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Special Issue "Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors"

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

Deadline for manuscript submissions: 31 January 2019

Special Issue Editors

Guest Editor
Dr. Christos Volos

Laboratory of Nonlinear Systems - Circuits & Complexity (LaNSCom)Department of Physics, Aristotle University of Thessaloniki, Greece
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Interests: Nonlinear Dynamics; Chaotic Systems; Nonlinear Circuits; Hidden Attractors
Guest Editor
Dr. Sajad Jafari

Department of Biomedical Engineering, Amirkabir University of Technology, Iran
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Interests: Chaos; Nonlinear Dynamics; Optimization
Guest Editor
Dr. Jacques Kengne

Department of Electrical Engineering, University of Dschang, Cameroon
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Interests: Department of Electrical Engineering, University of Dschang, Cameroon
Guest Editor
Dr. Jesus M. Munoz-Pacheco

Faculty of Electronics Sciences, Autonomous University of Puebla, Mexico
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Interests: Chaos; Synchronization; Analog Circuits; CAD; Evolutionary Algorithms
Guest Editor
Dr. Karthikeyan Rajagopal

Center for Nonlinear Dynamics, The PNG University of Technology, Lae, Papua New Guinea
Website | E-Mail
Interests: Optimal Control Theory; Artificial Intelligence; Adaptive Control; Neural Networks

Special Issue Information

Dear Colleagues,

In the last few years, entropy has been a basic and important concept in information theory. It is also often used as a measure of the degree of chaos in systems, e.g., Lyapunov exponents, fractal dimension, and entropy are usually used to describe the complexity of chaotic systems. Thus, it would be very interesting to study entropy in nonlinear systems. Additionally, there has been an increasing interest in a new classification of nonlinear dynamical systems including two kinds of attractors: Self-excited attractors and hidden attractors. Self-excited attractors can be localized straight forwardly by applying a standard computational procedure. Some interesting examples of systems with self-excited attractors are chaotic systems with different kinds of symmetry, with multi-scroll attractors, with multiple attractors, and with extreme multistability.

In systems with hidden attractors we have to develop a specific computational procedure to identify the hidden attractors due to the fact that the equilibrium points do not help in their localization. Some examples of this kind of systems are chaotic dynamical systems with no equilibrium points, with only stable equilibria, with curves of equilibria, with surfaces of equilibria, and with non-hyperbolic equilibria. There is evidence that hidden attractors play an important role in the various fields ranging from phase-locked loops, oscillators, describing convective fluid motion, model of drilling system, information theory, cryptography to multilevel DC/DC converter. Furthermore, hidden attractors may lead to unexpected and disastrous responses.

The Special Issue is dedicated to the presentation and discussion of the advanced topics of entropy in complex systems with hidden attractors and self-excited attractors. The contribution to the Special Issue should be focus on the aspects of nonlinear dynamics, entropy, and applications of nonlinear systems with hidden and self-excited attractors.

Dr. Christos Volos
Dr. Sajad Jafari
Dr. Jacques Kengne
Dr. Jesus M. Munoz-Pacheco
Dr. Karthikeyan Rajagopal
Guest Editors

Manuscript Submission Information

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

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

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1500 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Nonlinear systems
  • Complex systems
  • Chaos
  • Control
  • Entropy
  • Fractionalorder systems
  • Hidden attractors
  • Self-excited attractors
  • Synchronization

Published Papers (12 papers)

