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

Dynamics of heterogeneous Hopfield neural network with adaptive activation function based on memristor

Memristor and activation function are two important nonlinear factors of the memristive Hopfield neural network. The effects of different memristors on the dynamics of Hopfield neural networks have been studied by many researchers. However, less attention has been paid to the activation function. In this paper, we present a heterogeneous memristive Hopfield neural network with neurons using different activation functions. The activation functions include fixed activation functions and an adaptive activation function, where the adaptive activation function is based on a memristor. The theoretical and experimental study of the neural network's dynamics has been conducted using phase portraits, bifurcation diagrams, and Lyapunov exponents spectras. Numerical results show that complex dynamical behaviors such as multi-scroll chaos, transient chaos, state jumps and multi-type coexisting attractors can be observed in the heterogeneous memristive Hopfield neural network. In addition, the hardware implementation of memristive Hopfield neural network with adaptive activation function is designed and verified. The experimental results are in good agreement with those obtained using numerical simulations.

Multistability and fixed-time multisynchronization of switched neural networks with state-dependent switching rules

This paper presents theoretical results on the multistability and fixed-time synchronization of switched neural networks with multiple almost-periodic solutions and state-dependent switching rules. It is shown herein that the number, location, and stability of the almost-periodic solutions of the switched neural networks can be characterized by making use of the state-space partition. Two sets of sufficient conditions are derived to ascertain the existence of 3n exponentially stable almost-periodic solutions. Subsequently, this paper introduces the novel concept of fixed-time multisynchronization in switched neural networks associated with a range of almost-periodic parameters within multiple stable equilibrium states for the first time. Based on the multistability results, it is demonstrated that there are 3n synchronization manifolds, wherein n is the number of neurons. Additionally, an estimation for the settling time required for drive-response switched neural networks to achieve synchronization is provided. It should be noted that this paper considers stable equilibrium points (static multisynchronization), stable almost-periodic orbits (dynamical multisynchronization), and hybrid stable equilibrium states (hybrid multisynchronization) as special cases of multistability (multisynchronization). Two numerical examples are elaborated to substantiate the theoretical results.

Memristor-induced hyperchaos, multiscroll and extreme multistability in fractional-order HNN: Image encryption and FPGA implementation

Fractional-order differentiation (FOD) can record information from the past, present, and future. Compared with integer-order systems, FOD systems have higher complexity and more accurate ability to describe the real world. In this paper, two types of fractional-order memristors are proposed and one type is proved to have extreme multistability, local activity, and non-volatility. By using memristors to simulate the autapse of a neuron and to describe the phenomenon of electromagnetic induction caused by electromagnetic radiation, we establish a new 5D FOD memristive HNN (FOMHNN). Through dynamic simulation, rich dynamic behaviors are found, such as hyperchaos, multiscroll, extreme multistability, and "overclocking" behavior caused by order reduction. To the best of our knowledge, this is the first time that such rich dynamic behaviors are found in FOMHNN simultaneously. Based on this FOMHNN, a very efficient and secure image encryption scheme is designed. Security analysis shows that the encrypted Lena image has extremely low adjacent pixel correlation and high randomness, with information entropy of 7.9995. Despite discarding diffusion and scrambling, it has excellent plaintext sensitivity, with NCPR = 99.6095% and UACI = 33.4671%. Finally, this paper implements the proposed FOMHNN and image encryption on field programmable gate array (FPGA). To our knowledge, the related work of fully hardware implementation of fractional-order neural networks and image encryption schemes based on this is rare.

Coexistence and local stability of multiple equilibrium points for fractional-order state-dependent switched competitive neural networks with time-varying delays

This paper investigates the coexistence and local stability of multiple equilibrium points for a class of competitive neural networks with sigmoidal activation functions and time-varying delays, in which fractional-order derivative and state-dependent switching are involved at the same time. Some novel criteria are established to ensure that such n-neuron neural networks can have [Formula: see text] total equilibrium points and [Formula: see text] locally stable equilibrium points with m₁+m₂=n, based on the fixed-point theorem, the definition of equilibrium point in the sense of Filippov, the theory of fractional-order differential equation and Lyapunov function method. The investigation implies that the competitive neural networks with switching can possess greater storage capacity than the ones without switching. Moreover, the obtained results include the multistability results of both fractional-order switched Hopfield neural networks and integer-order switched Hopfield neural networks as special cases, thus generalizing and improving some existing works. Finally, four numerical examples are presented to substantiate the effectiveness of the theoretical analysis.

Asymptotic Stabilization of Delayed Linear Fractional-Order Systems Subject to State and Control Constraints

Studies have shown that fractional calculus can describe and characterize a practical system satisfactorily. Therefore, the stabilization of fractional-order systems is of great significance. The asymptotic stabilization problem of delayed linear fractional-order systems (DLFS) subject to state and control constraints is studied in this article. Firstly, the existence conditions for feedback controllers of DLFS subject to both state and control constraints are given. Furthermore, a sufficient condition for invariance of polyhedron set is established by using invariant set theory. A new Lyapunov function is constructed on the basis of the constraints, and some sufficient conditions for the asymptotic stability of DLFS are obtained. Then, the feedback controller and the corresponding solution algorithms are given to ensure the asymptotic stability under state and control input constraints. The proposed solution algorithm transforms the asymptotic stabilization problem into a linear/nonlinear programming (LP/NP) problem which is easy to solve from the perspective of computation. Finally, three numerical examples are offered to illustrate the effectiveness of the proposed method.

Chaos Control for a Fractional-Order Jerk System via Time Delay Feedback Controller and Mixed Controller

In this study, we propose a novel fractional-order Jerk system. Experiments show that, under some suitable parameters, the fractional-order Jerk system displays a chaotic phenomenon. In order to suppress the chaotic behavior of the fractional-order Jerk system, we design two control strategies. Firstly, we design an appropriate time delay feedback controller to suppress the chaos of the fractional-order Jerk system. The delay-independent stability and bifurcation conditions are established. Secondly, we design a suitable mixed controller, which includes a time delay feedback controller and a fractional-order PDσ controller, to eliminate the chaos of the fractional-order Jerk system. The sufficient condition ensuring the stability and the creation of Hopf bifurcation for the fractional-order controlled Jerk system is derived. Finally, computer simulations are executed to verify the feasibility of the designed controllers. The derived results of this study are absolutely new and possess potential application value in controlling chaos in physics. Moreover, the research approach also enriches the chaos control theory of fractional-order dynamical system.