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 4 n equilibrium points, 3 4 n 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.