Entropy

Editorial

4 pages, 169 KiB

Open AccessEditorial

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

by Christos K. Volos, Sajad Jafari, Jacques Kengne, Jesus M. Munoz-Pacheco and Karthikeyan Rajagopal

Entropy 2019, 21(4), 370; https://doi.org/10.3390/e21040370 - 5 Apr 2019

Cited by 11 | Viewed by 3354

Abstract

In the last few years, entropy has been a fundamental and essential concept in information theory [...] Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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12 pages, 1659 KiB

Open AccessArticle

Bogdanov Map for Modelling a Phase-Conjugated Ring Resonator

by Vicente Aboites, David Liceaga, Rider Jaimes-Reátegui and Juan Hugo García-López

Entropy 2019, 21(4), 384; https://doi.org/10.3390/e21040384 - 10 Apr 2019

Cited by 1 | Viewed by 3050

Abstract

In this paper, we propose using paraxial matrix optics to describe a ring-phase conjugated resonator that includes an intracavity chaos-generating element; this allows the system to behave in phase space as a Bogdanov Map. Explicit expressions for the intracavity chaos-generating matrix elements were [...] Read more.

In this paper, we propose using paraxial matrix optics to describe a ring-phase conjugated resonator that includes an intracavity chaos-generating element; this allows the system to behave in phase space as a Bogdanov Map. Explicit expressions for the intracavity chaos-generating matrix elements were obtained. Furthermore, computer calculations for several parameter configurations were made; rich dynamic behavior among periodic orbits high periodicity and chaos were observed through bifurcation diagrams. These results confirm the direct dependence between the parameters present in the intracavity chaos-generating element. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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21 pages, 14391 KiB

Open AccessArticle

Dynamics and Entropy Analysis for a New 4-D Hyperchaotic System with Coexisting Hidden Attractors

by Licai Liu, Chuanhong Du, Xiefu Zhang, Jian Li and Shuaishuai Shi

Entropy 2019, 21(3), 287; https://doi.org/10.3390/e21030287 - 15 Mar 2019

Cited by 27 | Viewed by 3956

Abstract

This paper presents a new no-equilibrium 4-D hyperchaotic multistable system with coexisting hidden attractors. One prominent feature is that by varying the system parameter or initial value, the system can generate several nonlinear complex attractors: periodic, quasiperiodic, multiple topology chaotic, and hyperchaotic. The [...] Read more.

This paper presents a new no-equilibrium 4-D hyperchaotic multistable system with coexisting hidden attractors. One prominent feature is that by varying the system parameter or initial value, the system can generate several nonlinear complex attractors: periodic, quasiperiodic, multiple topology chaotic, and hyperchaotic. The dynamics and complexity of the proposed system were investigated through Lyapunov exponents (LEs), a bifurcation diagram, a Poincaré map, and spectral entropy (SE). The simulation and calculation results show that the proposed multistable system has very rich and complex hidden dynamic characteristics. Additionally, the circuit of the chaotic system is designed to verify the physical realizability of the system. This study provides new insights into uncovering the dynamic characteristics of the coexisting hidden attractors system and provides a new choice for nonlinear control or chaotic secure communication technology. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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10 pages, 2979 KiB

Open AccessArticle

Chaotic Map with No Fixed Points: Entropy, Implementation and Control

by Van Van Huynh, Adel Ouannas, Xiong Wang, Viet-Thanh Pham, Xuan Quynh Nguyen and Fawaz E. Alsaadi

Entropy 2019, 21(3), 279; https://doi.org/10.3390/e21030279 - 14 Mar 2019

Cited by 24 | Viewed by 4376

Abstract

A map without equilibrium has been proposed and studied in this paper. The proposed map has no fixed point and exhibits chaos. We have investigated its dynamics and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents’ [...] Read more.