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Research

Open AccessArticle The Complexity and Entropy Analysis for Service Game Model Based on Different Expectations and Optimal Pricing
Entropy 2018, 20(11), 858; https://doi.org/10.3390/e20110858
Received: 3 October 2018 / Revised: 3 November 2018 / Accepted: 5 November 2018 / Published: 8 November 2018
Cited by 1 | PDF Full-text (2843 KB) | HTML Full-text | XML Full-text
Abstract
The internet has provided a new means for manufacturers to reach consumers. On the background of the widespread multichannel sales in China, based on a literature review of the service game and multichannel supply chain, this paper builds a multichannel dynamic service game
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The internet has provided a new means for manufacturers to reach consumers. On the background of the widespread multichannel sales in China, based on a literature review of the service game and multichannel supply chain, this paper builds a multichannel dynamic service game model where the retailer operates an offline channel and the manufacturer operates an online channel and offers customers the option to buy online and pick up from the retailer’s store (BOPS). The manufacturer and the retailer take maximizing the channel profits as their business objectives and make channel service game under optimal pricing. We carry on theoretical analysis of the model and perform numerical simulations from the perspective of entropy theory, game theory, and chaotic dynamics. The results show that the stability of the system will weaken with the increase in service elasticity coefficient and that it is unaffected by the feedback parameter adjustment of the retailer. The BOPS channel strengthens the cooperation between the manufacturer and the retailer and moderates the conflict between the online and the offline channels. The system will go into chaotic state and cause the system’s entropy to increase when the manufacturer adjusts his/her service decision quickly. In a chaotic state, the system is sensitive to initial conditions and service input is difficult to predict; the manufacturer and retailer need more additional information to make the system clear or use the method of feedback control to delay or eliminate the occurrence of chaos. Full article
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Open AccessArticle The Co-existence of Different Synchronization Types in Fractional-order Discrete-time Chaotic Systems with Non–identical Dimensions and Orders
Entropy 2018, 20(9), 710; https://doi.org/10.3390/e20090710
Received: 25 August 2018 / Revised: 13 September 2018 / Accepted: 13 September 2018 / Published: 14 September 2018
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Abstract
This paper is concerned with the co-existence of different synchronization types for fractional-order discrete-time chaotic systems with different dimensions. In particular, we show that through appropriate nonlinear control, projective synchronization (PS), full state hybrid projective synchronization (FSHPS), and generalized synchronization (GS) can be
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This paper is concerned with the co-existence of different synchronization types for fractional-order discrete-time chaotic systems with different dimensions. In particular, we show that through appropriate nonlinear control, projective synchronization (PS), full state hybrid projective synchronization (FSHPS), and generalized synchronization (GS) can be achieved simultaneously. A second nonlinear control scheme is developed whereby inverse full state hybrid projective synchronization (IFSHPS) and inverse generalized synchronization (IGS) are shown to co-exist. Numerical examples are presented to confirm the findings. Full article
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Open AccessArticle Strange Attractors Generated by Multiple-Valued Static Memory Cell with Polynomial Approximation of Resonant Tunneling Diodes
Entropy 2018, 20(9), 697; https://doi.org/10.3390/e20090697
Received: 20 August 2018 / Revised: 7 September 2018 / Accepted: 9 September 2018 / Published: 12 September 2018
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Abstract
This paper brings analysis of the multiple-valued memory system (MVMS) composed by a pair of the resonant tunneling diodes (RTD). Ampere-voltage characteristic (AVC) of both diodes is approximated in operational voltage range as common in practice: by polynomial scalar function. Mathematical model of
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This paper brings analysis of the multiple-valued memory system (MVMS) composed by a pair of the resonant tunneling diodes (RTD). Ampere-voltage characteristic (AVC) of both diodes is approximated in operational voltage range as common in practice: by polynomial scalar function. Mathematical model of MVMS represents autonomous deterministic dynamical system with three degrees of freedom and smooth vector field. Based on the very recent results achieved for piecewise-linear MVMS numerical values of the parameters are calculated such that funnel and double spiral chaotic attractor is observed. Existence of such types of strange attractors is proved both numerically by using concept of the largest Lyapunov exponents (LLE) and experimentally by computer-aided simulation of designed lumped circuit using only commercially available active elements. Full article
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Open AccessArticle Optimization of Thurston’s Core Entropy Algorithm for Polynomials with a Critical Point of Maximal Order
Entropy 2018, 20(9), 695; https://doi.org/10.3390/e20090695
Received: 1 August 2018 / Revised: 5 September 2018 / Accepted: 5 September 2018 / Published: 11 September 2018
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Abstract
