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Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II

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

Deadline for manuscript submissions: closed (15 May 2020) | Viewed by 56484

Special Issue Editors

Dr. Christos Volos

E-Mail Website
Guest Editor
Laboratory of Nonlinear Systems, Circuits & Coplexity (LaNSCom), Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece
Interests: electrical and electronics engineering; mathematical modeling; control theory; engineering, applied and computational mathematics; numerical analysis; mathematical analysis; numerical modeling; modeling and simulation; robotics
Special Issues, Collections and Topics in MDPI journals
Dr. Sajad Jafari

E-Mail
Guest Editor
Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Interests: chaos; nonlinear dynamics; optimization
Special Issues, Collections and Topics in MDPI journals
Dr. Jesus M. Munoz-Pacheco

E-Mail Website
Guest Editor
Fractional-Order Systems and Nonlinear Circuits Group, Faculty of Electronics Sciences, Autonomous University of Puebla, Puebla 72570, Mexico
Interests: fractional order calculus; chaotic systems and their applications; hidden attractors; mathematical modeling; chaos theory; synchronization; random number generators
Special Issues, Collections and Topics in MDPI journals
Dr. Jacques Kengne

E-Mail Website
Guest Editor
Department of Electrical Engineering, University of Dschang, Dschang P.O. Box 134, Cameroon
Interests: chaos theory; nonlinear phenomena; nonlinear circuits; hidden attractors; synchronization
Special Issues, Collections and Topics in MDPI journals
Dr. Karthikeyan Rajagopal

E-Mail Website
Guest Editor
Center for Nonlinear Systems, Chennai Institute of Technology, Tamil Nadu 600069, India
Interests: optimal control theory; artificial intelligence; adaptive control; neural networks
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Entropy is a basic and important concept in information theory. It is also often used as a measure of the degree of chaos in dynamical systems, for example, Lyapunov exponents, fractal dimension, and entropy are usually used to describe the complexity of chaotic systems. Thus, it would be interesting to collect the latest advances in the field of studying entropy in nonlinear systems.

Additionally, in the last few years, there has been increasing interest in a new classification of nonlinear dynamical systems, including two kinds of attractors, namely: 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, multiple attractors, and extreme multistability. On the other hand, in systems with hidden attractors, we have to develop a specific computational procedure to identify the hidden attractors because of the fact that the equilibrium points do not help in their localization. Some examples of these kinds of systems are chaotic dynamical systems with no equilibrium points, with only stable equilibria, curves of equilibria, surfaces of equilibria, and non-hyperbolic equilibria. There is evidence that hidden attractors play an important role in the various fields, ranging from phase-locked loops, oscillators describing a convective fluid motion, models of drilling systems, information theory, and cryptography to multilevel DC/DC converters. Furthermore, hidden attractors may lead to unexpected and disastrous responses. So, it is very useful to find new tools in order to study entropy for hidden attractors.

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

Potential topics include, but are not limited to, the following:

  • Analytical–numerical methods for investing hidden oscillations
  • Bifurcation and chaos in complex systems
  • Chimera states, spiral waves, and pattern formation in networks of oscillators
  • Self-organization
  • Designing new nonlinear systems with desired features
  • Experimental study of nonlinear systems
  • Extreme multistability
  • Complex networks
  • Fractional order dynamical systems
  • Hidden attractors in complex systems
  • Entropy of hidden attractors
  • Networks of nonlinear oscillators (like neurons)
  • New methods of control and synchronization nonlinear systems
  • Information theory
  • Nonlinear dynamics and chaos in engineering applications
  • Nonlinear systems with an infinite number of equilibrium points
  • Nonlinear systems with a stable equilibrium
  • Nonlinear systems without equilibrium
  • Entropy-based cryptography
  • Novel computation algorithms for studying nonlinear systems
  • Oscillations and chaos in dynamic economic models
  • Quantum chaos
  • Related engineering applications
  • Self-excited attractors

Dr. Christos Volos
Dr. Sajad Jafari
Dr. Jesus M. Munoz-Pacheco
Dr. Jacques Kengne
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 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.

