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Research Frontier in Chaos Theory and Complex Networks
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Research Frontier in Chaos Theory and Complex Networks

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

Deadline for manuscript submissions: closed (31 May 2018) | Viewed by 73160

Special Issue Editors

Prof. Dr. Guanrong (Ron) Chen

E-Mail Website
Guest Editor
Department of Electrical Engineering, City University of Hong Kong, Hong Kong, China
Interests: nonlinear dynamics; complex networks and control systems
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Marius-F. Danca

E-Mail Website
Guest Editor
1. Babes-Bolyai University, Cluj-Napoca, STAR-UBB Institute, 400084 Cluj-Napoca, Romania
2. Romanian Institute of Science and Technology, 400487 Cluj-Napoca, Romania
Interests: nonlinear dynamics; continuous/non-smooth chaotic; dynamical systems of integer/fractional order; chaotic hidden attractors; fractals
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Xiaosong Yang

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
Interests: nonlinear system; asymptotic stability; dynamics analysis; bifurcation and chaos
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Genaro J. Martinez

E-Mail Website
Guest Editor
Unconventional Computing Center, University of the West of England (UWE Bristol), Frenchay Campus, Coldharbour Lane, Bristol BS16 1QY, UK
Interests: cellular automata; unconventional and natural computing; computer sciences; artificial life; complex systems; dynamical systems; robotics
Dr. Hai Yu

E-Mail Website
Guest Editor
Software College, Northeastern University, Shenyang, Liaoning, China
Interests: chaos theory; complex networks; communications; video coding; digital chaotic ciphers
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In recent years, as natural and social sciences are rapidly evolving, classical chaos theory and modern complex networks studies are coming to the influence among each other with great developments. In particular, the notion of complex networks is becoming a new interdiscipline. Network science as the whole, which focuses on studying both qualitative and quantitative complex networks, has gradually merged with basic research and real-world applications of chaos theory, forming one of the most active fields in cognitive science, data science, cloud computing, social sciences, artificial intelligence (AI), and so on.

The theme of this Special Issue is on state-of-the-art advancements in chaos theory and complex networks, as well as their interactions, with applications in dynamical systems, nonlinear circuits, information processing, communications, image and signal processing, systems biology, finance and economy dynamics, etc. The issue will be an excellent reflection of current research efforts and progress in the promising field of chaos theory and complex networks.

Prof. Dr. Guanrong Chen
Prof. Dr. Marius-F. Danca
Prof. Dr. Xiaosong Yang
Prof. Dr. Genaro J. Martinez
Dr. Hai Yu
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • chaos
  • complex network
  • complexity
  • cellular automata
  • entropy
  • information theory
  • nonlinearity
  • nonlinear dynamics
  • systems biology

Published Papers (18 papers)

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Editorial

Jump to: Research

4 pages, 168 KiB  
Editorial
Research Frontier in Chaos Theory and Complex Networks
by Guanrong Chen, Marius-F. Danca, Xiaosong Yang, Genaro J. Martinez and Hai Yu
Entropy 2018, 20(10), 734; https://0-doi-org.brum.beds.ac.uk/10.3390/e20100734 - 25 Sep 2018
Cited by 10 | Viewed by 2827
Abstract
In recent years, as natural and social sciences are rapidly evolving, classical chaos theory
and modern complex networks studies are gradually interacting each other with a great joined
development [...] Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)