This paper discusses some properties of the topological entropy systems generated by polynomials of degree d in their Hubbard tree. An optimization of Thurston’s core entropy algorithm is developed for a family of polynomials of degree d. Full article
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Open AccessArticle A New Chaotic System with Stable Equilibrium: Entropy Analysis, Parameter Estimation, and Circuit Design
Entropy 2018, 20(9), 670; https://doi.org/10.3390/e20090670
Received: 12 July 2018 / Revised: 1 August 2018 / Accepted: 2 August 2018 / Published: 5 September 2018
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Abstract
In this paper, we introduce a new, three-dimensional chaotic system with one stable equilibrium. This system is a multistable dynamic system in which the strange attractor is hidden. We investigate its dynamic properties through equilibrium analysis, a bifurcation diagram and Lyapunov exponents. Such
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In this paper, we introduce a new, three-dimensional chaotic system with one stable equilibrium. This system is a multistable dynamic system in which the strange attractor is hidden. We investigate its dynamic properties through equilibrium analysis, a bifurcation diagram and Lyapunov exponents. Such multistable systems are important in engineering. We perform an entropy analysis, parameter estimation and circuit design using this new system to show its feasibility and ability to be used in engineering applications. Full article
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Open AccessArticle A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors
Entropy 2018, 20(8), 564; https://doi.org/10.3390/e20080564
Received: 2 July 2018 / Revised: 23 July 2018 / Accepted: 25 July 2018 / Published: 28 July 2018
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Abstract
In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden
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In this work, a new fractional-order chaotic system with a single parameter and four nonlinearities is introduced. One striking feature is that by varying the system parameter, the fractional-order system generates several complex dynamics: self-excited attractors, hidden attractors, and the coexistence of hidden attractors. In the family of self-excited chaotic attractors, the system has four spiral-saddle-type equilibrium points, or two nonhyperbolic equilibria. Besides, for a certain value of the parameter, a fractional-order no-equilibrium system is obtained. This no-equilibrium system presents a hidden chaotic attractor with a `hurricane’-like shape in the phase space. Multistability is also observed, since a hidden chaotic attractor coexists with a periodic one. The chaos generation in the new fractional-order system is demonstrated by the Lyapunov exponents method and equilibrium stability. Moreover, the complexity of the self-excited and hidden chaotic attractors is analyzed by computing their spectral entropy and Brownian-like motions. Finally, a pseudo-random number generator is designed using the hidden dynamics. Full article
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Open AccessArticle Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems
Entropy 2018, 20(8), 556; https://doi.org/10.3390/e20080556
Received: 9 June 2018 / Revised: 23 July 2018 / Accepted: 24 July 2018 / Published: 27 July 2018
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Abstract
Designing a chaotic system with infinitely many attractors is a hot topic. In this paper, multiscale multivariate permutation entropy (MMPE) and multiscale multivariate Lempel–Ziv complexity (MMLZC) are employed to analyze the complexity of those self-reproducing chaotic systems with one-directional and two-directional infinitely many
[...] Read more.
Designing a chaotic system with infinitely many attractors is a hot topic. In this paper, multiscale multivariate permutation entropy (MMPE) and multiscale multivariate Lempel–Ziv complexity (MMLZC) are employed to analyze the complexity of those self-reproducing chaotic systems with one-directional and two-directional infinitely many chaotic attractors. The analysis results show that complexity of this class of chaotic systems is determined by the initial conditions. Meanwhile, the values of MMPE are independent of the scale factor, which is different from the algorithm of MMLZC. The analysis proposed here is helpful as a reference for the application of the self-reproducing systems. Full article
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Open AccessArticle Stochastic Entropy Solutions for Stochastic Nonlinear Transport Equations
Entropy 2018, 20(6), 395; https://doi.org/10.3390/e20060395
Received: 26 April 2018 / Revised: 15 May 2018 / Accepted: 21 May 2018 / Published: 23 May 2018
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Abstract
This paper considers the existence and uniqueness of stochastic entropy solution for a nonlinear transport equation with a stochastic perturbation. The uniqueness is based on the doubling variable method. For the existence, we develop a new scheme of parabolic approximation motivated by the
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This paper considers the existence and uniqueness of stochastic entropy solution for a nonlinear transport equation with a stochastic perturbation. The uniqueness is based on the doubling variable method. For the existence, we develop a new scheme of parabolic approximation motivated by the method of vanishing viscosity given by Feng and Nualart (J. Funct. Anal. 2008, 255, 313–373). Furthermore, we prove the continuous dependence of stochastic strong entropy solutions on the coefficient b and the nonlinear function f. Full article
Open AccessArticle A New Two-Dimensional Map with Hidden Attractors