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Editorial

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5 pages, 194 KiB  
Editorial
Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II
by Christos K. Volos, Sajad Jafari, Jesus M. Munoz-Pacheco, Jacques Kengne and Karthikeyan Rajagopal
Entropy 2020, 22(12), 1428; https://doi.org/10.3390/e22121428 - 18 Dec 2020
Cited by 3 | Viewed by 1720
Abstract
According to the pioneering work of Leonov and Kuznetsov [...] Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)

Research

Jump to: Editorial

16 pages, 1602 KiB  
Article
Chaos Control and Synchronization of a Complex Rikitake Dynamo Model
by Wenkai Pang, Zekang Wu, Yu Xiao and Cuimei Jiang
Entropy 2020, 22(6), 671; https://doi.org/10.3390/e22060671 - 17 Jun 2020
Cited by 5 | Viewed by 2450
Abstract
A novel chaotic system called complex Rikitake system is proposed. Dynamical properties, including symmetry, dissipation, stability of equilibria, Lyapunov exponents and bifurcation, are analyzed on the basis of theoretical analysis and numerical simulation. Further, based on feedback control method, the complex Rikitake system [...] Read more.
A novel chaotic system called complex Rikitake system is proposed. Dynamical properties, including symmetry, dissipation, stability of equilibria, Lyapunov exponents and bifurcation, are analyzed on the basis of theoretical analysis and numerical simulation. Further, based on feedback control method, the complex Rikitake system can be controlled to any equilibrium points. Additionally, this paper not only proves the existence of two types of synchronization schemes in the complex Rikitake system but also designs adaptive controllers to realize them. The proposed results are verified by numerical simulations. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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15 pages, 20271 KiB  
Article
Birhythmic Analog Circuit Maze: A Nonlinear Neurostimulation Testbed
by Ian D. Jordan and Il Memming Park
Entropy 2020, 22(5), 537; https://doi.org/10.3390/e22050537 - 11 May 2020
Cited by 6 | Viewed by 2965
Abstract
Brain dynamics can exhibit narrow-band nonlinear oscillations and multistability. For a subset of disorders of consciousness and motor control, we hypothesized that some symptoms originate from the inability to spontaneously transition from one attractor to another. Using external perturbations, such as electrical pulses [...] Read more.
Brain dynamics can exhibit narrow-band nonlinear oscillations and multistability. For a subset of disorders of consciousness and motor control, we hypothesized that some symptoms originate from the inability to spontaneously transition from one attractor to another. Using external perturbations, such as electrical pulses delivered by deep brain stimulation devices, it may be possible to induce such transition out of the pathological attractors. However, the induction of transition may be non-trivial, rendering the current open-loop stimulation strategies insufficient. In order to develop next-generation neural stimulators that can intelligently learn to induce attractor transitions, we require a platform to test the efficacy of such systems. To this end, we designed an analog circuit as a model for the multistable brain dynamics. The circuit spontaneously oscillates stably on two periods as an instantiation of a 3-dimensional continuous-time gated recurrent neural network. To discourage simple perturbation strategies, such as constant or random stimulation patterns from easily inducing transition between the stable limit cycles, we designed a state-dependent nonlinear circuit interface for external perturbation. We demonstrate the existence of nontrivial solutions to the transition problem in our circuit implementation. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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15 pages, 3506 KiB  
Article
Neural Computing Enhanced Parameter Estimation for Multi-Input and Multi-Output Total Non-Linear Dynamic Models
by Longlong Liu, Di Ma, Ahmad Taher Azar and Quanmin Zhu
Entropy 2020, 22(5), 510; https://doi.org/10.3390/e22050510 - 30 Apr 2020
Cited by 16 | Viewed by 2689
Abstract
In this paper, a gradient descent algorithm is proposed for the parameter estimation of multi-input and multi-output (MIMO) total non-linear dynamic models. Firstly, the MIMO total non-linear model is mapped to a non-completely connected feedforward neural network, that is, the parameters of the [...] Read more.