Research

Jump to: Editorial

15 pages, 2770 KiB  
Article
On Chaos in the Fractional-Order Discrete-Time Unified System and Its Control Synchronization
by Amina-Aicha Khennaoui, Adel Ouannas, Samir Bendoukha, Xiong Wang and Viet-Thanh Pham
Entropy 2018, 20(7), 530; https://0-doi-org.brum.beds.ac.uk/10.3390/e20070530 - 15 Jul 2018
Cited by 28 | Viewed by 4348
Abstract
In this paper, we propose a fractional map based on the integer-order unified map. The chaotic behavior of the proposed map is analyzed by means of bifurcations plots, and experimental bounds are placed on the parameters and fractional order. Different control laws are [...] Read more.
In this paper, we propose a fractional map based on the integer-order unified map. The chaotic behavior of the proposed map is analyzed by means of bifurcations plots, and experimental bounds are placed on the parameters and fractional order. Different control laws are proposed to force the states to zero asymptotically and to achieve the complete synchronization of a pair of fractional unified maps with identical or nonidentical parameters. Numerical results are used throughout the paper to illustrate the findings. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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15 pages, 511 KiB  
Article
Inferring the Population Mean with Second-Order Information in Online Social Networks
by Saran Chen, Xin Lu, Zhong Liu and Zhongwei Jia
Entropy 2018, 20(6), 480; https://0-doi-org.brum.beds.ac.uk/10.3390/e20060480 - 20 Jun 2018
Cited by 2 | Viewed by 3447
Abstract
With the increasing use of online social networking platforms, online surveys are widely used in many fields, e.g., public health, business and sociology, to collect samples and to infer the population characteristics through self-reported data of respondents. Although the online surveys can protect [...] Read more.
With the increasing use of online social networking platforms, online surveys are widely used in many fields, e.g., public health, business and sociology, to collect samples and to infer the population characteristics through self-reported data of respondents. Although the online surveys can protect the privacy of respondents, self-reporting is challenged by a low response rate and unreliable answers when the survey contains sensitive questions, such as drug use, sexual behaviors, abortion or criminal activity. To overcome this limitation, this paper develops an approach that collects the second-order information of the respondents, i.e., asking them about the characteristics of their friends, instead of asking the respondents’ own characteristics directly. Then, we generate the inference about the population variable with the Hansen-Hurwitz estimator for the two classic sampling strategies (simple random sampling or random walk-based sampling). The method is evaluated by simulations on both artificial and real-world networks. Results show that the method is able to generate population estimates with high accuracy without knowing the respondents’ own characteristics, and the biases of estimates under various settings are relatively small and are within acceptable limits. The new method offers an alternative way for implementing surveys online and is expected to be able to collect more reliable data with improved population inference on sensitive variables. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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19 pages, 1205 KiB  
Article
Coupled Node Similarity Learning for Community Detection in Attributed Networks
by Fanrong Meng, Xiaobin Rui, Zhixiao Wang, Yan Xing and Longbing Cao
Entropy 2018, 20(6), 471; https://0-doi-org.brum.beds.ac.uk/10.3390/e20060471 - 17 Jun 2018
Cited by 19 | Viewed by 4707
Abstract
Attributed networks consist of not only a network structure but also node attributes. Most existing community detection algorithms only focus on network structures and ignore node attributes, which are also important. Although some algorithms using both node attributes and network structure information have [...] Read more.
Attributed networks consist of not only a network structure but also node attributes. Most existing community detection algorithms only focus on network structures and ignore node attributes, which are also important. Although some algorithms using both node attributes and network structure information have been proposed in recent years, the complex hierarchical coupling relationships within and between attributes, nodes and network structure have not been considered. Such hierarchical couplings are driving factors in community formation. This paper introduces a novel coupled node similarity (CNS) to involve and learn attribute and structure couplings and compute the similarity within and between nodes with categorical attributes in a network. CNS learns and integrates the frequency-based intra-attribute coupled similarity within an attribute, the co-occurrence-based inter-attribute coupled similarity between attributes, and coupled attribute-to-structure similarity based on the homophily property. CNS is then used to generate the weights of edges and transfer a plain graph to a weighted graph. Clustering algorithms detect community structures that are topologically well-connected and semantically coherent on the weighted graphs. Extensive experiments verify the effectiveness of CNS-based community detection algorithms on several data sets by comparing with the state-of-the-art node similarity measures, whether they involve node attribute information and hierarchical interactions, and on various levels of network structure complexity. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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10 pages, 643 KiB  
Article
Analytic Solution for a Complex Network of Chaotic Oscillators
by Jonathan N. Blakely, Marko S. Milosavljevic and Ned J. Corron
Entropy 2018, 20(6), 468; https://0-doi-org.brum.beds.ac.uk/10.3390/e20060468 - 16 Jun 2018
Cited by 3 | Viewed by 3362
Abstract
Chaotic evolution is generally too irregular to be captured in an analytic solution. Nonetheless, some dynamical systems do have such solutions enabling more rigorous analysis than can be achieved with numerical solutions. Here, we introduce a method of coupling solvable chaotic oscillators that [...] Read more.
Chaotic evolution is generally too irregular to be captured in an analytic solution. Nonetheless, some dynamical systems do have such solutions enabling more rigorous analysis than can be achieved with numerical solutions. Here, we introduce a method of coupling solvable chaotic oscillators that maintains solvability. In fact, an analytic solution is given for an entire network of coupled oscillators. Importantly, a valid chaotic solution is shown even when the coupling topology is complex and the population of oscillators is heterogeneous. We provide a specific example of a solvable chaotic network with star topology and a hub that oscillates much faster than its leaves. We present analytic solutions as the coupling strength is varied showing states of varying degrees of global organization. The covariance of the network is derived explicity from the analytic solution characterizing the degree of synchronization across the network as the coupling strength varies. This example suggests that analytic solutions may constitute a new tool in the study of chaotic network dynamics generally. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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17 pages, 5833 KiB  
Article
A Novel Delay Linear Coupling Logistics Map Model for Color Image Encryption
by Shouliang Li, Weikang Ding, Benshun Yin, Tongfeng Zhang and Yide Ma
Entropy 2018, 20(6), 463; https://0-doi-org.brum.beds.ac.uk/10.3390/e20060463 - 14 Jun 2018
Cited by 26 | Viewed by 4815
Abstract
With the popularity of the Internet, the transmission of images has become more frequent. It is of great significance to study efficient and secure image encryption algorithms. Based on traditional Logistic maps and consideration of delay, we propose a new one-dimensional (1D) delay [...] Read more.
With the popularity of the Internet, the transmission of images has become more frequent. It is of great significance to study efficient and secure image encryption algorithms. Based on traditional Logistic maps and consideration of delay, we propose a new one-dimensional (1D) delay and linearly coupled Logistic chaotic map (DLCL) in this paper. Time delay is a common phenomenon in various complex systems in nature, and it will greatly change the dynamic characteristics of the system. The map is analyzed in terms of trajectory, Lyapunov exponent (LE) and Permutation entropy (PE). The results show that this map has wide chaotic range, better ergodicity and larger maximum LE in comparison with some existing chaotic maps. A new method of color image encryption is put forward based on DLCL. In proposed encryption algorithm, after various analysis, it has good encryption performance, and the key used for scrambling is related to the original image. It is illustrated by simulation results that the ciphered images have good pseudo randomness through our method. The proposed encryption algorithm has large key space and can effectively resist differential attack and chosen plaintext attack. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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15 pages, 848 KiB  
Article
Recommending Queries by Extracting Thematic Experiences from Complex Search Tasks
by Yuli Zhao, Yin Zhang, Bin Zhang, Kening Gao and Pengfei Li
Entropy 2018, 20(6), 459; https://0-doi-org.brum.beds.ac.uk/10.3390/e20060459 - 13 Jun 2018
Cited by 4 | Viewed by 3028
Abstract
Since complex search tasks are usually divided into subtasks, providing subtask-oriented query recommendations is an effective way to support complex search tasks. Currently, most subtask-oriented query recommendation methods extract subtasks from plain form search logs consisting of only queries and clicks, providing limited [...] Read more.
