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.

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.

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.

Event-triggered adaptive neural networks control for fractional-order nonstrict-feedback nonlinear systems with unmodeled dynamics and input saturation.

The event-triggered adaptive neural networks control is investigated in this paper for a class of fractional-order systems (FOSs) with unmodeled dynamics and input saturation. Firstly, in order to obtain an auxiliary signal and then avoid the state variables of unmodeled dynamics directly appearing in the designed controller, the notion of exponential input-to-state practical stability (ISpS) and some related lemmas for integer-order systems are extended to the ones for FOSs. Then, based on the traditional event-triggered mechanism, we propose a novel adaptive event-triggered mechanism (AETM) in this paper, in which the threshold parameters can be adjusted dynamically according to the tracking performance. Besides, different from the previous works where the derivative of hyperbolic tangent function tanh(⋅) needs to have positive lower bound, a new type of auxiliary signal is introduced in this paper to handle the effect of input saturation and thus this limitation is released. Finally, two numerical examples and some comparisons are provided to illustrate our proposed controllers.

Novel bifurcation results for a delayed fractional-order quaternion-valued neural network.

This paper reports the innovative results on the stability and bifurcation for a delayed fractional-order quaternion-valued neural network(FOQVNN). Delay-stimulated bifurcation criteria of the developed FOQVNN are attained. Then, the bifurcation diagrams are perfectly exhibited to authenticate the veracity of the bifurcation results. Besides, the stability zone is more larger of the addressed FOQVNN in comparison with its counterpart if other parameters are intercalated. It further witnesses that the amplitudes of bifurcation oscillation get bigger with the augmentation of time delay. It discloses that the bifurcation phenomena engender earlier as the order incrementally magnifies. The exactness and merits of the achieved analytic results are eventually substantiated by a simulation example.

Multiple Mittag-Leffler stability of fractional-order competitive neural networks with Gaussian activation functions.

In this paper, we explore the coexistence and dynamical behaviors of multiple equilibrium points for fractional-order competitive neural networks with Gaussian activation functions. By virtue of the geometrical properties of activation functions, the fixed point theorem and the theory of fractional-order differential equation, some sufficient conditions are established to guarantee that such n-neuron neural networks have exactly 3k equilibrium points with 0≤k≤n, among which 2k equilibrium points are locally Mittag-Leffler stable. The obtained results cover both multistability and mono-stability of fractional-order neural networks and integer-order neural networks. Two illustrative examples with their computer simulations are presented to verify the theoretical analysis.

Coexistence and local µ-stability of multiple equilibrium points for memristive neural networks with nonmonotonic piecewise linear activation functions and unbounded time-varying delays.

In this paper, the coexistence and dynamical behaviors of multiple equilibrium points are discussed for a class of memristive neural networks (MNNs) with unbounded time-varying delays and nonmonotonic piecewise linear activation functions. By means of the fixed point theorem, nonsmooth analysis theory and rigorous mathematical analysis, it is proven that under some conditions, such n-neuron MNNs can have 5n equilibrium points located in ℜn, and 3n of them are locally µ-stable. As a direct application, some criteria are also obtained on the multiple exponential stability, multiple power stability, multiple log-stability and multiple log-log-stability. All these results reveal that the addressed neural networks with activation functions introduced in this paper can generate greater storage capacity than the ones with Mexican-hat-type activation function. Numerical simulations are presented to substantiate the theoretical results.

Dynamical Behaviors of Multiple Equilibria in Competitive Neural Networks With Discontinuous Nonmonotonic Piecewise Linear Activation Functions.

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.

Multistability and Instability of Neural Networks With Discontinuous Nonmonotonic Piecewise Linear Activation Functions.

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.

Multistability of memristive Cohen-Grossberg neural networks with non-monotonic piecewise linear activation functions and time-varying delays.

The problem of coexistence and dynamical behaviors of multiple equilibrium points is addressed for a class of memristive Cohen-Grossberg neural networks with non-monotonic piecewise linear activation functions and time-varying delays. By virtue of the fixed point theorem, nonsmooth analysis theory and other analytical tools, some sufficient conditions are established to guarantee that such n-dimensional memristive Cohen-Grossberg neural networks can have 5(n) equilibrium points, among which 3(n) equilibrium points are locally exponentially stable. It is shown that greater storage capacity can be achieved by neural networks with the non-monotonic activation functions introduced herein than the ones with Mexican-hat-type activation function. In addition, unlike most existing multistability results of neural networks with monotonic activation functions, those obtained 3(n) locally stable equilibrium points are located both in saturated regions and unsaturated regions. The theoretical findings are verified by an illustrative example with computer simulations.

Multistability of neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays.

This paper is concerned with the problem of coexistence and dynamical behaviors of multiple equilibrium points for neural networks with discontinuous non-monotonic piecewise linear activation functions and time-varying delays. The fixed point theorem and other analytical tools are used to develop certain sufficient conditions that ensure that the n-dimensional discontinuous neural networks with time-varying delays can have at least 5(n) equilibrium points, 3(n) of which are locally stable and the others are unstable. The importance of the derived results is that it reveals that the discontinuous neural networks can have greater storage capacity than the continuous ones. Moreover, different from the existing results on multistability of neural networks with discontinuous activation functions, the 3(n) locally stable equilibrium points obtained in this paper are located in not only saturated regions, but also unsaturated regions, due to the non-monotonic structure of discontinuous activation functions. A numerical simulation study is conducted to illustrate and support the derived theoretical results.

Multistability of second-order competitive neural networks with nondecreasing saturated activation functions.

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.