In this paper, a gradient descent algorithm is proposed for the parameter estimation of multi-input and multi-output (MIMO) total non-linear dynamic models. Firstly, the MIMO total non-linear model is mapped to a non-completely connected feedforward neural network, that is, the parameters of the total non-linear model are mapped to the connection weights of the neural network. Then, based on the minimization of network error, a weight-updating algorithm, that is, an estimation algorithm of model parameters, is proposed with the convergence conditions of a non-completely connected feedforward network. In further determining the variables of the model set, a method of model structure detection is proposed for selecting a group of important items from the whole variable candidate set. In order to verify the usefulness of the parameter identification process, we provide a virtual bench test example for the numerical analysis and user-friendly instructions for potential applications. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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20 pages, 32574 KiB  
Article
Modification of the Logistic Map Using Fuzzy Numbers with Application to Pseudorandom Number Generation and Image Encryption
by Lazaros Moysis, Christos Volos, Sajad Jafari, Jesus M. Munoz-Pacheco, Jacques Kengne, Karthikeyan Rajagopal and Ioannis Stouboulos
Entropy 2020, 22(4), 474; https://doi.org/10.3390/e22040474 - 20 Apr 2020
Cited by 37 | Viewed by 3795
Abstract
A modification of the classic logistic map is proposed, using fuzzy triangular numbers. The resulting map is analysed through its Lyapunov exponent (LE) and bifurcation diagrams. It shows higher complexity compared to the classic logistic map and showcases phenomena, like antimonotonicity and crisis. [...] Read more.
A modification of the classic logistic map is proposed, using fuzzy triangular numbers. The resulting map is analysed through its Lyapunov exponent (LE) and bifurcation diagrams. It shows higher complexity compared to the classic logistic map and showcases phenomena, like antimonotonicity and crisis. The map is then applied to the problem of pseudo random bit generation, using a simple rule to generate the bit sequence. The resulting random bit generator (RBG) successfully passes the National Institute of Standards and Technology (NIST) statistical tests, and it is then successfully applied to the problem of image encryption. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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32 pages, 16399 KiB  
Article
Fractional-Order Chaotic Memory with Wideband Constant Phase Elements
by Jiri Petrzela
Entropy 2020, 22(4), 422; https://doi.org/10.3390/e22040422 - 09 Apr 2020
Cited by 15 | Viewed by 2823
Abstract
This paper provides readers with three partial results that are mutually connected. Firstly, the gallery of the so-called constant phase elements (CPE) dedicated for the wideband applications is presented. CPEs are calculated for 9° (decimal orders) and 10° phase steps including ¼, ½, [...] Read more.
This paper provides readers with three partial results that are mutually connected. Firstly, the gallery of the so-called constant phase elements (CPE) dedicated for the wideband applications is presented. CPEs are calculated for 9° (decimal orders) and 10° phase steps including ¼, ½, and ¾ orders, which are the most used mathematical orders between zero and one in practice. For each phase shift, all necessary numerical values to design fully passive RC ladder two-terminal circuits are provided. Individual CPEs are easily distinguishable because of a very high accuracy; maximal phase error is less than 1.5° in wide frequency range beginning with 3 Hz and ending with 1 MHz. Secondly, dynamics of ternary memory composed by a series connection of two resonant tunneling diodes is investigated and, consequently, a robust chaotic behavior is discovered and reported. Finally, CPEs are directly used for realization of fractional-order (FO) ternary memory as lumped chaotic oscillator. Existence of structurally stable strange attractors for different orders is proved, both by numerical analyzed and experimental measurement. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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8 pages, 4645 KiB  
Article
Investigation of Early Warning Indexes in a Three-Dimensional Chaotic System with Zero Eigenvalues
by Lianyu Chen, Fahimeh Nazarimehr, Sajad Jafari, Esteban Tlelo-Cuautle and Iqtadar Hussain
Entropy 2020, 22(3), 341; https://doi.org/10.3390/e22030341 - 17 Mar 2020
