Multistability Analysis Of Quaternion-Valued Neural Networks With Time-Varying Delays

In recent years,some valuable results have been obtained about multistability of neural networks.However,most of the results are based on real-valued neural network models and complex-valued neural network models.The quaternion is composed of a real part and three imaginary parts,which is a generalization of complex number and has the advantage of being able to be handled as a whole in the study of multidimensional data.Since the multiplication of quaternions does not satisfy the commutation law,the existing theory and methods of dynamic analysis for real-valued and complex-valued neural networks cannot be directly applied to quaternion-valued neural networks,so the study of quaternion-valued neural networks is more challenging and meaningful.This dissertation discusses the multistability for several types of quaternion-valued neural network models with time-varying delays,including quaternion-valued competitive neural networks with time-varying delays and quaternion-valued Hopfield neural networks with time-varying delays.Based on the quaternion algebra theory,Lyapunov function method,Brouwer's fixed point theorem and inequality technique,the multistability of quaternion-valued neural networks with time-varying delays is studied.This dissertation is divided into four chapters and the main contents are summarized as follows.In the first chapter,the current research situations are briefed,which concern the multistability of classical neural networks and the stability of quaternion-valued neural networks.The main contents and the main contributions of this dissertation are also expounded according to the above-mentioned analysis.In the second chapter,we investigate the coexistence and dynamical behaviors of multiple e-quilibrium points for the quaternion-valued competitive neural network with continuous Gaussian-wavelet-type activation function and time-varying delays.By analyzing the geometric characteristics of the continuous Gaussian-wavelet-type activation function and applying the Brouwer's fixed point theorem,some sufficient conditions are established to guarantee that such n-neuron quaternion-valued competitive neural networks with time-varying delays have 5⁴ⁿequilibrium points.The sufficient con-ditions for 3⁴ⁿequilibrium points to be locally stable are obtained based on the Lyapunov function method and the inequality technique.Finally,two illustrative examples with their computer simula-tions are presented to verify the correctness and validity of the theory.In the third chapter,we discuss the multiple?-stability of quaternion-valued Hopfield neural networks with Gaussian activation functions and time-varying delays.By using the geometrical properties of Gaussian activation functions,the theory of quaternion algebra and the Brouwer's fixed point theorem,we present some sufficient conditions to ensure such n-neuron quaternion-valued Hop-field neural networks with time-varying delays have 3⁴ⁿequilibrium points.The sufficient conditions for 2⁴ⁿequilibrium points to be locally?-stable are obtained by applying the Lyapunov function method and the inequality technique.Finally,two examples with their computer simulations are provided to show the effectiveness of the theoretical results.In the last chapter,the research work of this dissertation is summarized,and the prospect for the future work is made.