Since complex search tasks are usually divided into subtasks, providing subtask-oriented query recommendations is an effective way to support complex search tasks. Currently, most subtask-oriented query recommendation methods extract subtasks from plain form search logs consisting of only queries and clicks, providing limited clues to identify subtasks. Meanwhile, for several decades, the Computer Human Interface (CHI)/Human Computer Interaction (HCI) communities have been working on new complex search tools for the purpose of supporting rich user interactions beyond just queries and clicks, and thus providing rich form search logs with more clues for subtask identification. In this paper, we researched the provision of subtask-oriented query recommendations by extracting thematic experiences from the rich form search logs of complex search tasks logged in a proposed visual data structure. We introduce the tree structure of the visual data structure and propose a visual-based subtask identification method based on the visual data structure. We then introduce a personalized PageRank-based method to recommend queries by ranking nodes on the network from the identified subtasks. We evaluated the proposed methods in experiments consisting of informative and tentative search tasks. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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17 pages, 447 KiB  
Article
Quantifying the Effects of Topology and Weight for Link Prediction in Weighted Complex Networks
by Bo Liu, Shuang Xu, Ting Li, Jing Xiao and Xiao-Ke Xu
Entropy 2018, 20(5), 363; https://0-doi-org.brum.beds.ac.uk/10.3390/e20050363 - 13 May 2018
Cited by 20 | Viewed by 4211
Abstract
In weighted networks, both link weight and topological structure are significant characteristics for link prediction. In this study, a general framework combining null models is proposed to quantify the impact of the topology, weight correlation and statistics on link prediction in weighted networks. [...] Read more.
In weighted networks, both link weight and topological structure are significant characteristics for link prediction. In this study, a general framework combining null models is proposed to quantify the impact of the topology, weight correlation and statistics on link prediction in weighted networks. Three null models for topology and weight distribution of weighted networks are presented. All the links of the original network can be divided into strong and weak ties. We can use null models to verify the strong effect of weak or strong ties. For two important statistics, we construct two null models to measure their impacts on link prediction. In our experiments, the proposed method is applied to seven empirical networks, which demonstrates that this model is universal and the impact of the topology and weight distribution of these networks in link prediction can be quantified by it. We find that in the USAir, the Celegans, the Gemo, the Lesmis and the CatCortex, the strong ties are easier to predict, but there are a few networks whose weak edges can be predicted more easily, such as the Netscience and the CScientists. It is also found that the weak ties contribute more to link prediction in the USAir, the NetScience and the CScientists, that is, the strong effect of weak ties exists in these networks. The framework we proposed is versatile, which is not only used to link prediction but also applicable to other directions in complex networks. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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12 pages, 8814 KiB  
Article
Lyapunov Exponents of a Discontinuous 4D Hyperchaotic System of Integer or Fractional Order
by Marius-F. Danca
Entropy 2018, 20(5), 337; https://0-doi-org.brum.beds.ac.uk/10.3390/e20050337 - 03 May 2018
Cited by 17 | Viewed by 4642
Abstract
In this paper, the dynamics of local finite-time Lyapunov exponents of a 4D hyperchaotic system of integer or fractional order with a discontinuous right-hand side and as an initial value problem, are investigated graphically. It is shown that a discontinuous system of integer [...] Read more.
In this paper, the dynamics of local finite-time Lyapunov exponents of a 4D hyperchaotic system of integer or fractional order with a discontinuous right-hand side and as an initial value problem, are investigated graphically. It is shown that a discontinuous system of integer or fractional order cannot be numerically integrated using methods for continuous differential equations. A possible approach for discontinuous systems is presented. To integrate the initial value problem of fractional order or integer order, the discontinuous system is continuously approximated via Filippov’s regularization and Cellina’s Theorem. The Lyapunov exponents of the approximated system of integer or fractional order are represented as a function of two variables: as a function of two parameters, or as a function of the fractional order and one parameter, respectively. The obtained three-dimensional representation leads to comprehensive conclusions regarding the nature, differences and sign of the Lyapunov exponents in both integer order and fractional order cases. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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19 pages, 13876 KiB  
