Fractional-order Hopfield Neural Networks Research Articles

Asymptotic stability of fractional-order Hopfield neural networks with event-triggered delayed impulses and switching effects

This paper examines the asymptotic stability of nonlinear fractional-order switched systems (FOSSs) under a mode-dependent event-triggered delayed impulsive mechanism (MDETDIM). The impulses and switched signals are asynchronous. A novel MDETDIM is proposed to determine the impulsive sequence, which can prevent the Zeno phenomenon. Lyapunov-based asymptotic stability conditions for general FOSSs are derived using the proposed MDETDIM. The theoretical results are then applied to a fractional-order Hopfield neural network (FOHNN) with event-based delayed impulses and switching effects. Two examples are provided to demonstrate the effectiveness of our proposed results.

Fractional-order-induced symmetric multi-scroll chaotic attractors and double bubble bifurcations in a memristive coupled Hopfield neural networks

The multi-state memristor is a type of memristor capable of memorizing multiple conductance states. This special feature is successfully used to mimic the behavior of neural synapses and simulate the influences of electromagnetic induction and radiation in artificial neural networks such as Hopfield neural networks. This paper introduces and examines a novel fractional-order Hopfield neural network model consisting of two non-identical sub-Hopfield neural networks coupled by a multi-stable flux controlled memristor (MCFHNN). The pinched hysteresis loops of the multi-stable flux controlled memristor are analyzed via numerical simulations. The equilibrium points of MCFHNN model and their stability are investigated. The introduced MCFHNN model is solved by using Adomian Decomposition Method (ADM). By using some examination methods such as spectral entropy complexity, bifurcation diagrams, phase portraits, maximum Lyapunov exponent and basins of attraction, we analyze the dynamic behaviors of MCFHNN model associated with coupling strength, fractional-order and initial states. Numerical results demonstrate that the proposed MCFHNN model is capable to develop abundant and complicated dynamics including, chaos, multiple transient transition behaviors, double bubble bifurcations, initial-boosted behavior and symmetric multi-scroll chaotic attractors. It is found that the complexity results are in agreement with those obtained from the bifurcation diagrams. This proves that the complexity can well reflect the dynamic behaviors of MCFHNN model. In order to support theoretical and numerical results, the designed MCFHNN model is implemented with the help of an Arduino-Due microcontroller board based on the Atmel SAM3X8E ARM Cortex-M3 processor. The results are in very good agreement with those obtained from computational simulations.

Multi-scroll and coexisting attractors in a Hopfield neural network under electromagnetic induction and external stimuli

The application of external stimuli to biological neurons is a valuable tool for investigating neuronal properties, understanding neural circuitry, and developing therapeutic interventions for neurological disorders. In this article, we propose a discrete fractional Hopfield neural network model consisting of four neurons to explore the influence of external stimuli in the presence of electromagnetic induction and radiation. To incorporate the electromagnetic induction between connected neurons, we construct and employ a discrete fractional sine memristor. Additionally, we introduce a multi-level pulse function to the sine memristor element to examine the chaotic dynamics of the neural network model. The qualitative behavior of the network model is demonstrated through stability analysis and bifurcation diagrams showcasing chaos. The study also focuses on understanding the coexisting behavior of the neural network model in the presence and absence of external stimuli. Moreover, we investigate the generation of multi-scroll attractors by varying the level of the pulse function, which is introduced to electromagnetic induction. Numerical simulations reveal that increasing the level of the multi-pulse function doubles the number of scrolls in the attractors when external stimuli are present. The findings presented in this article contribute to our understanding of discrete fractional memristors and shed light on the dynamical behavior of neurons and their electrical activity in the brain. Innovation within the discrete fractional-order Hopfield neural networks realm entails the creation and utilization of fresh ideas, methodologies, and strategies that harness fractional-order dynamics to confront diverse hurdles and enhance the effectiveness of Hopfield networks. Discrete fractional-order Hopfield neural networks have the capacity to propel an array of applications forward, spanning artificial intelligence, machine learning, control systems, and optimization, showcasing their potential for substantial progress in various domains.

