Extreme Multistability of a Fractional-Order Discrete-Time Neural Network

At present, the extreme multistability of fractional order neural networks are gaining much interest from researchers. In this paper, by utilizing the fractional ℑ-Caputo operator, a simple fractional order discrete-time neural network with three neurons is introduced. The dynamic of this model are experimentally investigated via the maximum Lyapunov exponent, phase portraits, and bifurcation diagrams. Numerical simulation demonstrates that the new network has various types of coexisting attractors. Moreover, it is of note that the interesting phenomena of extreme multistability is discovered, i.e., the coexistence of symmetric multiple attractors.

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Dynamics Analysis of a New Fractional-Order Hopfield Neural Network with Delay and Its Generalized Projective Synchronization

Entropy ◽

10.3390/e21010001 ◽

2018 ◽

Vol 21 (1) ◽

pp. 1 ◽

Cited By ~ 4

Author(s):

Han-Ping Hu ◽

Jia-Kun Wang ◽

Fei-Long Xie

Keyword(s):

Neural Network ◽

Fractional Order ◽

Complex Dynamics ◽

Three Dimensional ◽

Hopfield Neural Network ◽

Projective Synchronization ◽

Phase Portraits ◽

Largest Lyapunov Exponent ◽

Bifurcation Diagrams ◽

Intermittent Chaos

In this paper, a new three-dimensional fractional-order Hopfield-type neural network with delay is proposed. The system has a unique equilibrium point at the origin, which is a saddle point with index two, hence unstable. Intermittent chaos is found in this system. The complex dynamics are analyzed both theoretically and numerically, including intermittent chaos, periodicity, and stability. Those phenomena are confirmed by phase portraits, bifurcation diagrams, and the Largest Lyapunov exponent. Furthermore, a synchronization method based on the state observer is proposed to synchronize a class of time-delayed fractional-order Hopfield-type neural networks.

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Nonlinear Dynamics and Chaos in Fractional-Order Hopfield Neural Networks with Delay

Advances in Mathematical Physics ◽

10.1155/2013/657245 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9 ◽

Cited By ~ 8

Author(s):

Xia Huang ◽

Zhen Wang ◽

Yuxia Li

Keyword(s):

Neural Network ◽

Neural Networks ◽

Fractional Order ◽

Hopfield Neural Network ◽

Hopfield Neural Networks ◽

Phase Portraits ◽

Chaotic Attractors ◽

Chaotic Motions ◽

Dynamical Behaviors ◽

Nonlinear Dynamics And Chaos

A fractional-order two-neuron Hopfield neural network with delay is proposed based on the classic well-known Hopfield neural networks, and further, the complex dynamical behaviors of such a network are investigated. A great variety of interesting dynamical phenomena, including single-periodic, multiple-periodic, and chaotic motions, are found to exist. The existence of chaotic attractors is verified by the bifurcation diagram and phase portraits as well.

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Various Types of Coexisting Attractors in a New 4D Autonomous Chaotic System

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127417501425 ◽

2017 ◽

Vol 27 (09) ◽

pp. 1750142 ◽

Cited By ~ 48

Author(s):

Qiang Lai ◽

Akif Akgul ◽

Xiao-Wen Zhao ◽

Huiqin Pei

Keyword(s):

Numerical Simulation ◽

Dynamic Behavior ◽

Chaotic System ◽

Limit Cycles ◽

Electronic Circuit ◽

Strange Attractors ◽

Bifurcation Diagrams ◽

Coexisting Attractors ◽

Signum Function ◽

Multiple Coexisting Attractors

An unique 4D autonomous chaotic system with signum function term is proposed in this paper. The system has four unstable equilibria and various types of coexisting attractors appear. Four-wing and four-scroll strange attractors are observed in the system and they will be broken into two coexisting butterfly attractors and two coexisting double-scroll attractors with the variation of the parameters. Numerical simulation shows that the system has various types of multiple coexisting attractors including two butterfly attractors with four limit cycles, two double-scroll attractors with a limit cycle, four single-scroll strange attractors, four limit cycles with regard to different parameters and initial values. The coexistence of the attractors is determined by the bifurcation diagrams. The chaotic and hyperchaotic properties of the attractors are verified by the Lyapunov exponents. Moreover, we present an electronic circuit to experimentally realize the dynamic behavior of the system.

