networks with Nonlinear Coupling
Entropy 2015, 17(5), 3097-3109; doi:10.3390/e17053097
Received: 5 March 2015 / Revised: 27 April 2015 / Accepted: 5 May 2015 / Published: 11 May 2015
Cited by 6 | PDF Full-text (289 KB) | HTML Full-text | XML Full-text
Abstract
This paper studies the problem of exponential outer synchronization between two uncertain nonlinearly coupled complex networks with time delays. In order to synchronize uncertain complex networks, an adaptive control scheme is designed based on the Lyapunov stability theorem. Simultaneously, the unknown system parameters
[...] Read more.
This paper studies the problem of exponential outer synchronization between two uncertain nonlinearly coupled complex networks with time delays. In order to synchronize uncertain complex networks, an adaptive control scheme is designed based on the Lyapunov stability theorem. Simultaneously, the unknown system parameters of uncertain complex networks are identified when exponential outer synchronization occurs. Finally, numerical examples are provided to demonstrate the feasibility and effectiveness of the theoretical results. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle A Community-Based Approach to Identifying Influential Spreaders
Entropy 2015, 17(4), 2228-2252; doi:10.3390/e17042228
Received: 18 December 2014 / Accepted: 1 April 2015 / Published: 14 April 2015
Cited by 2 | PDF Full-text (8247 KB) | HTML Full-text | XML Full-text
Abstract
Identifying influential spreaders in complex networks has a significant impact on understanding and control of spreading process in networks. In this paper, we introduce a new centrality index to identify influential spreaders in a network based on the community structure of the network.
[...] Read more.
Identifying influential spreaders in complex networks has a significant impact on understanding and control of spreading process in networks. In this paper, we introduce a new centrality index to identify influential spreaders in a network based on the community structure of the network. The community-based centrality (CbC) considers both the number and sizes of communities that are directly linked by a node. We discuss correlations between CbC and other classical centrality indices. Based on simulations of the single source of infection with the Susceptible-Infected-Recovered (SIR) model, we find that CbC can help to identify some critical influential nodes that other indices cannot find. We also investigate the stability of CbC. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Image Encryption Using Chebyshev Map and Rotation Equation
Entropy 2015, 17(4), 2117-2139; doi:10.3390/e17042117
Received: 28 December 2014 / Revised: 18 March 2015 / Accepted: 30 March 2015 / Published: 9 April 2015
PDF Full-text (2319 KB) | HTML Full-text | XML Full-text
Abstract
We propose a novel image encryption algorithm based on two pseudorandom bit generators: Chebyshev map based and rotation equation based. The first is used for permutation, and the second one for substitution operations. Detailed security analysis has been provided on the novel image
[...] Read more.
We propose a novel image encryption algorithm based on two pseudorandom bit generators: Chebyshev map based and rotation equation based. The first is used for permutation, and the second one for substitution operations. Detailed security analysis has been provided on the novel image encryption algorithm using visual testing, key space evaluation, histogram analysis, information entropy calculation, correlation coefficient analysis, differential analysis, key sensitivity test, and computational and complexity analysis. Based on the theoretical and empirical results the novel image encryption scheme demonstrates an excellent level of security. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Target Detection and Ranging through Lossy Media using Chaotic Radar
Entropy 2015, 17(4), 2082-2093; doi:10.3390/e17042082
Received: 23 January 2015 / Revised: 11 March 2015 / Accepted: 31 March 2015 / Published: 8 April 2015
PDF Full-text (3002 KB) | HTML Full-text | XML Full-text
Abstract
A chaotic radar system has been developed for through-wall detection and ranging of targets. The chaotic signal generated by an improved Colpitts oscillator is designed as a probe signal. Ranging to target is achieved by the cross-correlation between the time-delayed reflected return signal
[...] Read more.