Entropy 2018, 20(5), 322; https://doi.org/10.3390/e20050322
Received: 31 January 2018 / Revised: 21 March 2018 / Accepted: 24 April 2018 / Published: 27 April 2018
Cited by 5 | PDF Full-text (5288 KB) | HTML Full-text | XML Full-text
Abstract
The investigations of hidden attractors are mainly in continuous-time dynamic systems, and there are a few investigations of hidden attractors in discrete-time dynamic systems. The classical chaotic attractors of the Logistic map, Tent map, Henon map, Arnold’s cat map, and other widely-known chaotic
[...] Read more.
The investigations of hidden attractors are mainly in continuous-time dynamic systems, and there are a few investigations of hidden attractors in discrete-time dynamic systems. The classical chaotic attractors of the Logistic map, Tent map, Henon map, Arnold’s cat map, and other widely-known chaotic attractors are those excited from unstable fixed points. In this paper, the hidden dynamics of a new two-dimensional map inspired by Arnold’s cat map is investigated, and the existence of fixed points and their stabilities are studied in detail. Full article
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Open AccessArticle A Novel Algorithm to Improve Digital Chaotic Sequence Complexity through CCEMD and PE
Entropy 2018, 20(4), 295; https://doi.org/10.3390/e20040295
Received: 18 March 2018 / Revised: 10 April 2018 / Accepted: 12 April 2018 / Published: 18 April 2018
Cited by 1 | PDF Full-text (4904 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, a three-dimensional chaotic system with a hidden attractor is introduced. The complex dynamic behaviors of the system are analyzed with a Poincaré cross section, and the equilibria and initial value sensitivity are analyzed by the method of numerical simulation. Further,
[...] Read more.
In this paper, a three-dimensional chaotic system with a hidden attractor is introduced. The complex dynamic behaviors of the system are analyzed with a Poincaré cross section, and the equilibria and initial value sensitivity are analyzed by the method of numerical simulation. Further, we designed a new algorithm based on complementary ensemble empirical mode decomposition (CEEMD) and permutation entropy (PE) that can effectively enhance digital chaotic sequence complexity. In addition, an image encryption experiment was performed with post-processing of the chaotic binary sequences by the new algorithm. The experimental results show good performance of the chaotic binary sequence. Full article
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Open AccessArticle A New Chaotic System with a Self-Excited Attractor: Entropy Measurement, Signal Encryption, and Parameter Estimation
Entropy 2018, 20(2), 86; https://doi.org/10.3390/e20020086
Received: 28 December 2017 / Revised: 19 January 2018 / Accepted: 21 January 2018 / Published: 27 January 2018
Cited by 19 | PDF Full-text (8699 KB) | HTML Full-text | XML Full-text
Abstract
In this paper, we introduce a new chaotic system that is used for an engineering application of the signal encryption. It has some interesting features, and its successful implementation and manufacturing were performed via a real circuit as a random number generator. In
[...] Read more.
In this paper, we introduce a new chaotic system that is used for an engineering application of the signal encryption. It has some interesting features, and its successful implementation and manufacturing were performed via a real circuit as a random number generator. In addition, we provide a parameter estimation method to extract chaotic model parameters from the real data of the chaotic circuit. The parameter estimation method is based on the attractor distribution modeling in the state space, which is compatible with the chaotic system characteristics. Here, a Gaussian mixture model (GMM) is used as a main part of cost function computations in the parameter estimation method. To optimize the cost function, we also apply two recent efficient optimization methods: WOA (Whale Optimization Algorithm), and MVO (Multi-Verse Optimizer) algorithms. The results show the success of the parameter estimation procedure. Full article
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Open AccessArticle A New Chaotic System with Multiple Attractors: Dynamic Analysis, Circuit Realization and S-Box Design
Entropy 2018, 20(1), 12; https://doi.org/10.3390/e20010012
Received: 17 November 2017 / Revised: 22 December 2017 / Accepted: 25 December 2017 / Published: 27 December 2017
Cited by 2 | PDF Full-text (5274 KB) | HTML Full-text | XML Full-text
Abstract
This paper reports about a novel three-dimensional chaotic system with three nonlinearities. The system has one stable equilibrium, two stable equilibria and one saddle node, two saddle foci and one saddle node for different parameters. One salient feature of this novel system is
[...] Read more.
This paper reports about a novel three-dimensional chaotic system with three nonlinearities. The system has one stable equilibrium, two stable equilibria and one saddle node, two saddle foci and one saddle node for different parameters. One salient feature of this novel system is its multiple attractors caused by different initial values. With the change of parameters, the system experiences mono-stability, bi-stability, mono-periodicity, bi-periodicity, one strange attractor, and two coexisting strange attractors. The complex dynamic behaviors of the system are revealed by analyzing the corresponding equilibria and using the numerical simulation method. In addition, an electronic circuit is given for implementing the chaotic attractors of the system. Using the new chaotic system, an S-Box is developed for cryptographic operations. Moreover, we test the performance of this produced S-Box and compare it to the existing S-Box studies. Full article
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