Cited by 6 | Viewed by 2307
Abstract
A rare three-dimensional chaotic system with all eigenvalues equal to zero is proposed, and its dynamical properties are investigated. The chaotic system has one equilibrium point at the origin. Numerical analysis shows that the equilibrium point is unstable. Bifurcation analysis of the system [...] Read more.
A rare three-dimensional chaotic system with all eigenvalues equal to zero is proposed, and its dynamical properties are investigated. The chaotic system has one equilibrium point at the origin. Numerical analysis shows that the equilibrium point is unstable. Bifurcation analysis of the system shows various dynamics in a period-doubling route to chaos. We highlight that from the evaluation of the entropy, bifurcation points can be predicted by identifying early warning signals. In this manner, bifurcation points of the system are analyzed using Shannon and Kolmogorov-Sinai entropy. The results are compared with Lyapunov exponents. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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15 pages, 4325 KiB  
Article
Synchronization of a Non-Equilibrium Four-Dimensional Chaotic System Using a Disturbance-Observer-Based Adaptive Terminal Sliding Mode Control Method
by Shaojie Wang, Amin Yousefpour, Abdullahi Yusuf, Hadi Jahanshahi, Raúl Alcaraz, Shaobo He and Jesus M. Munoz-Pacheco
Entropy 2020, 22(3), 271; https://doi.org/10.3390/e22030271 - 27 Feb 2020
Cited by 35 | Viewed by 2736
Abstract
In this paper, dynamical behavior and synchronization of a non-equilibrium four-dimensional chaotic system are studied. The system only includes one constant term and has hidden attractors. Some dynamical features of the governing system, such as invariance and symmetry, the existence of attractors and [...] Read more.
In this paper, dynamical behavior and synchronization of a non-equilibrium four-dimensional chaotic system are studied. The system only includes one constant term and has hidden attractors. Some dynamical features of the governing system, such as invariance and symmetry, the existence of attractors and dissipativity, chaotic flow with a plane of equilibria, and offset boosting of the chaotic attractor, are stated and discussed and a new disturbance-observer-based adaptive terminal sliding mode control (ATSMC) method with input saturation is proposed for the control and synchronization of the chaotic system. To deal with unexpected noises, an extended Kalman filter (EKF) is implemented along with the designed controller. Through the concept of Lyapunov stability, the proposed control technique guarantees the finite time convergence of the uncertain system in the presence of disturbances and control input limits. Furthermore, to decrease the chattering phenomena, a genetic algorithm is used to optimize the controller parameters. Finally, numerical simulations are presented to demonstrate the performance of the designed control scheme in the presence of noise, disturbances, and control input saturation. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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11 pages, 547 KiB  
Article
Dynamic Effects Arise Due to Consumers’ Preferences Depending on Past Choices
by Sameh S. Askar and A. Al-khedhairi
Entropy 2020, 22(2), 173; https://doi.org/10.3390/e22020173 - 03 Feb 2020
Cited by 4 | Viewed by 1837
Abstract
We analyzed a dynamic duopoly game where players adopt specific preferences. These preferences are derived from Cobb–Douglas utility function with the assumption that they depend on past choices. For this paper, we investigated two possible cases for the suggested game. The first case [...] Read more.
We analyzed a dynamic duopoly game where players adopt specific preferences. These preferences are derived from Cobb–Douglas utility function with the assumption that they depend on past choices. For this paper, we investigated two possible cases for the suggested game. The first case considers only focusing on the action done by one player. This action reduces the game’s map to a one-dimensional map, which is the logistic map. Using analytical and numerical simulation, the stability of fixed points of this map is studied. In the second case, we focus on the actions applied by both players. The fixed points, in this case, are calculated, and their stability is discussed. The conditions of stability are provided in terms of the game’s parameters. Numerical simulation is carried out to give local and global investigations of the chaotic behavior of the game’s map. In addition, we use a statistical measure, such as entropy, to get more evidences on the regularity and predictability of time series associated with this case. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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15 pages, 826 KiB  