Synchronization of Fractional-Order Delayed Neural Networks Using Dynamic-Free Adaptive Sliding Mode Control

In this work, a dynamic-free adaptive sliding mode control (adaptive-SMC) methodology for the synchronization of a specific class of chaotic delayed fractional-order neural network systems in the presence of input saturation is proposed. By incorporating the frequency distributed model (FDM) and the fractional version of the Lyapunov stability theory, a dynamic-free adaptive SMC methodology is designed to effectively overcome the inherent chaotic behavior exhibited by the delayed FONNSs to achieve synchronization. Notably, the decoupling of the control laws from the nonlinear/linear dynamical components of the system is ensured, taking advantage of the norm-boundedness property of the states in chaotic systems. The effectiveness of the suggested adaptive-SMC method for chaos synchronization in delayed fractional-order Hopfield neural network systems is validated through numerical simulations, demonstrating its robustness and efficiency. The proposed dynamic-free adaptive-SMC approach, incorporating the FDM and fractional Lyapunov stability theorem, offers a promising solution for synchronizing chaotic delayed FONNSs with input saturation, with potential applications in various domains requiring synchronization of such systems.

ASB-CS: Adaptive sparse basis compressive sensing model and its application to medical image encryption

Recent advances in intelligent wearable devices have brought tremendous chances for the development of healthcare monitoring system. However, the data collected by various sensors in it are user-privacy-related information. Once the individuals’ privacy is subjected to attacks, it can potentially cause serious hazards. For this reason, a feasible solution built upon the compression-encryption architecture is proposed. In this scheme, we design an Adaptive Sparse Basis Compressive Sensing (ASB-CS) model by leveraging Singular Value Decomposition (SVD) manipulation, while performing a rigorous proof of its effectiveness. Additionally, incorporating the Parametric Deformed Exponential Rectified Linear Unit (PDE-ReLU) memristor, a new fractional-order Hopfield neural network model is introduced as a pseudo-random number generator for the proposed cryptosystem, which has demonstrated superior properties in many aspects, such as hyperchaotic dynamics and multistability. To be specific, a plain medical image is subjected to the ASB-CS model and bidirectional diffusion manipulation under the guidance of the key-controlled cipher flows to yield the corresponding cipher image without visual semantic features. Ultimately, the simulation results and analysis demonstrate that the proposed scheme is capable of withstanding multiple security attacks and possesses balanced performance in terms of compressibility and robustness.

Dynamical Analysis of the Incommensurate Fractional-Order Hopfield Neural Network System and Its Digital Circuit Realization

Dynamical analysis of the incommensurate fractional-order neural network is a novel topic in the field of chaos research. This article investigates a Hopfield neural network (HNN) system in view of incommensurate fractional orders. Using the Adomian decomposition method (ADM) algorithm, the solution of the incommensurate fractional-order Hopfield neural network (FOHNN) system is solved. The equilibrium point of the system is discussed, and the dissipative characteristics are verified and discussed. By varying the order values of the proposed system, different dynamical behaviors of the incommensurate FOHNN system are explored and discussed via bifurcation diagrams, the Lyapunov exponent spectrum, complexity, etc. Finally, using the DSP platform to implement the system, the results are in good agreement with those of the simulation. The actual results indicate that the system shows many complex and interesting phenomena, such as attractor coexistence and an inversion property, with dynamic changes of the order of q0, q1, and q2. These phenomena provide important insights for simulating complex neural system states in pathological conditions and provide the theoretical basis for the later study of incommensurate fractional-order neural network systems.

CHAOTIC IMAGE ENCRYPTION WITH HOPFIELD NEURAL NETWORK

With modern cryptography evolves, some sensitive information needs to be protected with secure and efficient algorithms. In this context, we found that Hopfield neural network (HNN) has stronger memory and can generate luxuriant kinetic behavior, especially with the introduction of fractional-order operators. Therefore, we propose a chaotic image encryption based on the fractional-order HNN (FO-HNN), where FO-HNN appears as a key generator. To de-correlate the correlation between pixels, a spatial permutation strategy is designed first, and then a new diffusion technique based on a Three-input logic valve is adopted to guide the diffusion process. Simulation results and security analysis show that the HNN-based image cryptosystem has superior security performance.

Multistability of State-Dependent Switched Fractional-Order Hopfield Neural Networks With Mexican-Hat Activation Function and Its Application in Associative Memories.

The multistability and its application in associative memories are investigated in this article for state-dependent switched fractional-order Hopfield neural networks (FOHNNs) with Mexican-hat activation function (AF). Based on the Brouwer's fixed point theorem, the contraction mapping principle and the theory of fractional-order differential equations, some sufficient conditions are established to ensure the existence, exact existence and local stability of multiple equilibrium points (EPs) in the sense of Filippov, in which the positively invariant sets are also estimated. In particular, the analysis concerning the existence and stability of EPs is quite different from those in the literature because the considered system involves both fractional-order derivative and state-dependent switching. It should be pointed out that, compared with the results in the literature, the total number of EPs and stable EPs increases from 5l1 3l2 and 3l1 2l2 to 7l1 5l2 and 4l1 3l2 , respectively, where 0 ≤ l1 + l2 ≤ n with n being the system dimension. Besides, a new method is designed to realize associative memories for grayscale and color images by introducing a deviation vector, which, in comparison with the existing works, not only improves the utilization efficiency of EPs, but also reduces the system dimension and computational burden. Finally, the effectiveness of the theoretical results is illustrated by four numerical simulations.

