Multistability analysis of quaternion-valued neural networks with cosine activation functions *

It is significant to design a system with high storage capacity for associative memory and pattern recognition. To address this issue, this paper first proposes a quaternion-valued neural network (QVNN) model with multiple equilibrium points in which the cosine function is used as the activation function of QVNN. Then, based on the Brouwer fixed point theorem and the geometric properties of the activation function, sufficient conditions for QVNN to have unique equilibrium points, finite equilibrium points, and countable infinite equilibrium points are obtained, respectively. Furthermore, sufficient conditions for the exponential stability of equilibrium points are derived, and the attraction basins of the stable equilibrium points are given. Finally, two numerical examples are given to confirm the validity of the proposed theoretical results.

Automated summary

This paper proposes a design for a system with high storage capacity for associative memory and pattern recognition. A quaternion-valued neural network (QVNN) model with multiple equilibrium points is introduced, using the cosine function as the activation function of QVNN. Sufficient conditions are then derived for QVNN to have unique, finite, and countable infinite equilibrium points based on the Brouwer fixed point theorem and the geometric properties of the activation function. The paper also establishes sufficient conditions for the exponential stability of equilibrium points and provides the attraction basins of stable equilibrium points. Two numerical examples are presented to validate the proposed theoretical results.