Article
Stabilization of Port Hamiltonian Chaotic Systems with Hidden Attractors by Adaptive Terminal Sliding Mode Control
by Ahmad Taher Azar and Fernando E. Serrano
Entropy 2020, 22(1), 122; https://doi.org/10.3390/e22010122 - 19 Jan 2020
Cited by 24 | Viewed by 3064
Abstract
In this study, the design of an adaptive terminal sliding mode controller for the stabilization of port Hamiltonian chaotic systems with hidden attractors is proposed. This study begins with the design methodology of a chaotic oscillator with a hidden attractor implementing the topological [...] Read more.
In this study, the design of an adaptive terminal sliding mode controller for the stabilization of port Hamiltonian chaotic systems with hidden attractors is proposed. This study begins with the design methodology of a chaotic oscillator with a hidden attractor implementing the topological framework for its respective design. With this technique it is possible to design a 2-D chaotic oscillator, which is then converted into port-Hamiltonia to track and analyze these models for the stabilization of the hidden chaotic attractors created by this analysis. Adaptive terminal sliding mode controllers (ATSMC) are built when a Hamiltonian system has a chaotic behavior and a hidden attractor is detected. A Lyapunov approach is used to formulate the adaptive device controller by creating a control law and the adaptive law, which are used online to make the system states stable while at the same time suppressing its chaotic behavior. The empirical tests obtaining the discussion and conclusions of this thesis should verify the theoretical findings. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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16 pages, 2255 KiB  
Article
Image Parallel Encryption Technology Based on Sequence Generator and Chaotic Measurement Matrix
by Jiayin Yu, Shiyu Guo, Xiaomeng Song, Yaqin Xie and Erfu Wang
Entropy 2020, 22(1), 76; https://doi.org/10.3390/e22010076 - 06 Jan 2020
Cited by 14 | Viewed by 2754
Abstract
In this paper, a new image encryption transmission algorithm based on the parallel mode is proposed. This algorithm aims to improve information transmission efficiency and security based on existing hardware conditions. To improve efficiency, this paper adopts the method of parallel compressed sensing [...] Read more.
In this paper, a new image encryption transmission algorithm based on the parallel mode is proposed. This algorithm aims to improve information transmission efficiency and security based on existing hardware conditions. To improve efficiency, this paper adopts the method of parallel compressed sensing to realize image transmission. Compressed sensing can perform data sampling and compression at a rate much lower than the Nyquist sampling rate. To enhance security, this algorithm combines a sequence signal generator with chaotic cryptography. The initial sensitivity of chaos, used in a measurement matrix, makes it possible to improve the security of an encryption algorithm. The cryptographic characteristics of chaotic signals can be fully utilized by the flexible digital logic circuit. Simulation experiments and analyses show that the algorithm achieves the goal of improving transmission efficiency and has the capacity to resist illegal attacks. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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19 pages, 5951 KiB  
Article
Low-Element Image Restoration Based on an Out-of-Order Elimination Algorithm
by Yaqin Xie, Jiayin Yu, Xinwu Chen, Qun Ding and Erfu Wang
Entropy 2019, 21(12), 1192; https://doi.org/10.3390/e21121192 - 04 Dec 2019
Cited by 3 | Viewed by 2171
Abstract
To reduce the consumption of receiving devices, a number of devices at the receiving end undergo low-element treatment (the number of devices at the receiving end is less than that at the transmitting ends). The underdetermined blind-source separation system is a classic low-element [...] Read more.
