Multistability of state-dependent switching neural networks with discontinuous nonmonotonic piecewise linear activation functions

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 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.

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.