Multistability of Quaternion-Valued Recurrent Neural Networks with Discontinuous Nonmonotonic Piecewise Nonlinear Activation Functions

In this article, the coexistence and dynamical behaviors of multiple equilibrium points for quaternion-valued neural networks (QVNNs) are investigated, whose activation functions are discontinuous and nonmonotonic piecewise nonlinear. According to the Hamilton rules, the QVNNs can be divided into four real-valued parts. By utilizing the Brouwer's Fixed Point Theorem and property of strictly diagonally dominant matrices, some sufficient conditions are derived to ensure that the QVNNs have at least 5(4n) equilibrium points, 3(4n) of them are locally exponentially stable, and the others are unstable. It is shown that the number of stable equilibria in QVNNs is more than that in the real-valued ones. Finally, a numerical simulation is presented to clarify the theoretical analysis is valid.

Automated summary

This article investigates the coexistence and dynamical behaviors of multiple equilibrium points for quaternion-valued neural networks with discontinuous and nonmonotonic piecewise nonlinear activation functions. By utilizing mathematical theorems and matrix properties, sufficient conditions are derived to ensure the existence of multiple stable equilibrium points in the network. Numerical simulations are provided to validate the theoretical analysis.