To reduce the consumption of receiving devices, a number of devices at the receiving end undergo low-element treatment (the number of devices at the receiving end is less than that at the transmitting ends). The underdetermined blind-source separation system is a classic low-element model at the receiving end. Blind signal extraction in an underdetermined system remains an ill-posed problem, as it is difficult to extract all the source signals. To realize fewer devices at the receiving end without information loss, this paper proposes an image restoration method for underdetermined blind-source separation based on an out-of-order elimination algorithm. Firstly, a chaotic system is used to perform hidden transmission of source signals, where the source signals can hardly be observed and confidentiality is guaranteed. Secondly, empirical mode decomposition is used to decompose and complement the missing observed signals, and the fast independent component analysis (FastICA) algorithm is used to obtain part of the source signals. Finally, all the source signals are successfully separated using the out-of-order elimination algorithm and the FastICA algorithm. The results show that the performance of the underdetermined blind separation algorithm is related to the configuration of the transceiver antenna. When the signal is 3 × 4 antenna configuration, the algorithm in this paper is superior to the comparison algorithm in signal recovery, and its separation performance is better for a lower degree of missing array elements. The end result is that the algorithms discussed in this paper can effectively and completely extract all the source signals. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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24 pages, 28547 KiB  
Article
A High Spectral Entropy (SE) Memristive Hidden Chaotic System with Multi-Type Quasi-Periodic and its Circuit
by Licai Liu, Chuanhong Du, Lixiu Liang and Xiefu Zhang
Entropy 2019, 21(10), 1026; https://doi.org/10.3390/e21101026 - 22 Oct 2019
Cited by 23 | Viewed by 3314
Abstract
As a new type of nonlinear electronic component, a memristor can be used in a chaotic system to increase the complexity of the system. In this paper, a flux-controlled memristor is applied to an existing chaotic system, and a novel five-dimensional chaotic system [...] Read more.
As a new type of nonlinear electronic component, a memristor can be used in a chaotic system to increase the complexity of the system. In this paper, a flux-controlled memristor is applied to an existing chaotic system, and a novel five-dimensional chaotic system with high complexity and hidden attractors is proposed. Analyzing the nonlinear characteristics of the system, we can find that the system has new chaotic attractors and many novel quasi-periodic limit cycles; the unique attractor structure of the Poincaré map also reflects the complexity and novelty of the hidden attractor for the system; the system has a very high complexity when measured through spectral entropy. In addition, under different initial conditions, the system exhibits the coexistence of chaotic attractors with different topologies, quasi-periodic limit cycles, and chaotic attractors. At the same time, an interesting transient chaos phenomenon, one kind of novel quasi-periodic, and weak chaotic hidden attractors are found. Finally, we realize the memristor model circuit and the proposed chaotic system use off-the-shelf electronic components. The experimental results of the circuit are consistent with the numerical simulation, which shows that the system is physically achievable and provides a new option for the application of memristive chaotic systems. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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14 pages, 6336 KiB  
Article
Entropy Analysis and Image Encryption Application Based on a New Chaotic System Crossing a Cylinder
by Alaa Kadhim Farhan, Nadia M.G. Al-Saidi, Abeer Tariq Maolood, Fahimeh Nazarimehr and Iqtadar Hussain
Entropy 2019, 21(10), 958; https://doi.org/10.3390/e21100958 - 30 Sep 2019
Cited by 42 | Viewed by 3742
Abstract
Designing chaotic systems with specific features is a hot topic in nonlinear dynamics. In this study, a novel chaotic system is presented with a unique feature of crossing inside and outside of a cylinder repeatedly. This new system is thoroughly analyzed by the [...] Read more.