A chaotic radar system has been developed for through-wall detection and ranging of targets. The chaotic signal generated by an improved Colpitts oscillator is designed as a probe signal. Ranging to target is achieved by the cross-correlation between the time-delayed reflected return signal and the replica of the transmitted chaotic signal. In this paper, we explore the performance of the chaotic radar system for target detection and ranging through lossy media. Experimental results show that the designed chaotic radar has the advantages of high range resolution, unambiguous correlation profile, and can be used for through wall target detection and sensing. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Synchronicity from Synchronized Chaos
Entropy 2015, 17(4), 1701-1733; doi:10.3390/e17041701
Received: 24 December 2014 / Revised: 10 March 2015 / Accepted: 16 March 2015 / Published: 27 March 2015
Cited by 1 | PDF Full-text (1145 KB) | HTML Full-text | XML Full-text
Abstract
The synchronization of loosely-coupled chaotic oscillators, a phenomenon investigated intensively for the last two decades, may realize the philosophical concept of “synchronicity”—the commonplace notion that related eventsmysteriously occur at the same time. When extended to continuous media and/or large discrete arrays, and when
[...] Read more.
The synchronization of loosely-coupled chaotic oscillators, a phenomenon investigated intensively for the last two decades, may realize the philosophical concept of “synchronicity”—the commonplace notion that related eventsmysteriously occur at the same time. When extended to continuous media and/or large discrete arrays, and when general (non-identical) correspondences are considered between states, intermittent synchronous relationships indeed become ubiquitous. Meaningful synchronicity follows naturally if meaningful events are identified with coherent structures, defined by internal synchronization between remote degrees of freedom; a condition that has been posited as necessary for synchronizability with an external system. The important case of synchronization between mind and matter is realized if mind is analogized to a computer model, synchronizing with a sporadically observed system, as in meteorological data assimilation. Evidence for the ubiquity of synchronization is reviewed along with recent proposals that: (1) synchronization of different models of the same objective process may be an expeditious route to improved computational modeling and may also describe the functioning of conscious brains; and (2) the nonlocality in quantum phenomena implied by Bell’s theorem may be explained in a variety of deterministic (hidden variable) interpretations if the quantum world resides on a generalized synchronization “manifold”. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Research on the Stability of Open Financial System
Entropy 2015, 17(4), 1734-1754; doi:10.3390/e17041734
Received: 6 February 2015 / Revised: 19 March 2015 / Accepted: 24 March 2015 / Published: 27 March 2015
Cited by 1 | PDF Full-text (1511 KB) | HTML Full-text | XML Full-text
Abstract
We propose a new herd mechanism and embed it into an open financial market system, which allows traders to get in and out of the system based on some transition rates. Moreover, the novel mechanism can avoid the volatility disappearance when the population
[...] Read more.
We propose a new herd mechanism and embed it into an open financial market system, which allows traders to get in and out of the system based on some transition rates. Moreover, the novel mechanism can avoid the volatility disappearance when the population scale increases. There are three kinds of heterogeneous agents in the system: optimistic, pessimistic and fundamental. Interactions especially occur among three different groups of agents instead of two, which makes the artificial financial market more close to the real one. By the simulation results of this complex system, we can explain stylized facts like volatility clustering and find the key parameters of market bubbles and market collapses. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Distributed Extreme Learning Machine for Nonlinear Learning over Network
Entropy 2015, 17(2), 818-840; doi:10.3390/e17020818
Received: 26 December 2014 / Revised: 20 January 2015 / Accepted: 6 February 2015 / Published: 12 February 2015
Cited by 1 | PDF Full-text (419 KB) | HTML Full-text | XML Full-text
Abstract
Distributed data collection and analysis over a network are ubiquitous, especially over a wireless sensor network (WSN). To our knowledge, the data model used in most of the distributed algorithms is linear. However, in real applications, the linearity of systems is not always
[...] Read more.