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Dynamics analysis of fractional-order Hopfield neural networks

International Journal of Biomathematics ◽

10.1142/s1793524520500837 ◽

2020 ◽

pp. 2050083

Author(s):

Iqbal M. Batiha ◽

Ramzi B. Albadarneh ◽

Shaher Momani ◽

Iqbal H. Jebril

Keyword(s):

Neural Network ◽

Lyapunov Exponents ◽

Fractional Order ◽

Hopfield Neural Network ◽

Phase Portraits ◽

Dynamics Analysis ◽

Fractional Order Systems ◽

Runge Kutta Method ◽

Intermittent Chaos ◽

Predictor Corrector

This paper proposes fractional-order systems for Hopfield Neural Network (HNN). The so-called Predictor–Corrector Adams–Bashforth–Moulton Method (PCABMM) has been implemented for solving such systems. Graphical comparisons between the PCABMM and the Runge–Kutta Method (RKM) solutions for the classical HNN reveal that the proposed technique is one of the powerful tools for handling these systems. To determine all Lyapunov exponents for them, the Benettin–Wolf algorithm has been involved in the PCABMM. Based on such algorithm, the Lyapunov exponents as a function of a given parameter and as another function of the fractional-order have been described, the intermittent chaos for these systems has been explored. A new result related to the Mittag–Leffler stability of some nonlinear Fractional-order Hopfield Neural Network (FoHNN) systems has been shown. Besides, the description and the dynamic analysis of those phenomena have been discussed and verified theoretically and numerically via illustrating the phase portraits and the Lyapunov exponents’ diagrams.

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Computational Analysis and Bifurcation of Regular and Chaotic Ca2+ Oscillations

Mathematics ◽

10.3390/math9243324 ◽

2021 ◽

Vol 9 (24) ◽

pp. 3324

Author(s):

Xinxin Qie ◽

Quanbao Ji

Keyword(s):

Numerical Simulation ◽

Center Manifold ◽

Computational Analysis ◽

System Model ◽

Phase Portraits ◽

Bifurcation Parameter ◽

Bifurcation Diagrams ◽

Saddle Node Bifurcation ◽

The Stability ◽

Torus Bifurcation

This study investigated the stability and bifurcation of a nonlinear system model developed by Marhl et al. based on the total Ca2+ concentration among three different Ca2+ stores. In this study, qualitative theories of center manifold and bifurcation were used to analyze the stability of equilibria. The bifurcation parameter drove the system to undergo two supercritical bifurcations. It was hypothesized that the appearance and disappearance of Ca2+ oscillations are driven by them. At the same time, saddle-node bifurcation and torus bifurcation were also found in the process of exploring bifurcation. Finally, numerical simulation was carried out to determine the validity of the proposed approach by drawing bifurcation diagrams, time series, phase portraits, etc.

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HYPERCHAOTIC DYNAMICS OF A NEW FRACTIONAL DISCRETE-TIME SYSTEM

Fractals ◽

10.1142/s0218348x2140034x ◽

2021 ◽

pp. 2140034

Author(s):

AMINA-AICHA KHENNAOUI ◽

ADEL OUANNAS ◽

SHAHER MOMANI ◽

ZOHIR DIBI ◽

GIUSEPPE GRASSI ◽

...

Keyword(s):

Discrete Time ◽

Three Dimensional ◽

Phase Portraits ◽

Bifurcation Diagrams ◽

Discrete Time System ◽

Time System ◽

Discrete Time Systems ◽

Simulation Results ◽

Time Systems ◽

First Time

In recent years, some efforts have been devoted to nonlinear dynamics of fractional discrete-time systems. A number of papers have so far discussed results related to the presence of chaos in fractional maps. However, less results have been published to date regarding the presence of hyperchaos in fractional discrete-time systems. This paper aims to bridge the gap by introducing a new three-dimensional fractional map that shows, for the first time, complex hyperchaotic behaviors. A detailed analysis of the map dynamics is conducted via computation of Lyapunov exponents, bifurcation diagrams, phase portraits, approximated entropy and [Formula: see text] complexity. Simulation results confirm the effectiveness of the approach illustrated herein.

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Dynamics Analysis and Control of a Five-Term Fractional-Order System

Discrete Dynamics in Nature and Society ◽

10.1155/2018/8284121 ◽

2018 ◽

Vol 2018 ◽

pp. 1-10

Author(s):

Li-xin Yang ◽

Xiao-jun Liu

Keyword(s):

Fractional Order ◽

Equilibrium Points ◽

Fractional Order System ◽

Phase Portraits ◽

Order System ◽

Bifurcation Diagrams ◽

Compound Structure ◽

Dynamical Properties ◽

The Stability ◽

And Control

This paper proposes a new fractional-order chaotic system with five terms. Firstly, basic dynamical properties of the fractional-order system are investigated in terms of the stability of equilibrium points, Jacobian matrices theoretically. Furthermore, rich dynamics with interesting characteristics are demonstrated by phase portraits, bifurcation diagrams numerically. Besides, the control problem of the new fractional-order system is discussed via numerical simulations. Our results demonstrate that the new fractional-order system has compound structure.

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Chaotic Behavior Analysis of a New Incommensurate Fractional-Order Hopfield Neural Network System

Complexity ◽

10.1155/2021/3394666 ◽

2021 ◽

Vol 2021 ◽

pp. 1-11

Author(s):

Nadjette Debbouche ◽

Adel Ouannas ◽

Iqbal M. Batiha ◽

Giuseppe Grassi ◽

Mohammed K. A. Kaabar ◽

...

