Multistability of Fuzzy Neural Networks With Rectified Linear Units and State-Dependent Switching Rules

This article presents theoretical results on the multistability of fuzzy neural networks with rectified linear units and a state-dependent switching rule. Because of the boundlessness of state activation and multifariousness of state-dependent switching, such fuzzy neural networks exhibit very rich and complex dynamics. We show that there are up to $3^{n}-2^{n}-1$ stable equilibria in an $n$-neuron switched fuzzy neural network, substantially more than recurrent neural networks without switching. Based on the properties of positive invariant set, we derive seven sets of sufficient conditions to ensure the multistability of switched fuzzy neural networks with rectified linear units. We elaborate on three numerical examples to illustrate the theoretical results and a potential application in associative memories.