This paper studies the finite-time multistability of impulsive Hopfield neural networks with a general class of activation functions. First, the existence of Πni=1(2Mi+1) equilibrium points and Πni=1(Mi+1) invariant sets in such n-neuron neural networks can be guaranteed by applying the Brouwer’s fixed-point theorem as well as upper and lower functions method. Furthermore, it is demonstrated that these equilibrium points and invariant sets remain valid for the same neural networks when subjected to an appropriate controller. Then, on the basis of Lyapunov function method and impulsive control theory, two finite-time multistability theorems are established for Hopfield neural networks under distinct impulse scenarios: stabilizing impulses and destabilizing impulses. The settling time estimations for determining the local finite-time stability of Πni=1(Mi+1) equilibrium points are developed by designing general impulsive sequences, which reveal that the settling time is dependent on initial state, impulsive effects and control parameters. From the perspective of impulsive effects, the introduced stabilizing impulses in neural networks not only accelerate the convergence rate but also yield tighter upper bound of settling time estimation relative to impulse-free systems. In stark contrast, destabilizing impulses significantly degrade the convergence performance while resulting in more conservative upper bound of settling time estimation. Finally, theoretical results are shown to be effective by two illustrative examples and two associative memory applications of grayscale image.

This article investigates the prespecified performance consensus problem for a class of nonlinear multiagent systems (MASs) with unknown actuator faults. By employing a sensor-triggered mechanism and neural estimation algorithm, a novel leader-follower consensus protocol is devised for the nonlinear MASs. The developed sensor event-triggered mechanism comprises two parts, the first one is sensor event-triggered sampling, and the second one is event-triggered information transmission. Due to the presence of the sensor-triggered mechanism, the system states cannot be available in real time. In order to solve this challenge, a signal decomposition and compensation strategy is constructed to balance the intermittent sensor-sampled signals and the real system inputs. Furthermore, the considered actuator faults in each follower are not limited to be finite, the time, frequency and mode of the faults are also unknown. To address the unknown actuator faults in the nonlinear MASs, a resilient fault management mechanism is developed for each follower. Based on the managed actuator faults dynamics, some bounded estimation signals are constructed and the issue of "explosion of complexity" in the backstepping design procedure is eliminated through the application of nonlinear filters with compensation terms. Finally, simulation results are given to illustrate the effectiveness of developed control protocol.

In this article, we contribute to dynamic event-triggered-based leader-follower quantized consensus design for fractional-order (FO) nonlinear multi-agent systems (MASs) under asymmetric time-varying state constraints. Firstly, we propose a novel dynamic event-triggered mechanism (ETM) to fully utilize the channel resources between the controller and actuator. Subsequently, by means of the characteristics of quantization nonlinearities and the architecture of FO nonlinear MASs, the influence of quantized inputs is eliminated. In the consensus protocol design procedure, the radial basis function neural networks (RBF-NNs) technology is employed to dynamically estimate the uncertain functions existing in the system. Then, by employing a bivariate FO derivative lemma and some convex time-varying barrier Lyapunov functions (BLFs) with two variables, a dynamic event-triggered-based leader-follower quantized consensus protocol is constructed. Exhaustive theoretical analysis manifests that the tracking errors of the FO nonlinear MASs converge to a small region, all signals in the FO nonlinear MASs are bounded, and the asymmetric time-varying state constraints can be achieved regardless of the presence of the quantized inputs and dynamic event-triggered communication. Finally, a chaotic Duffing FO nonlinear MASs is employed to demonstrate the efficacy of the developed consensus protocol.

This paper provides theoretical analysis of the fixed-time multi-almost-periodicity in switched fuzzy neural networks, employing multi-controller strategies and a state-dependent switching mechanism. Utilizing the Ascoli-Arzela theorem, the properties of$M$-matrix, Lyapunov functions method, and some inequality techniques, we establish some sufficient conditions to ascertain that the number of exponentially stable almost-periodic solutions can be up to$4^{n}$, where$n$is the number of neurons. Furthermore, we design various controllers to achieve the fixed-time stability for various almost-periodic solutions located in the positive invariant sets. Then, the settling time for the switched fuzzy networks to achieve multi-almost-periodicity is estimated. It is noteworthy to state that this paper considers fixed-time multiperiodicity and fixed-time multistability as special cases of fixed-time multi-almost-periodicity. Two numerical examples are presented to demonstrate the theoretical results.

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 and to and , respectively, where with 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.