A map without equilibrium has been proposed and studied in this paper. The proposed map has no fixed point and exhibits chaos. We have investigated its dynamics and shown its chaotic behavior using tools such as return map, bifurcation diagram and Lyapunov exponents’ diagram. Entropy of this new map has been calculated. Using an open micro-controller platform, the map is implemented, and experimental observation is presented. In addition, two control schemes have been proposed to stabilize and synchronize the chaotic map. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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12 pages, 2107 KiB

Open AccessArticle

Adaptive Synchronization of Fractional-Order Complex Chaotic system with Unknown Complex Parameters

by Ruoxun Zhang, Yongli Liu and Shiping Yang

Entropy 2019, 21(2), 207; https://doi.org/10.3390/e21020207 - 21 Feb 2019

Cited by 16 | Viewed by 3500

Abstract

This paper investigates the problem of synchronization of fractional-order complex-variable chaotic systems (FOCCS) with unknown complex parameters. Based on the complex-variable inequality and stability theory for fractional-order complex-valued system, a new scheme is presented for adaptive synchronization of FOCCS with unknown complex parameters. [...] Read more.

This paper investigates the problem of synchronization of fractional-order complex-variable chaotic systems (FOCCS) with unknown complex parameters. Based on the complex-variable inequality and stability theory for fractional-order complex-valued system, a new scheme is presented for adaptive synchronization of FOCCS with unknown complex parameters. The proposed scheme not only provides a new method to analyze fractional-order complex-valued system but also significantly reduces the complexity of computation and analysis. Theoretical proof and simulation results substantiate the effectiveness of the presented synchronization scheme. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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Figure 1
Dynamic behaviors of the fractional-order complex Lorenz-like System with commensurate order <math display="inline"><semantics> <mi>α</mi> </semantics></math> (<math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>10</mn> <mo>+</mo> <mi>i</mi> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo> </mo> <mi>c</mi> <mo>=</mo> <mn>16</mn> <mo>+</mo> <mn>0.3</mn> <mi>i</mi> </mrow> </semantics></math>). (a) maximal Lyapunov exponent; (b) bifurcation diagram. Full article ">Figure 2
Chaotic attractors of fractional-order complex Lorenz-like system with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>10</mn> <mo>+</mo> <mi>i</mi> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo> </mo> <mi>c</mi> <mo>=</mo> <mn>16</mn> <mo>+</mo> <mn>0.3</mn> <mi>i</mi> </mrow> </semantics></math> and commensurate order <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>. Full article ">Figure 3
Dynamic behaviors of the fractional-order complex Lorenz-like System with commensurate order 0.95 (<math display="inline"><semantics> <mrow> <msup> <mi>a</mi> <mi>r</mi> </msup> <mo>=</mo> <mn>10</mn> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo> </mo> <mi>c</mi> <mo>=</mo> <mn>16</mn> <mo>+</mo> <mn>0.3</mn> <mi>i</mi> </mrow> </semantics></math>). (a) maximal Lyapunov exponent; (b) bifurcation diagram. Full article ">Figure 4
The state trajectories of fractional-order complex Lorenz-like system with <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>10</mn> <mo>+</mo> <mi>i</mi> <mo>,</mo> <mo> </mo> <mi>b</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo> </mo> <mi>c</mi> <mo>=</mo> <mn>16</mn> <mo>+</mo> <mn>0.3</mn> <mi>i</mi> </mrow> </semantics></math> and commensurate order <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math>. Full article ">Figure 5
Synchronization errors e1, e2, e3 of fractional-order complex Lorenz-like chaotic system. Full article ">Figure 6
Estimated complex parameters of fractional-order complex Lorenz-like chaotic system. Full article ">

15 pages, 3574 KiB

Open AccessEditor’s ChoiceArticle

Entropy Analysis and Neural Network-Based Adaptive Control of a Non-Equilibrium Four-Dimensional Chaotic System with Hidden Attractors

by Hadi Jahanshahi, Maryam Shahriari-Kahkeshi, Raúl Alcaraz, Xiong Wang, Vijay P. Singh and Viet-Thanh Pham

Entropy 2019, 21(2), 156; https://doi.org/10.3390/e21020156 - 7 Feb 2019

Cited by 84 | Viewed by 4675

Abstract

Today, four-dimensional chaotic systems are attracting considerable attention because of their special characteristics. This paper presents a non-equilibrium four-dimensional chaotic system with hidden attractors and investigates its dynamical behavior using a bifurcation diagram, as well as three well-known entropy measures, such as approximate [...] Read more.

