Existence and Global Exponential Stability of Pseudo Almost Periodic Solutions for Neutral Type Quaternion-Valued Neural Networks with Delays in the Leakage Term on Time Scales

Abstract

We propose a class of neutral type quaternion-valued neural networks with delays in the leakage term on time scales that can unify the discrete-time and the continuous-time neural networks. In order to avoid the difficulty brought by the noncommutativity of quaternion multiplication, we first decompose the quaternion-valued system into four real-valued systems. Then, by applying the exponential dichotomic theory of linear dynamic equations on time scales, Banach's fixed point theorem, the theory of calculus on time scales, and inequality techniques, we obtain some sufficient conditions on the existence and global exponential stability of pseudo almost periodic solutions for this class of neural networks. Our results are completely new even for both the case of the neural networks governed by differential equations and the case of the neural networks governed by difference equations and show that, under a simple condition, the continuous-time quaternion-valued network and its corresponding discrete-time quaternionvalued network have the same dynamical behavior for the pseudo almost periodicity. Finally, a numerical example is given to illustrate the feasibility of our results.

Existence and Global Exponential Stability of Almost Automorphic Solution for Clifford-Valued High-Order Hopfield Neural Networks with Leakage Delays

“…Therefore, neural networks with various delays have been extensively studied. In particular, since K. Gopalsamy [25] first studied the stability of neural networks with leakage delays, a lot of research has been done on neural networks with leakage delays [6,11,12,26]. However, there is no research on the Clifford-valued neural network with leakage delays.…”

Weighted pseudo-almost periodic solutions and global exponential synchronization for delayed QVCNNs

“…In recent years, quaternion-valued neural networks, which can be seen as a generic extension of complex-valued neural networks or real-valued neural networks, have been found many practical applications and have been widely concerned [36,37]. Since the application of neural networks depends on their dynamics, some papers have been devoted to the study of the dynamical behaviors for quaternion-valued neural networks ( [38][39][40][41][42][43]). However, up to now, there are still no results about weighted pseudo-almost periodic solutions and synchronization of QVCNNs.…”

Existence and global exponential stability of anti‐periodic solutions for delayed quaternion‐valued cellular neural networks with impulsive effects

“…So far, only a few of dynamical behaviors of QVNNs have been studied. () For examples, in Liu et al, the global μ ‐stability criteria for quaternion‐valued neural networks with unbounded time‐varying delays is studied; in Liu et al, based on Halanay inequality instead of Lyapunov function, some sufficient conditions are derived to guarantee the global exponential stability for QVNNs; in Zhang et al, the exponential convergence is proved directly accompanied with the existence and uniqueness of the equilibrium point to the consider system; in Li and Qin, by using the Mahwin's continuation theorem of coincidence degree theory and constructing a suitable Lyapunov function, the existence and global exponential stability of periodic solutions for a class of DQVCNNs are established; and in Li and Meng, by applying the exponential dichotomic theory of linear dynamic equations on time scales, Banach's fixed‐point theorem, and inequality techniques, some sufficient conditions on the existence and global exponential stability of pseudo almost periodic solutions for a class of QVNNs with delays on time scales are obtained.…”