Distributed data collection and analysis over a network are ubiquitous, especially over a wireless sensor network (WSN). To our knowledge, the data model used in most of the distributed algorithms is linear. However, in real applications, the linearity of systems is not always guaranteed. In nonlinear cases, the single hidden layer feedforward neural network (SLFN) with radial basis function (RBF) hidden neurons has the ability to approximate any continuous functions and, thus, may be used as the nonlinear learning system. However, confined by the communication cost, using the distributed version of the conventional algorithms to train the neural network directly is usually prohibited. Fortunately, based on the theorems provided in the extreme learning machine (ELM) literature, we only need to compute the output weights of the SLFN. Computing the output weights itself is a linear learning problem, although the input-output mapping of the overall SLFN is still nonlinear. Using the distributed algorithmto cooperatively compute the output weights of the SLFN, we obtain a distributed extreme learning machine (dELM) for nonlinear learning in this paper. This dELM is applied to the regression problem and classification problem to demonstrate its effectiveness and advantages. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Parameters Estimation of Uncertain Fractional-Order Chaotic Systems via a Modified Artificial Bee Colony Algorithm
Entropy 2015, 17(2), 692-709; doi:10.3390/e17020692
Received: 25 December 2014 / Revised: 23 January 2015 / Accepted: 27 January 2015 / Published: 4 February 2015
PDF Full-text (196 KB) | HTML Full-text | XML Full-text
Abstract
Parameter estimation for fractional-order chaotic systems has been an interesting and important issue in theory and various fields of application. In this paper, fractional orders, as well as systematic parameters of fractional-order chaotic systems are considered by treating fractional orders as additional parameters.
[...] Read more.
Parameter estimation for fractional-order chaotic systems has been an interesting and important issue in theory and various fields of application. In this paper, fractional orders, as well as systematic parameters of fractional-order chaotic systems are considered by treating fractional orders as additional parameters. The parameter estimation is transformed into a multidimensional optimization problem, and an effective modified artificial bee colony algorithm is proposed to solve this problem. Numerical simulations are conducted on two typical fractional-order chaotic systems to show the effectiveness of the proposed scheme. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Modeling and Analyzing the Interaction between Network Rumors and Authoritative Information
Entropy 2015, 17(1), 471-482; doi:10.3390/e17010471
Received: 8 October 2014 / Accepted: 14 January 2015 / Published: 19 January 2015
Cited by 4 | PDF Full-text (740 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we propose a novel two-stage rumor spreading Susceptible-Infected-Authoritative-Removed (SIAR) model for complex homogeneous and heterogeneous networks. The interaction Markov chains (IMC) mean-field equations based on the SIAR model are derived to describe the dynamic interaction between the rumors and authoritative
[...] Read more.
In this paper, we propose a novel two-stage rumor spreading Susceptible-Infected-Authoritative-Removed (SIAR) model for complex homogeneous and heterogeneous networks. The interaction Markov chains (IMC) mean-field equations based on the SIAR model are derived to describe the dynamic interaction between the rumors and authoritative information. We use a Monte Carlo simulation method to characterize the dynamics of the Susceptible-Infected-Removed (SIR) and SIAR models, showing that the SIAR model with consideration of authoritative information gives a more realistic description of propagation features of rumors than the SIR model. The simulation results demonstrate that the critical threshold λc of the SIAR model has the tiniest increase than the threshold of SIR model. The sooner the authoritative information is introduced, the less negative impact the rumors will bring. We also get the result that heterogeneous networks are more prone to the spreading of rumors. Additionally, the inhibition of rumor spreading, as one of the characteristics of the new SIAR model itself, is instructive for later studies on the rumor spreading models and the controlling strategies. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle An Image Encryption Scheme Based on Hyperchaotic Rabinovich and Exponential Chaos Maps
Entropy 2015, 17(1), 181-196; doi:10.3390/e17010181
Received: 31 October 2014 / Accepted: 24 December 2014 / Published: 8 January 2015
Cited by 7 | PDF Full-text (2654 KB) | HTML Full-text | XML Full-text
Abstract
This paper proposes a new four-dimensional hyperchaotic map based on the Rabinovich system to realize chaotic encryption in higher dimension and improve the security. The chaotic sequences generated by Runge-Kutta method are combined with the chaotic sequences generated by an exponential chaos map
[...] Read more.