Designing chaotic systems with specific features is a hot topic in nonlinear dynamics. In this study, a novel chaotic system is presented with a unique feature of crossing inside and outside of a cylinder repeatedly. This new system is thoroughly analyzed by the help of the bifurcation diagram, Lyapunov exponents’ spectrum, and entropy measurement. Bifurcation analysis of the proposed system with two initiation methods reveals its multistability. As an engineering application, the system’s efficiency is tested in image encryption. The complexity of the chaotic attractor of the proposed system makes it a proper choice for encryption. States of the chaotic attractor are used to shuffle the rows and columns of the image, and then the shuffled image is XORed with the states of chaotic attractor. The unpredictability of the chaotic attractor makes the encryption method very safe. The performance of the encryption method is analyzed using the histogram, correlation coefficient, Shannon entropy, and encryption quality. The results show that the encryption method using the proposed chaotic system has reliable performance. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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37 pages, 39282 KiB  
Article
New Nonlinear Active Element Dedicated to Modeling Chaotic Dynamics with Complex Polynomial Vector Fields
by Jiri Petrzela and Roman Sotner
Entropy 2019, 21(9), 871; https://doi.org/10.3390/e21090871 - 06 Sep 2019
Cited by 6 | Viewed by 3168
Abstract
This paper describes evolution of new active element that is able to significantly simplify the design process of lumped chaotic oscillator, especially if the concept of analog computer or state space description is adopted. The major advantage of the proposed active device lies [...] Read more.
This paper describes evolution of new active element that is able to significantly simplify the design process of lumped chaotic oscillator, especially if the concept of analog computer or state space description is adopted. The major advantage of the proposed active device lies in the incorporation of two fundamental mathematical operations into a single five-port voltage-input current-output element: namely, differentiation and multiplication. The developed active device is verified inside three different synthesis scenarios: circuitry realization of a third-order cyclically symmetrical vector field, hyperchaotic system based on the Lorenz equations and fourth- and fifth-order hyperjerk function. Mentioned cases represent complicated vector fields that cannot be implemented without the necessity of utilizing many active elements. The captured oscilloscope screenshots are compared with numerically integrated trajectories to demonstrate good agreement between theory and measurement. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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17 pages, 6736 KiB  
Article
Image Encryption Scheme with Compressed Sensing Based on New Three-Dimensional Chaotic System
by Yaqin Xie, Jiayin Yu, Shiyu Guo, Qun Ding and Erfu Wang
Entropy 2019, 21(9), 819; https://doi.org/10.3390/e21090819 - 22 Aug 2019
Cited by 47 | Viewed by 3824
Abstract
In this paper, a new three-dimensional chaotic system is proposed for image encryption. The core of the encryption algorithm is the combination of chaotic system and compressed sensing, which can complete image encryption and compression at the same time. The Lyapunov exponent, bifurcation [...] Read more.
In this paper, a new three-dimensional chaotic system is proposed for image encryption. The core of the encryption algorithm is the combination of chaotic system and compressed sensing, which can complete image encryption and compression at the same time. The Lyapunov exponent, bifurcation diagram and complexity of the new three-dimensional chaotic system are analyzed. The performance analysis shows that the chaotic system has two positive Lyapunov exponents and high complexity. In the encryption scheme, a new chaotic system is used as the measurement matrix for compressed sensing, and Arnold is used to scrambling the image further. The proposed method has better reconfiguration ability in the compressible range of the algorithm compared with other methods. The experimental results show that the proposed encryption scheme has good encryption effect and image compression capability. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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19 pages, 12122 KiB  
Article
Coexisting Attractors and Multistability in a Simple Memristive Wien-Bridge Chaotic Circuit
by Yixuan Song, Fang Yuan and Yuxia Li
Entropy 2019, 21(7), 678; https://doi.org/10.3390/e21070678 - 11 Jul 2019
Cited by 32 | Viewed by 3511
Abstract
In this paper, a new voltage-controlled memristor is presented. The mathematical expression of this memristor has an absolute value term, so it is called an absolute voltage-controlled memristor. The proposed memristor is locally active, which is proved by its DC V–I [...] Read more.