Keyword(s):

Neural Network ◽

Fractional Order ◽

Complex Dynamics ◽

Chaotic Behavior ◽

Hopfield Neural Network ◽

Phase Portraits ◽

Chaotic Attractors ◽

Neural Network System ◽

The Stability ◽

Arbitrary Location

This study intends to examine different dynamics of the chaotic incommensurate fractional-order Hopfield neural network model. The stability of the proposed incommensurate-order model is analyzed numerically by continuously varying the values of the fractional-order derivative and the values of the system parameters. It turned out that the formulated system using the Caputo differential operator exhibits many rich complex dynamics, including symmetry, bistability, and coexisting chaotic attractors. On the other hand, it has been detected that by adapting the corresponding controlled constants, such systems possess the so-called offset boosting of three variables. Besides, the resultant periodic and chaotic attractors can be scattered in several forms, including 1D line, 2D lattice, and 3D grid, and even in an arbitrary location of the phase space. Several numerical simulations are implemented, and the obtained findings are illustrated through constructing bifurcation diagrams, computing Lyapunov exponents, calculating Lyapunov dimensions, and sketching the phase portraits in 2D and 3D projections.

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Identification and Machine Learning Prediction of Nonlinear Behavior in a Robotic Arm System

Symmetry ◽

10.3390/sym13081445 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1445

Author(s):

Cheng-Chi Wang ◽

Yong-Quan Zhu

Keyword(s):

Neural Network ◽

Lyapunov Exponent ◽

Chaotic Motion ◽

Chaotic Behavior ◽

Phase Portraits ◽

Robotic Arm ◽

Maximum Lyapunov Exponent ◽

Double Pendulum ◽

Chaotic Motions ◽

Robotic Arms

In this study, the subject of investigation was the dynamic double pendulum crank mechanism used in a robotic arm. The arm is driven by a DC motor though the crank system and connected to a fixed side with a mount that includes a single spring and damping. Robotic arms are now widely used in industry, and the requirements for accuracy are stringent. There are many factors that can cause the induction of nonlinear or asymmetric behavior and even excite chaotic motion. In this study, bifurcation diagrams were used to analyze the dynamic response, including stable symmetric orbits and periodic and chaotic motions of the system under different damping and stiffness parameters. Behavior under different parameters was analyzed and verified by phase portraits, the maximum Lyapunov exponent, and Poincaré mapping. Firstly, to distinguish instability in the system, phase portraits and Poincaré maps were used for the identification of individual images, and the maximum Lyapunov exponents were used for prediction. GoogLeNet and ResNet-50 were used for image identification, and the results were compared using a convolutional neural network (CNN). This widens the convolutional layer and expands pooling to reduce network training time and thickening of the image; this deepens the network and strengthens performance. Secondly, the maximum Lyapunov exponent was used as the key index for the indication of chaos. Gaussian process regression (GPR) and the back propagation neural network (BPNN) were used with different amounts of data to quickly predict the maximum Lyapunov exponent under different parameters. The main finding of this study was that chaotic behavior occurs in the robotic arm system and can be more efficiently identified by ResNet-50 than by GoogLeNet; this was especially true for Poincaré map diagnosis. The results of GPR and BPNN model training on the three types of data show that GPR had a smaller error value, and the GPR-21 × 21 model was similar to the BPNN-51 × 51 model in terms of error and determination coefficient, showing that GPR prediction was better than that of BPNN. The results of this study allow the formation of a highly accurate prediction and identification model system for nonlinear and chaotic motion in robotic arms.

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Fractional-Order Discrete-Time SIR Epidemic Model with Vaccination: Chaos and Complexity

Mathematics ◽

10.3390/math10020165 ◽

2022 ◽

Vol 10 (2) ◽

pp. 165

Author(s):

Zai-Yin He ◽

Abderrahmane Abbes ◽

Hadi Jahanshahi ◽

Naif D. Alotaibi ◽

Ye Wang

Keyword(s):

Fractional Order ◽

Discrete Time ◽

Epidemic Model ◽

Chaotic Behavior ◽

Dynamical Behavior ◽

Maximum Lyapunov Exponent ◽

Sir Epidemic Model ◽

Sir Epidemic ◽

Reasonable Range ◽

Fractional Orders

This research presents a new fractional-order discrete-time susceptible-infected-recovered (SIR) epidemic model with vaccination. The dynamical behavior of the suggested model is examined analytically and numerically. Through using phase attractors, bifurcation diagrams, maximum Lyapunov exponent and the 0−1 test, it is verified that the newly introduced fractional discrete SIR epidemic model vaccination with both commensurate and incommensurate fractional orders has chaotic behavior. The discrete fractional model gives more complex dynamics for incommensurate fractional orders compared to commensurate fractional orders. The reasonable range of commensurate fractional orders is between γ = 0.8712 and γ = 1, while the reasonable range of incommensurate fractional orders is between γ2 = 0.77 and γ2 = 1. Furthermore, the complexity analysis is performed using approximate entropy (ApEn) and C0 complexity to confirm the existence of chaos. Finally, simulations were carried out on MATLAB to verify the efficacy of the given findings.

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