This paper proposes a new four-dimensional hyperchaotic map based on the Rabinovich system to realize chaotic encryption in higher dimension and improve the security. The chaotic sequences generated by Runge-Kutta method are combined with the chaotic sequences generated by an exponential chaos map to generate key sequences. The key sequences are used for image encryption. The security test results indicate that the new hyperchaotic system has high security and complexity. The comparison between the new hyperchaotic system and the several low-dimensional chaotic systems shows that the proposed system performs more efficiently. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Finite-Time Synchronization of Chaotic Complex Networks with Stochastic Disturbance
Entropy 2015, 17(1), 39-51; doi:10.3390/e17010039
Received: 16 October 2014 / Accepted: 22 December 2014 / Published: 30 December 2014
Cited by 2 | PDF Full-text (248 KB) | HTML Full-text | XML Full-text
Abstract
This paper is concerned with the problem of finite-time synchronization in complex networks with stochastic noise perturbations. By using a novel finite-time ℒ -operator differential inequality and other inequality techniques, some novel sufficient conditions are obtained to ensure finite-time stochastic synchronization for the complex
[...] Read more.
This paper is concerned with the problem of finite-time synchronization in complex networks with stochastic noise perturbations. By using a novel finite-time ℒ -operator differential inequality and other inequality techniques, some novel sufficient conditions are obtained to ensure finite-time stochastic synchronization for the complex networks concerned, where the coupling matrix need not be symmetric. The effects of control parameters on synchronization speed and time are also analyzed, and the synchronization time in this paper is shorter than that in the existing literature. The results here are also applicable to both directed and undirected weighted networks without any information of the coupling matrix. Finally, an example with numerical simulations is given to demonstrate the effectiveness of the proposed method. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle A Color Image Encryption Algorithm Based on a Fractional-Order Hyperchaotic System
Entropy 2015, 17(1), 28-38; doi:10.3390/e17010028
Received: 24 October 2014 / Accepted: 13 December 2014 / Published: 23 December 2014
Cited by 2 | PDF Full-text (4116 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, a new color image encryption algorithm based on a fractional-order hyperchaotic system is proposed. Firstly, four chaotic sequences are generated by a fractional-order hyperchaotic system. The parameters of such a system, together with the initial value, are regarded as the
[...] Read more.
In this paper, a new color image encryption algorithm based on a fractional-order hyperchaotic system is proposed. Firstly, four chaotic sequences are generated by a fractional-order hyperchaotic system. The parameters of such a system, together with the initial value, are regarded as the secret keys and the plain image is encrypted by performing the XOR and shuffling operations simultaneously. The proposed encryption scheme is described in detail with security analyses, including correlation analysis, histogram analysis, differential attacks, and key sensitivity analysis. Experimental results show that the proposed encryption scheme has big key space, and high sensitivity to keys properties, and resists statistical analysis and differential attacks, so it has high security and is suitable for color image encryption. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Open AccessArticle Generation and Nonlinear Dynamical Analyses of Fractional-Order Memristor-Based Lorenz Systems
Entropy 2014, 16(12), 6240-6253; doi:10.3390/e16126240
Received: 1 October 2014 / Revised: 9 November 2014 / Accepted: 21 November 2014 / Published: 28 November 2014
Cited by 6 | PDF Full-text (14220 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, four fractional-order memristor-based Lorenz systems with the flux-controlled memristor characterized by a monotone-increasing piecewise linear function, a quadratic nonlinearity, a smooth continuous cubic nonlinearity and a quartic nonlinearity are presented, respectively. The nonlinear dynamics are analyzed by using numerical simulation
[...] Read more.