Article
A Novel Fractional-Order Chaotic Phase Synchronization Model for Visual Selection and Shifting
by Xiaoran Lin, Shangbo Zhou, Hongbin Tang, Ying Qi and Xianzhong Xie
Entropy 2018, 20(4), 251; https://0-doi-org.brum.beds.ac.uk/10.3390/e20040251 - 04 Apr 2018
Cited by 9 | Viewed by 3235
Abstract
Visual information processing is one of the fields of cognitive informatics. In this paper, a two-layer fractional-order chaotic network, which can simulate the mechanism of visual selection and shifting, is established. Unlike other object selection models, the proposed model introduces control units to [...] Read more.
Visual information processing is one of the fields of cognitive informatics. In this paper, a two-layer fractional-order chaotic network, which can simulate the mechanism of visual selection and shifting, is established. Unlike other object selection models, the proposed model introduces control units to select object. The first chaotic network layer of the model is used to implement image segmentation. A control layer is added as the second layer, consisting of a central neuron, which controls object selection and shifting. To implement visual selection and shifting, a strategy is proposed that can achieve different subnets corresponding to the objects in the first layer synchronizing with the central neuron at different time. The central unit acting as the central nervous system synchronizes with different subnets (hybrid systems), implementing the mechanism of visual selection and shifting in the human system. The proposed model corresponds better with the human visual system than the typical model of visual information encoding and transmission and provides new possibilities for further analysis of the mechanisms of the human cognitive system. The reasonability of the proposed model is verified by experiments using artificial and natural images. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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28 pages, 1819 KiB  
Article
2D Tsallis Entropy for Image Segmentation Based on Modified Chaotic Bat Algorithm
by Zhiwei Ye, Juan Yang, Mingwei Wang, Xinlu Zong, Lingyu Yan and Wei Liu
Entropy 2018, 20(4), 239; https://0-doi-org.brum.beds.ac.uk/10.3390/e20040239 - 30 Mar 2018
Cited by 14 | Viewed by 5971
Abstract
Image segmentation is a significant step in image analysis and computer vision. Many entropy based approaches have been presented in this topic; among them, Tsallis entropy is one of the best performing methods. However, 1D Tsallis entropy does not consider make use of [...] Read more.
Image segmentation is a significant step in image analysis and computer vision. Many entropy based approaches have been presented in this topic; among them, Tsallis entropy is one of the best performing methods. However, 1D Tsallis entropy does not consider make use of the spatial correlation information within the neighborhood results might be ruined by noise. Therefore, 2D Tsallis entropy is proposed to solve the problem, and results are compared with 1D Fisher, 1D maximum entropy, 1D cross entropy, 1D Tsallis entropy, fuzzy entropy, 2D Fisher, 2D maximum entropy and 2D cross entropy. On the other hand, due to the existence of huge computational costs, meta-heuristics algorithms like genetic algorithm (GA), particle swarm optimization (PSO), ant colony optimization algorithm (ACO) and differential evolution algorithm (DE) are used to accelerate the 2D Tsallis entropy thresholding method. In this paper, considering 2D Tsallis entropy as a constrained optimization problem, the optimal thresholds are acquired by maximizing the objective function using a modified chaotic Bat algorithm (MCBA). The proposed algorithm has been tested on some actual and infrared images. The results are compared with that of PSO, GA, ACO and DE and demonstrate that the proposed method outperforms other approaches involved in the paper, which is a feasible and effective option for image segmentation. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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13 pages, 1790 KiB  
Article
An Approach for the Generation of an Nth-Order Chaotic System with Hyperbolic Sine
by Jizhao Liu, Jun Ma, Jing Lian, Pengbin Chang and Yide Ma
Entropy 2018, 20(4), 230; https://0-doi-org.brum.beds.ac.uk/10.3390/e20040230 - 27 Mar 2018
Cited by 14 | Viewed by 4314
Abstract
Chaotic systems with hyperbolic sine nonlinearity have attracted the attention of researchers in the last two years. This paper introduces a new approach for generating a class of simple chaotic systems with hyperbolic sine. With nth-order ordinary differential equations (ODEs), any desirable order [...] Read more.
