Asymptotic behavior of Clifford-valued dynamic systems with D-operator on time scales

Chaouki Aouiti,Jinde Cao,Imen Ben Gharbia,Xiaodi Li

doi:10.1186/s13662-021-03266-3

Abstract

In this paper, a general class of Clifford-valued neutral high-order neural network (HNN) with D-operator on time scales is investigated. In this model, time-varying delays and continuously distributed delays are taken into account. As an extension of the real-valued neural network, the Clifford-valued neural network, which includes a familiar complex-valued neural network and a quaternion-valued neural network as special cases, has been an active research field recently. By utilizing this novel method, which incorporates the differential inequality techniques and the fixed point theorem and time-scale theory of computation, we derive a few sufficient conditions to ensure the existence, uniqueness, and exponential stability of the pseudo almost periodic (PAP) solution of the considered model. The results in this paper are new, even if time scale mathbb{T}=mathbb{R} or mathbb{T}=mathbb{Z}, and complementary to the previously existing works. Furthermore, an example and its numerical simulations are included to demonstrate the validity and advantage of the obtained results.

Highlights

1 Introduction The well-known Hopfield neural network (HNN) has been widely studied since it was introduced by John Hopfield in [1]
Dads et al in [7] noted that it will be very valuable to investigate the dynamics of pseudo almost periodic (PAP) systems with delays
In [14], the authors investigated the weighted PAP functions on time scales with applications to delayed cellular neural networks

Summary

Introduction

The well-known Hopfield neural network (HNN) has been widely studied since it was introduced by John Hopfield in [1]. In [14], the authors investigated the weighted PAP functions on time scales with applications to delayed cellular neural networks. In [12], the authors studied global exponential stability of unique equilibrium for complex-valued neural networks on time scales. Our contributions lie in four aspects: (a) The study of the existence, the uniqueness, and the global exponential stability of pseudo almost periodic solutions for neutral Clifford-valued HHNNs with D-operator, time-varying delays, and infinite distributed delay on time scales is first advanced. 3 and 4, on the basis of the results presented in the previous sections, ∇-differential inequalities on time scales, and Banach’s fixed-point theorem, we present sufficient conditions that assure the existence and global exponential stability of PAP solutions to (1).

Notations

Conclusion and open problems