Today, four-dimensional chaotic systems are attracting considerable attention because of their special characteristics. This paper presents a non-equilibrium four-dimensional chaotic system with hidden attractors and investigates its dynamical behavior using a bifurcation diagram, as well as three well-known entropy measures, such as approximate entropy, sample entropy, and Fuzzy entropy. In order to stabilize the proposed chaotic system, an adaptive radial-basis function neural network (RBF-NN)–based control method is proposed to represent the model of the uncertain nonlinear dynamics of the system. The Lyapunov direct method-based stability analysis of the proposed approach guarantees that all of the closed-loop signals are semi-globally uniformly ultimately bounded. Also, adaptive learning laws are proposed to tune the weight coefficients of the RBF-NN. The proposed adaptive control approach requires neither the prior information about the uncertain dynamics nor the parameters value of the considered system. Results of simulation validate the performance of the proposed control method. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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18 pages, 4267 KiB

Open AccessArticle

Dynamics and Complexity of a New 4D Chaotic Laser System

by Hayder Natiq, Mohamad Rushdan Md Said, Nadia M. G. Al-Saidi and Adem Kilicman

Entropy 2019, 21(1), 34; https://doi.org/10.3390/e21010034 - 7 Jan 2019

Cited by 51 | Viewed by 4702

Abstract

Derived from Lorenz-Haken equations, this paper presents a new 4D chaotic laser system with three equilibria and only two quadratic nonlinearities. Dynamics analysis, including stability of symmetric equilibria and the existence of coexisting multiple Hopf bifurcations on these equilibria, are investigated, and the [...] Read more.

Derived from Lorenz-Haken equations, this paper presents a new 4D chaotic laser system with three equilibria and only two quadratic nonlinearities. Dynamics analysis, including stability of symmetric equilibria and the existence of coexisting multiple Hopf bifurcations on these equilibria, are investigated, and the complex coexisting behaviors of two and three attractors of stable point and chaotic are numerically revealed. Moreover, a conducted research on the complexity of the laser system reveals that the complexity of the system time series can locate and determine the parameters and initial values that show coexisting attractors. To investigate how much a chaotic system with multistability behavior is suitable for cryptographic applications, we generate a pseudo-random number generator (PRNG) based on the complexity results of the laser system. The randomness test results show that the generated PRNG from the multistability regions fail to pass most of the statistical tests. Full article

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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16 pages, 2843 KiB

Open AccessArticle

The Complexity and Entropy Analysis for Service Game Model Based on Different Expectations and Optimal Pricing

by Yimin Huang, Xingli Chen, Qiuxiang Li and Xiaogang Ma

Entropy 2018, 20(11), 858; https://doi.org/10.3390/e20110858 - 8 Nov 2018

Cited by 9 | Viewed by 4451

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 [...] Read more.

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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13 pages, 408 KiB

Open AccessArticle

The Co-existence of Different Synchronization Types in Fractional-order Discrete-time Chaotic Systems with Non–identical Dimensions and Orders

by Samir Bendoukha, Adel Ouannas, Xiong Wang, Amina-Aicha Khennaoui, Viet-Thanh Pham, Giuseppe Grassi and Van Van Huynh

Entropy 2018, 20(9), 710; https://doi.org/10.3390/e20090710 - 14 Sep 2018

Cited by 23 | Viewed by 3837

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 [...] Read more.

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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Figure 1
Phase space plot for the fractional Hénon map with <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <msub> <mi>b</mi> <mn>1</mn> </msub> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>1.4</mn> <mo>,</mo> <mspace width="4.pt"/> <mn>0.3</mn> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>υ</mi> <mo>=</mo> <mn>0.984</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <msub> <mi>x</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0</mn> <mo>,</mo> <mn>0</mn> </mfenced> </mrow> </semantics></math>. Full article ">Figure 2
Phase portraits for the fractional generalized Hénon map with <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>,</mo> <mspace width="4.pt"/> <msub> <mi>b</mi> <mn>2</mn> </msub> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0.99</mn> <mo>,</mo> <mspace width="4.pt"/> <mn>0.2</mn> </mfenced> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>υ</mi> <mo>=</mo> <mn>0.984</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mfenced separators="" open="(" close=")"> <msub> <mi>y</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>y</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>y</mi> <mn>3</mn> </msub> <mrow> <mo>(</mo> <mn>0</mn> <mo>)</mo> </mrow> </mfenced> <mo>=</mo> <mfenced separators="" open="(" close=")"> <mn>0.1</mn> <mo>,</mo> <mn>0.2</mn> <mo>,</mo> <mn>0.5</mn> </mfenced> </mrow> </semantics></math>. Full article ">Figure 3
The evolution of errors over time for Example 1. Full article ">Figure 4
The evolution of errors over time for Example 2. Full article ">

23 pages, 11259 KiB

Open AccessArticle

Strange Attractors Generated by Multiple-Valued Static Memory Cell with Polynomial Approximation of Resonant Tunneling Diodes

by Jiri Petrzela

Entropy 2018, 20(9), 697; https://doi.org/10.3390/e20090697 - 12 Sep 2018

Cited by 22 | Viewed by 4986

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 [...] Read more.

