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

Complexity ◽  
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.

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

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
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.

scholarly journals A New Fractional-Order Chaotic Complex System and Its Antisynchronization

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Cuimei Jiang ◽  
Shutang Liu ◽  
Chao Luo
Keyword(s):  
Complex System ◽  
Fractional Order ◽  
Lorenz System ◽  
Chaotic Behavior ◽  
Phase Portraits ◽  
Chaotic Attractors ◽  
Fractional Order Systems ◽  
Chaotic Complex System ◽  
The Stability ◽  
Complex Lorenz System

We propose a new fractional-order chaotic complex system and study its dynamical properties including symmetry, equilibria and their stability, and chaotic attractors. Chaotic behavior is verified with phase portraits, bifurcation diagrams, the histories, and the largest Lyapunov exponents. And we find that chaos exists in this system with orders less than 5 by numerical simulation. Additionally, antisynchronization of different fractional-order chaotic complex systems is considered based on the stability theory of fractional-order systems. This new system and the fractional-order complex Lorenz system can achieve antisynchronization. Corresponding numerical simulations show the effectiveness and feasibility of the scheme.

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

Entropy ◽  
2018 ◽  
Vol 21 (1) ◽  
pp. 1 ◽  
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.

Dynamics analysis of fractional-order Hopfield neural networks

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.

scholarly journals Stability Analysis of  Fractional Order Memristor Synapse-coupled Hopfield Neural Network with Ring Structure

2021 ◽  
Author(s):  
Leila Eftekhari ◽  
Mohammad Amirian
Keyword(s):  
Neural Network ◽  
Fractional Order ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Hopfield Neural Network ◽  
Ring Structure ◽  
Embedded Memory ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Memristor Synapse

Abstract A memristor is a non-linear two-terminal electrical element that incorporates memory features and nanoscale properties, enabling us to design very high-density artificial neural networks. To examine the embedded memory property, we should use mathematical frameworks like fractional calculus, which is capable of doing so. Here, we first present a fractional-order memristor synapse-coupling Hopfield neural network on two neurons and then extend the model to a neural network with a ring structure that consists of $n$ sub-network neurons. Necessary and sufficient conditions for the stability of equilibrium points are investigated, highlighting the dependency of the stability on the fractional-order value and the number of neurons. Numerical simulations and bifurcation analysis, along with Lyapunov exponents, are given in the two-neuron case that substantiates the theoretical findings, suggesting possible routes towards chaos when the fractional order of the system increases. In the $n$-neuron case also, it is revealed that the stability depends on the structure and number of sub-networks.

Stability and Synchronization of a Fractional Neutral Higher-Order Neural Network System

2020 ◽  
Vol 21 (5) ◽  
pp. 443-458
Author(s):  
N.-E. Tatar
Keyword(s):  
Neural Network ◽  
Fractional Order ◽  
Neutral Type ◽  
Uniform Boundedness ◽  
Higher Order ◽  
Network System ◽  
Activation Functions ◽  
Network Systems ◽  
Neural Network System ◽  
The Stability

AbstractWe discuss the stability and synchronization of some neural network systems with more than one feature. They are of higher-order, of fractional order and also involving delays of neutral type. Each one of these features presents substantial difficulties to overcome. We prove stability and synchronization of Mittag-Leffler type. This rate is fairly reasonable in case of fractional order. This leads us to prove a neutral fractional version of the well-known Halanay inequality which is interesting by itself. Another feature of the present work is the treatment of unbounded activation functions. The condition of uniform boundedness of the activation functions was commonly used in the literature.

scholarly journals Dynamic Behavior Analysis and Robust Synchronization of a Novel Fractional-Order Chaotic System with Multiwing Attractors

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Chenhui Wang
Keyword(s):  
Fractional Order ◽  
Chaotic Systems ◽  
Chaotic Behavior ◽  
Chaotic Attractors ◽  
Stability Criteria ◽  
Minimum Order ◽  
3 Dimensional ◽  
Robust Synchronization ◽  
New Type ◽  
The Stability

To enrich the types of multiwing chaotic attractors in fractional-order chaotic systems (FOCSs), a new type of 3-dimensional FOCSs is designed in this study. The most important contribution of this FOCS consists in the coexistence of multiple multiwing chaotic attractors, including 2-wing, 3-wing, and 4-wing attractors. It is also indicated that the minimum order that the system can exhibit chaotic behavior is 0.84. Then, based on certain fractional stability criteria, a robust synchronization controller is derived for this kind of FOCSs with multiwing chaotic attractors and parametric uncertainties, and the stability of the synchronization error is proven strictly. Meanwhile, the theoretical analysis is tested by simulation results.

Effects of Low and High Neuron Activation Gradients on the Dynamics of a Simple 3D Hopfield Neural Network

2020 ◽  
Vol 30 (11) ◽  
pp. 2050159 ◽  
Author(s):  
Sami Doubla Isaac ◽  
Z. Tabekoueng Njitacke ◽  
J. Kengne
Keyword(s):  
Neural Network ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Power Spectra ◽  
Hopfield Neural Network ◽  
Basins Of Attraction ◽  
Dynamical Analysis ◽  
Phase Portraits ◽  
Neuron Activation ◽  
Wide Range

In this paper, the effects of low and fast response speeds of neuron activation gradient of a simple 3D Hopfield neural network are explored. It consists of analyzing the effects of low and high neuron activation gradients on the dynamics. By considering an imbalance of the neuron activation gradients, different electrical activities are induced in the network, which enable the occurrence of several nonlinear behaviors. The significant sensitivity of nontrivial equilibrium points to the activation gradients of the first and second neurons relative to that of the third neuron is reported. The dynamical analysis of the model is done in a wide range of the activation gradient of the second neuron. In this range, the model presents areas of periodic behavior, chaotic behavior and periodic window behavior through complex bifurcations. Interesting behaviors such as the coexistences of two, four, six and eight disconnected attractors, as well as the phenomenon of coexisting antimonotonicity, are reported. These singular results are obtained by using nonlinear dynamics analysis tools such as bifurcation diagrams and largest Lyapunov exponents, phase portraits, power spectra and basins of attraction. Finally, some analog results obtained from PSpice-based simulations further verify the numerical results.

scholarly journals Observer-based adaptive neural network backstepping sliding mode control for switched fractional order uncertain nonlinear systems with unmeasured states

2021 ◽  
pp. 002029402110211
Author(s):  
Tao Chen ◽  
Damin Cao ◽  
Jiaxin Yuan ◽  
Hui Yang
Keyword(s):  
Neural Network ◽  
Nonlinear Systems ◽  
Sliding Mode Control ◽  
Fractional Order ◽  
Sliding Mode ◽  
Tracking Error ◽  
Adaptive Neural Network ◽  
Dynamic Surface ◽  
Mode Control ◽  
The Stability

This paper proposes an observer-based adaptive neural network backstepping sliding mode controller to ensure the stability of switched fractional order strict-feedback nonlinear systems in the presence of arbitrary switchings and unmeasured states. To avoid “explosion of complexity” and obtain fractional derivatives for virtual control functions continuously, the fractional order dynamic surface control (DSC) technology is introduced into the controller. An observer is used for states estimation of the fractional order systems. The sliding mode control technology is introduced to enhance robustness. The unknown nonlinear functions and uncertain disturbances are approximated by the radial basis function neural networks (RBFNNs). The stability of system is ensured by the constructed Lyapunov functions. The fractional adaptive laws are proposed to update uncertain parameters. The proposed controller can ensure convergence of the tracking error and all the states remain bounded in the closed-loop systems. Lastly, the feasibility of the proposed control method is proved by giving two examples.

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