In this paper, four fractional-order memristor-based Lorenz systems with the flux-controlled memristor characterized by a monotone-increasing piecewise linear function, a quadratic nonlinearity, a smooth continuous cubic nonlinearity and a quartic nonlinearity are presented, respectively. The nonlinear dynamics are analyzed by using numerical simulation methods, including phase portraits, bifurcation diagrams, the largest Lyapunov exponent and power spectrum diagrams. Some interesting phenomena, such as inverse period-doubling bifurcation and intermittent chaos, are found to exist in the proposed systems. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Figures

Open AccessArticle The Property of Chaotic Orbits with Lower Positions of Numerical Solutions in the Logistic Map
Entropy 2014, 16(11), 5618-5632; doi:10.3390/e16115618
Received: 23 September 2014 / Revised: 20 October 2014 / Accepted: 22 October 2014 / Published: 27 October 2014
PDF Full-text (223 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we introduce an iterative method with lower positions of true numerical solutions located in the real orbit in order to investigate the property of the logistic map. The basic structure of the logistic map is presented, which consists of the
[...] Read more.
In this paper, we introduce an iterative method with lower positions of true numerical solutions located in the real orbit in order to investigate the property of the logistic map. The basic structure of the logistic map is presented, which consists of the root gene position, the common gene position and the individual gene position. The ergodicity and randomness of the logistic map are dependent on the individual gene position. We find that the lower positions of the true numerical solutions in the real orbits have the property of a half-life. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)
Figures

Open AccessArticle Research and Development of a Chaotic Signal Synchronization Error Dynamics-Based Ball Bearing Fault Diagnostor
Entropy 2014, 16(10), 5358-5376; doi:10.3390/e16105358
Received: 30 August 2014 / Revised: 5 October 2014 / Accepted: 8 October 2014 / Published: 15 October 2014
Cited by 4 | PDF Full-text (2759 KB) | HTML Full-text | XML Full-text
Abstract
This paper describes the fault diagnosis in the operation of industrial ball bearings. In order to cluster the very small differential signals of the four classic fault types of the ball bearing system, the chaos synchronization (CS) concept is used in this study
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This paper describes the fault diagnosis in the operation of industrial ball bearings. In order to cluster the very small differential signals of the four classic fault types of the ball bearing system, the chaos synchronization (CS) concept is used in this study as the chaos system is very sensitive to a system’s variation such as initial conditions or system parameters. In this study, the Chen-Lee chaotic system was used to load the normal and fault signals of the bearings into the chaos synchronization error dynamics system. The fractal theory was applied to determine the fractal dimension and lacunarity from the CS error dynamics. Extenics theory was then applied to distinguish the state of the bearing faults. This study also compared the proposed method with discrete Fourier transform and wavelet packet analysis. According to the results, it is shown that the proposed chaos synchronization method combined with extenics theory can separate the characteristics (fractal dimension vs. lacunarity) completely. Therefore, it has a better fault diagnosis rate than the two traditional signal processing methods, i.e., Fourier transform and wavelet packet analysis combined with extenics theory. Full article
(This article belongs to the Special Issue Recent Advances in Chaos Theory and Complex Networks)

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Title: Dynamical Systems from Networks Constructed from Time Series"
Authors: Lvlin Hou 1 and Michael Small 2
Affiliations: 1 National University of Defence Technology, China
2 The University of Western Australia
Abstract: Several methods exist to construct complex networks from time series. In general these methods claim to construct complex networks which preserve certain properties of the underlying time series. In this paper we test this assertion by developing an algorithm to realise dynamical systems from these complex networks in such a way that trajectories of these dynamical systems are time series that preserve certain statistical properties of the original time series (and hence the underlying true dynamical system). Using this algorithm we are able to demonstrate that the so-called $k$-nearet neighbour adaptive algorithm for generating networks out-performs methods based of $\epsilon$-ball recurrence plots. For such networks - and with suitable choice of parameter values, which we provide - the time series generated by this method function as a nonlinear surrogate generation algorithm. Hence, one can use this approach to test the null hypothesis of a noise driven nonlinear periodic orbit in observed time series data - with the alternative hypothesis of deterministic chaos.

Author: Marc Brachet
Affiliation: Centre National de la Recherche Scientifique, Laboratoire de Physique Statistique, Université Pierre-et-Marie-Curie Paris 06, Université Paris Diderot, Ecole Normale Supérieure, 75231 Paris Cedex 05, France
E-Mail
: brachet@phys.ens.fr

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