Chaotic systems with hyperbolic sine nonlinearity have attracted the attention of researchers in the last two years. This paper introduces a new approach for generating a class of simple chaotic systems with hyperbolic sine. With nth-order ordinary differential equations (ODEs), any desirable order of chaotic systems with hyperbolic sine nonlinearity can be easily constructed. Fourth-order, fifth-order, and tenth-order chaotic systems are taken as examples to verify the theory. To achieve simplicity of the electrical circuit, two back-to-back diodes represent hyperbolic sine nonlinearity without any multiplier or subcircuits. Thus, these systems can achieve both physical simplicity and analytic complexity at the same time. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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19 pages, 820 KiB  
Article
Output-Feedback Control for Discrete-Time Spreading Models in Complex Networks
by Luis A. Alarcón Ramos, Roberto Bernal Jaquez and Alexander Schaum
Entropy 2018, 20(3), 204; https://0-doi-org.brum.beds.ac.uk/10.3390/e20030204 - 19 Mar 2018
Cited by 5 | Viewed by 3943
Abstract
The problem of stabilizing the spreading process to a prescribed probability distribution over a complex network is considered, where the dynamics of the nodes in the network is given by discrete-time Markov-chain processes. Conditions for the positioning and identification of actuators and sensors [...] Read more.
The problem of stabilizing the spreading process to a prescribed probability distribution over a complex network is considered, where the dynamics of the nodes in the network is given by discrete-time Markov-chain processes. Conditions for the positioning and identification of actuators and sensors are provided, and sufficient conditions for the exponential stability of the desired distribution are derived. Simulations results for a network of N = 10 6 corroborate our theoretical findings. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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9 pages, 249 KiB  
Article
Some Iterative Properties of ( F 1 , F 2 ) -Chaos in Non-Autonomous Discrete Systems
by Xiao Tang, Guanrong Chen and Tianxiu Lu
Entropy 2018, 20(3), 188; https://0-doi-org.brum.beds.ac.uk/10.3390/e20030188 - 12 Mar 2018
Cited by 13 | Viewed by 3102
Abstract
This paper is concerned with invariance ( F 1 , F 2 ) -scrambled sets under iterations. The main results are an extension of the compound invariance of Li–Yorke chaos and distributional chaos. New definitions of [...] Read more.
This paper is concerned with invariance ( F 1 , F 2 ) -scrambled sets under iterations. The main results are an extension of the compound invariance of Li–Yorke chaos and distributional chaos. New definitions of ( F 1 , F 2 ) -scrambled sets in non-autonomous discrete systems are given. For a positive integer k, the properties P ( k ) and Q ( k ) of Furstenberg families are introduced. It is shown that, for any positive integer k, for any s ∈ [ 0 , 1 ] , Furstenberg family M ¯ ( s ) has properties P ( k ) and Q ( k ) , where M ¯ ( s ) denotes the family of all infinite subsets of Z + whose upper density is not less than s. Then, the following conclusion is obtained. D is an ( M ¯ ( s ) , M ¯ ( t ) ) -scrambled set of ( X , f 1 , ∞ ) if and only if D is an ( M ¯ ( s ) , M ¯ ( t ) ) -scrambled set of ( X , f 1 , ∞ [ m ] ) . Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
14 pages, 955 KiB  
Article
Adaptive Synchronization of Fractional-Order Complex-Valued Neural Networks with Discrete and Distributed Delays
by Li Li, Zhen Wang, Junwei Lu and Yuxia Li
Entropy 2018, 20(2), 124; https://0-doi-org.brum.beds.ac.uk/10.3390/e20020124 - 13 Feb 2018
Cited by 20 | Viewed by 3848
Abstract
In this paper, the synchronization problem of fractional-order complex-valued neural networks with discrete and distributed delays is investigated. Based on the adaptive control and Lyapunov function theory, some sufficient conditions are derived to ensure the states of two fractional-order complex-valued neural networks with [...] Read more.
In this paper, the synchronization problem of fractional-order complex-valued neural networks with discrete and distributed delays is investigated. Based on the adaptive control and Lyapunov function theory, some sufficient conditions are derived to ensure the states of two fractional-order complex-valued neural networks with discrete and distributed delays achieve complete synchronization rapidly. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of the theoretical results. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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16 pages, 2660 KiB  
Article
Synchronization in Fractional-Order Complex-Valued Delayed Neural Networks
by Weiwei Zhang, Jinde Cao, Dingyuan Chen and Fuad E. Alsaadi
Entropy 2018, 20(1), 54; https://0-doi-org.brum.beds.ac.uk/10.3390/e20010054 - 12 Jan 2018