Study on the Complex Dynamical Behavior of the Fractional-Order Hopfield Neural Network System and Its Implementation

The complex dynamics analysis of fractional-order neural networks is a cutting-edge topic in the field of neural network research. In this paper, a fractional-order Hopfield neural network (FOHNN) system is proposed, which contains four neurons. Using the Adomian decomposition method, the FOHNN system is solved. The dissipative characteristics of the system are discussed, as well as the equilibrium point is resolved. The characteristics of the dynamics through the phase diagram, the bifurcation diagram, the Lyapunov exponential spectrum, and the Lyapunov dimension of the system are investigated. The circuit of the system was also designed, based on the Multisim simulation platform, and the simulation of the circuit was realized. The simulation results show that the proposed FOHNN system exhibits many interesting phenomena, which provides more basis for the study of complex brain working patterns, and more references for the design, as well as the hardware implementation of the realized fractional-order neural network circuit.

A Multistable Memristor and Its Application in Fractional-Order Hopfield Neural Network

A Multistable Memristor and Its Application in Fractional-Order Hopfield Neural Network

Design and implementation of a new fractional-order Hopfield neural network system

In this work, a novel chaotic system of fractional-order based on the model of Hopfield Neural Network (HNN) is proposed. The numerical solutions of the 4-neurons-based HNN fractional-order chaotic system are obtained by using the Adomain decomposition method. The dynamical performances of the 4-neurons-based HNN fractional-order chaotic system are explored through attractor trajectories, bifurcation diagrams, Lyapunov exponents, SE complexity and chaotic diagram based on SE complexity. In addition, the 4-neurons-based HNN fractional-order chaotic system is implemented based on the Multisim platform. The experimental results indicate that the 4-neurons-based HNN fractional-order chaotic system has rich dynamic behavior, and the influence of different order on the dynamical properties of the system are particularly great. This research will provide theoretical foundation and experimental basis for the hardware implementation and application of the 4-neurons-based HNN fractional-order chaotic system.

Asymptotic stability and synchronization of fractional order Hopfield neural networks with unbounded delay

In this paper, the asymptotic stability and synchronization of fractional order Hopfield neural networks (FOHNNs) with time‐varying delay are investigated. First, we propose an extended fractional Halanay inequality, which plays an essential role in the proof of the main results. Using the Banach fixed point theorem, the existence and uniqueness of the equilibrium point of the system is proved. Through the extended Halanay inequality and a useful fractional derivative inequality, the sufficient conditions of the asymptotic stability and synchronization are given for FOHNNs. Finally, two numerical examples are presented to illustrate the effectiveness of the theoretical results.

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

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.

Exact coexistence and locally asymptotic stability of multiple equilibria for fractional-order delayed Hopfield neural networks with Gaussian activation function

This paper explores the multistability issue for fractional-order Hopfield neural networks with Gaussian activation function and multiple time delays. First, several sufficient criteria are presented for ensuring the exact coexistence of 3n equilibria, based on the geometric characteristics of Gaussian function, the fixed point theorem and the contraction mapping principle. Then, different from the existing methods used in the multistability analysis of fractional-order neural networks without time delays, it is shown that 2n of 3n total equilibria are locally asymptotically stable, by applying the theory of fractional-order linear delayed system and constructing suitable Lyapunov function. The obtained results improve and extend some existing multistability works for classical integer-order neural networks and fractional-order neural networks without time delays. Finally, an illustrative example with comprehensive computer simulations is given to demonstrate the theoretical results.

A new model of Hopfield network with fractional-order neurons for parameter estimation

In this work, we study an application of fractional-order Hopfield neural networks for optimization problem solving. The proposed network was simulated using a semi-analytical method based on Adomian decomposition,, and it was applied to the on-line estimation of time-varying parameters of nonlinear dynamical systems. Through simulations, it was demonstrated how fractional-order neurons influence the convergence of the Hopfield network, improving the performance of the parameter identification process if compared with integer-order implementations. Two different approaches for computing fractional derivatives were considered and compared as a function of the fractional-order of the derivatives: the Caputo and the Caputo–Fabrizio definitions. Simulation results related to different benchmarks commonly adopted in the literature are reported to demonstrate the suitability of the proposed architecture in the field of on-line parameter estimation.