In this paper, a new voltage-controlled memristor is presented. The mathematical expression of this memristor has an absolute value term, so it is called an absolute voltage-controlled memristor. The proposed memristor is locally active, which is proved by its DC V–I (Voltage–Current) plot. A simple three-order Wien-bridge chaotic circuit without inductor is constructed on the basis of the presented memristor. The dynamical behaviors of the simple chaotic system are analyzed in this paper. The main properties of this system are coexisting attractors and multistability. Furthermore, an analog circuit of this chaotic system is realized by the Multisim software. The multistability of the proposed system can enlarge the key space in encryption, which makes the encryption effect better. Therefore, the proposed chaotic system can be used as a pseudo-random sequence generator to provide key sequences for digital encryption systems. Thus, the chaotic system is discretized and implemented by Digital Signal Processing (DSP) technology. The National Institute of Standards and Technology (NIST) test and Approximate Entropy analysis of the proposed chaotic system are conducted in this paper. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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13 pages, 1407 KiB  
Article
A Class of Quadratic Polynomial Chaotic Maps and Their Fixed Points Analysis
by Chuanfu Wang and Qun Ding
Entropy 2019, 21(7), 658; https://doi.org/10.3390/e21070658 - 04 Jul 2019
Cited by 12 | Viewed by 3020
Abstract
When chaotic systems are used in different practical applications, such as chaotic secure communication and chaotic pseudorandom sequence generators, a large number of chaotic systems are strongly required. However, for a lack of a systematic construction theory, the construction of chaotic systems mainly [...] Read more.
When chaotic systems are used in different practical applications, such as chaotic secure communication and chaotic pseudorandom sequence generators, a large number of chaotic systems are strongly required. However, for a lack of a systematic construction theory, the construction of chaotic systems mainly depends on the exhaustive search of systematic parameters or initial values, especially for a class of dynamical systems with hidden chaotic attractors. In this paper, a class of quadratic polynomial chaotic maps is studied, and a general method for constructing quadratic polynomial chaotic maps is proposed. The proposed polynomial chaotic maps satisfy the Li–Yorke definition of chaos. This method can accurately control the amplitude of chaotic time series. Through the existence and stability analysis of fixed points, we proved that such class quadratic polynomial maps cannot have hidden chaotic attractors. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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14 pages, 6818 KiB  
Article
A Giga-Stable Oscillator with Hidden and Self-Excited Attractors: A Megastable Oscillator Forced by His Twin
by Thoai Phu Vo, Yeganeh Shaverdi, Abdul Jalil M. Khalaf, Fawaz E. Alsaadi, Tasawar Hayat and Viet-Thanh Pham
Entropy 2019, 21(5), 535; https://doi.org/10.3390/e21050535 - 25 May 2019
Cited by 10 | Viewed by 3640
Abstract
In this paper, inspired by a newly proposed two-dimensional nonlinear oscillator with an infinite number of coexisting attractors, a modified nonlinear oscillator is proposed. The original system has an exciting feature of having layer–layer coexisting attractors. One of these attractors is self-excited while [...] Read more.
In this paper, inspired by a newly proposed two-dimensional nonlinear oscillator with an infinite number of coexisting attractors, a modified nonlinear oscillator is proposed. The original system has an exciting feature of having layer–layer coexisting attractors. One of these attractors is self-excited while the rest are hidden. By forcing this system with its twin, a new four-dimensional nonlinear system is obtained which has an infinite number of coexisting torus attractors, strange attractors, and limit cycle attractors. The entropy, energy, and homogeneity of attractors’ images and their basin of attractions are calculated and reported, which showed an increase in the complexity of attractors when changing the bifurcation parameters. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors II)
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