Cited by 30 | Viewed by 3913
Abstract
This paper discusses the synchronization of fractional order complex valued neural networks (FOCVNN) at the presence of time delay. Synchronization criterions are achieved through the employment of a linear feedback control and comparison theorem of fractional order linear systems with delay. Feasibility and [...] Read more.
This paper discusses the synchronization of fractional order complex valued neural networks (FOCVNN) at the presence of time delay. Synchronization criterions are achieved through the employment of a linear feedback control and comparison theorem of fractional order linear systems with delay. Feasibility and effectiveness of the proposed system are validated through numerical simulations. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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20 pages, 2652 KiB  
Article
Searching for Chaos Evidence in Eye Movement Signals
by Katarzyna Harezlak and Pawel Kasprowski
Entropy 2018, 20(1), 32; https://0-doi-org.brum.beds.ac.uk/10.3390/e20010032 - 07 Jan 2018
Cited by 24 | Viewed by 4892
Abstract
Most naturally-occurring physical phenomena are examples of nonlinear dynamic systems, the functioning of which attracts many researchers seeking to unveil their nature. The research presented in this paper is aimed at exploring eye movement dynamic features in terms of the existence of chaotic [...] Read more.
Most naturally-occurring physical phenomena are examples of nonlinear dynamic systems, the functioning of which attracts many researchers seeking to unveil their nature. The research presented in this paper is aimed at exploring eye movement dynamic features in terms of the existence of chaotic nature. Nonlinear time series analysis methods were used for this purpose. Two time series features were studied: fractal dimension and entropy, by utilising the embedding theory. The methods were applied to the data collected during the experiment with “jumping point” stimulus. Eye movements were registered by means of the Jazz-novo eye tracker. One thousand three hundred and ninety two (1392) time series were defined, based on the horizontal velocity of eye movements registered during imposed, prolonged fixations. In order to conduct detailed analysis of the signal and identify differences contributing to the observed patterns of behaviour in time scale, fractal dimension and entropy were evaluated in various time series intervals. The influence of the noise contained in the data and the impact of the utilized filter on the obtained results were also studied. The low pass filter was used for the purpose of noise reduction with a 50 Hz cut-off frequency, estimated by means of the Fourier transform and all concerned methods were applied to time series before and after noise reduction. These studies provided some premises, which allow perceiving eye movements as observed chaotic data: characteristic of a space-time separation plot, low and non-integer time series dimension, and the time series entropy characteristic for chaotic systems. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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538 KiB  
Article
State Estimation for General Complex Dynamical Networks with Incompletely Measured Information
by Xinwei Wang, Guo-Ping Jiang and Xu Wu
Entropy 2018, 20(1), 5; https://0-doi-org.brum.beds.ac.uk/10.3390/e20010005 - 23 Dec 2017
Cited by 9 | Viewed by 3215
Abstract
Estimating uncertain state variables of a general complex dynamical network with randomly incomplete measurements of transmitted output variables is investigated in this paper. The incomplete measurements, occurring randomly through the transmission of output variables, always cause the failure of the state estimation process. [...] Read more.
Estimating uncertain state variables of a general complex dynamical network with randomly incomplete measurements of transmitted output variables is investigated in this paper. The incomplete measurements, occurring randomly through the transmission of output variables, always cause the failure of the state estimation process. Different from the existing methods, we propose a novel method to handle the incomplete measurements, which can perform well to balance the excessively deviated estimators under the influence of incomplete measurements. In particular, the proposed method has no special limitation on the node dynamics compared with many existing methods. By employing the Lyapunov stability theory along with the stochastic analysis method, sufficient criteria are deduced rigorously to ensure obtaining the proper estimator gains with known model parameters. Illustrative simulation for the complex dynamical network composed of chaotic nodes are given to show the validity and efficiency of the proposed method. Full article
(This article belongs to the Special Issue Research Frontier in Chaos Theory and Complex Networks)
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