Dynamics analysis of fractional-order Hopfield neural networks

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.

New Criteria on Finite-Time Stability of Fractional-Order Hopfield Neural Networks With Time Delays.

In this article, the finite-time stability (FTS) of fractional-order Hopfield neural networks with time delays (FHNNTDs) is studied. A widely used inequality in investigating the stability of the fractional-order neural networks is fractional-order Gronwall inequality related to the Mittag-Leffler function, which cannot be directly used to study the stability of the factional-order neural networks with time delays. In the existing works related to fractional-order Gronwall inequality with time delays, the order was divided into two cases: λ ∈ (0,0.5] and λ ∈ (0.5,+∞) . In this article, a new fractional-order Gronwall integral inequality with time delay and the unified form for all the fractional order is developed, which can be widely applied to investigate FTS of various fractional-order systems with time delays. Based on this new inequality, a new criterion for the FTS of FHNNTDs is derived. Compared with the existing criteria, in which fractional order λ ∈ (0,1) was divided into two cases, λ ∈ (0,0.5] and λ ∈ (0.5,1) , the obtained results in this article are presented in the unified form of fractional order λ ∈ (0,1) and convenient to verify. More importantly, the criteria in this article are less conservative than some existing ones. Finally, two numerical examples are given to demonstrate the validity of the proposed results.

Solving ill-posed problems faster using fractional-order Hopfield neural network

The Hopfield neural network (HNN) is one of the most used neural network architectures, it has been used to solve ill-posed problems with great success. Some works in recent years have incorporated fractional-order derivative with respect to time into original Hopfield neural network model. The motivation for this proposal is that it will be included nonlocal features in the original model. However, these studies were restricted to answer questions about the existence, uniqueness and stability of the equilibrium point for the neural network equation system used. The question that was addressed here is how the fractional-order Hopfield neural network (FHNN) can be used to solve ill-posed problems in Physical-Chemistry research field and how they compare to conventional model. For this purpose, fractional-order derivative with respect to time was included on HNN equations, obtained from Lyapunov function defined by 2-norm of the residual function. Therefore, the FHNN model used here was different from that used in previous work. Three prototype problems for the examination of the FHNN model were presented and discussed. In all of them, the solution found by FHNN model was achieved in less time when compared to HNN model. In this work, the Mittag-Leffler stability of the modified FHNN model is demonstrated and used to discuss how fractional order affects the solution. This work shows how the improvement in Hopfield neural network model, using fractional derivative order, gained competitive advantages over integer order Hopfield neural networks.

A fractional-order hopfield neural network chaotic system and its circuit realization

This paper investigated a novel fractional-order Hopfield neural network with complex function nonlinear terms based on Adomian decomposition method. The equilibrium points set and stability of the system are studied. The dynamic behaviors of the system are analyzed through bifurcation diagram, Lyapunov exponent spectrum and Lyapunov dimension. Meanwhile, the influence of the synaptic coupling strength and the order of the system are determined. Especially, complex phenomenon of coexisting attractors is observed in the fractional-order system. The results show that the novel fractional-order Hopfield neural network chaotic system has rich dynamic characteristics. In addition, we implement the system through a DSP platform and the results are in good agreement with those on simulation. It provides a theoretical foundation for the application of fractional-order Hopfield neural network in secure communication.

Dynamical analysis, sliding mode synchronization of a fractional-order memristor Hopfield neural network with parameter uncertainties and its non-fractional-order FPGA implementation

Recent developments in the applications of neural networks in various engineering and technology applications have motivated researchers to study the nonlinear behavior of such networks. In this work we investigate a fractional-order Hopfield neural network with memristor synaptic weight. The dynamical properties of the proposed system are examined and the memristor neural network shows hyperchaotic attractors in fractional orders with hidden oscillations. We also propose an adaptive sliding mode control technique to synchronize the proposed fractional-order systems with uncertainties. Numerical simulations are derived to show the effectiveness of the synchronization algorithm. Moreover, the designed chaotic memristor Hopfield neural network system is realized on FPGA using the 4th-order Runge–Kutta (RK4) numerical algorithm. The FPGA-based chaotic memristor HNN is coded in VHDL using the 32-bit IEEE-754-1985 floating point standard. The chaotic memristor neural network designed on FPGA is synthesized and tested using Xilinx ISE. The chip statistics of Xilinx XC6VLX240T-1-FF1156 kit obtained from Place & Route operation for the designed RK4-based system is presented. The operating frequency of newly modeled FPGA-based memristor neural network chaotic signal generator is 231.616 MHz.