This work explores the issue of multistability for a competitive neural network (NN) class with sigmoidal activation functions (AFs) involving state-dependent switching and fractional-order derivative. Specifically, first, we consider three different switching point locations, and establish some sufficient criteria ensuring that NNs with$n$-neurons can have, and only have,$5^{n_{1}}\cdot 3^{n_{2}}$equilibrium points (EPs) with$n_{1}+n_{2}=n$, by utilizing the geometric features of the sigmoidal functions, the fixed point theorem, the Filippov’s EP definition, and the contraction mapping theorem. Then, based on novel Lyapunov functions and by applying the fractional-order calculus theory, it is demonstrated that$3^{n_{1}}\cdot 2^{n_{2}}$out of$5^{n_{1}}\cdot 3^{n_{2}}$total EPs are locally stable. This work’s investigation reveals that competitive NNs with switching afford more storage capacity compared to the nonswitching case. Additionally, our results are valid for the integer-order and fractional-order switched NNs, improving and generalizing current works. Furthermore, two numerical examples and an application example of associative memory are provided to validate the effectiveness of the theoretical findings, and the way various fractional orders affect the NNs’ convergence speed is shown through simulations.

This paper addresses the problem of coexistence and dynamical behaviors of multiple equilibria for competitive neural networks. First, a general class of discontinuous nonmonotonic piecewise linear activation functions is introduced for competitive neural networks. Then based on the fixed point theorem and theory of strict diagonal dominance matrix, it is shown that under some conditions, such n -neuron competitive neural networks can have 5(n) equilibria, among which 3(n) equilibria are locally stable and the others are unstable. More importantly, it is revealed that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibria and locally stable equilibria than the ones with other activation functions, such as the continuous Mexican-hat-type activation function and discontinuous two-level activation function. Furthermore, the 3(n) locally stable equilibria given in this paper are located in not only saturated regions, but also unsaturated regions, which is different from the existing results on multistability of neural networks with multiple level activation functions. A simulation example is provided to illustrate and validate the theoretical findings.

This paper is concerned with stability analysis of multiple equilibria for recurrent neural networks. A new type of activation function, namely, discontinuous Mexican-hat-type activation function, is proposed for recurrent neural networks. Then with the aid of the fixed point theorem, some sufficient conditions for coexistent multiple equilibria are obtained to guarantee that such n-neuron recurrent neural networks can have at least 4nequilibria. In view of the theory of strict diagonal dominance matrix, further stability analysis reveals that 3nequilibria are locally exponentially stable. The new results considerably improve the existing multistability results in the literature.

In this paper, a general class of non-monotonic piecewise linear activation functions is introduced and then the coexistence and dynamical behaviors of multiple equilibrium points are studied for a class of memristive neural networks (MNNs). It is proven that under some conditions, such n-neuron MNNs can have 5 n equilibrium points located in $\Re^n$ , and 3 n of them are locally exponentially stable, by means of fixed point theorem, nonsmooth analysis theory and rigorous mathematical analysis. The investigation shows that the neural networks with non-monotonic piecewise linear activation functions introduced in this paper can have greater storage capacity than the ones with Mexican-hat-type activation function.

In this paper, we discuss the coexistence and dynamical behaviors of multiple equilibrium points for recurrent neural networks with a class of discontinuous nonmonotonic piecewise linear activation functions. It is proved that under some conditions, such n -neuron neural networks can have at least 5(n) equilibrium points, 3(n) of which are locally stable and the others are unstable, based on the contraction mapping theorem and the theory of strict diagonal dominance matrix. The investigation shows that the neural networks with the discontinuous activation functions introduced in this paper can have both more total equilibrium points and more locally stable equilibrium points than the ones with continuous Mexican-hat-type activation function or discontinuous two-level activation functions. An illustrative example with computer simulations is presented to verify the theoretical analysis.

In this paper, second-order interactions are introduced into competitive neural networks (NNs) and the multistability is discussed for second-order competitive NNs (SOCNNs) with nondecreasing saturated activation functions. Firstly, based on decomposition of state space, Cauchy convergence principle, and inequality technique, some sufficient conditions ensuring the local exponential stability of 2N equilibrium points are derived. Secondly, some conditions are obtained for ascertaining equilibrium points to be locally exponentially stable and to be located in any designated region. Thirdly, the theory is extended to more general saturated activation functions with 2r corner points and a sufficient criterion is given under which the SOCNNs can have (r+1)N locally exponentially stable equilibrium points. Even if there is no second-order interactions, the obtained results are less restrictive than those in some recent works. Finally, three examples with their simulations are presented to verify the theoretical analysis.