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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10 pages, 434 KiB

Open AccessArticle

Optimization of Thurston’s Core Entropy Algorithm for Polynomials with a Critical Point of Maximal Order

by Gamaliel Blé and Domingo González

Entropy 2018, 20(9), 695; https://doi.org/10.3390/e20090695 - 11 Sep 2018

Cited by 1 | Viewed by 3225

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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12 pages, 3930 KiB

Open AccessArticle

A New Chaotic System with Stable Equilibrium: Entropy Analysis, Parameter Estimation, and Circuit Design

by Tomasz Kapitaniak, S. Alireza Mohammadi, Saad Mekhilef, Fawaz E. Alsaadi, Tasawar Hayat and Viet-Thanh Pham

Entropy 2018, 20(9), 670; https://doi.org/10.3390/e20090670 - 5 Sep 2018

Cited by 30 | Viewed by 4670

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 [...] Read more.

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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18 pages, 12752 KiB

Open AccessArticle

A New Fractional-Order Chaotic System with Different Families of Hidden and Self-Excited Attractors

by Jesus M. Munoz-Pacheco, Ernesto Zambrano-Serrano, Christos Volos, Sajad Jafari, Jacques Kengne and Karthikeyan Rajagopal

Entropy 2018, 20(8), 564; https://doi.org/10.3390/e20080564 - 28 Jul 2018

Cited by 74 | Viewed by 5541

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 [...] Read more.

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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14 pages, 9380 KiB

Open AccessArticle

Multivariate Multiscale Complexity Analysis of Self-Reproducing Chaotic Systems

by Shaobo He, Chunbiao Li, Kehui Sun and Sajad Jafari

Entropy 2018, 20(8), 556; https://doi.org/10.3390/e20080556 - 27 Jul 2018

Cited by 46 | Viewed by 3705

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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17 pages, 347 KiB

Open AccessArticle

Stochastic Entropy Solutions for Stochastic Nonlinear Transport Equations

by Rongrong Tian and Yanbin Tang

Entropy 2018, 20(6), 395; https://doi.org/10.3390/e20060395 - 23 May 2018

Cited by 6 | Viewed by 2623

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 [...] Read more.

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

10 pages, 5288 KiB

Open AccessArticle

A New Two-Dimensional Map with Hidden Attractors

by Chuanfu Wang and Qun Ding

Entropy 2018, 20(5), 322; https://doi.org/10.3390/e20050322 - 27 Apr 2018

Cited by 43 | Viewed by 4989

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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14 pages, 4904 KiB

Open AccessArticle

A Novel Algorithm to Improve Digital Chaotic Sequence Complexity through CCEMD and PE

by Chunlei Fan, Zhigang Xie and Qun Ding

Entropy 2018, 20(4), 295; https://doi.org/10.3390/e20040295 - 18 Apr 2018

Cited by 7 | Viewed by 4196

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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23 pages, 8699 KiB

Open AccessArticle

A New Chaotic System with a Self-Excited Attractor: Entropy Measurement, Signal Encryption, and Parameter Estimation

by Guanghui Xu, Yasser Shekofteh, Akif Akgül, Chunbiao Li and Shirin Panahi

Entropy 2018, 20(2), 86; https://doi.org/10.3390/e20020086 - 27 Jan 2018

Cited by 83 | Viewed by 7312

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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5274 KiB

Open AccessArticle

A New Chaotic System with Multiple Attractors: Dynamic Analysis, Circuit Realization and S-Box Design

by Qiang Lai, Akif Akgul, Chunbiao Li, Guanghui Xu and Ünal Çavuşoğlu

Entropy 2018, 20(1), 12; https://doi.org/10.3390/e20010012 - 27 Dec 2017

Cited by 96 | Viewed by 7474

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

(This article belongs to the Special Issue Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors)

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

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