Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays

Abstract

In this paper, we are concerned with a class of quaternion-valued cellular neural networks with time-varying transmission delays and leakage delays. By applying a continuation theorem of coincidence degree theory and the Wirtinger inequality as well as constructing a suitable Lyapunov functional, sufficient conditions are derived to ensure the existence and global exponential stability of anti-periodic solutions via direct approaches. Our results are completely new. Finally, numerical examples are also provided to show the effectiveness of our results.

Besicovitch almost periodic solutions for fractional‐order quaternion‐valued neural networks with discrete and distributed delays

“…[16][17][18][19][20][21][22] What we need to point out here is that periodic and almost periodic oscillations are significant dynamics of non-autonomous neural networks. [23][24][25][26][27][28][29][30][31][32][33] However, the almost periodic oscillation of fractional-order neural networks is still rarely studied. In addition, Besicovitch almost periodic oscillation is a kind of oscillation that is more complicated than almost periodic oscillations in other senses.…”

“…And various dynamics of fractional‐order neural networks have been studied a lot 16–22 . What we need to point out here is that periodic and almost periodic oscillations are significant dynamics of non‐autonomous neural networks 23–33 . However, the almost periodic oscillation of fractional‐order neural networks is still rarely studied.…”

Degree, quaternions and periodic solutions

“…The study of periodic solutions of quaternionic valued ordinary differential equations using topological degree techniques has been initiated in [1], and was followed by the interesting contributions of Zoladek [2] and Wilczynsky [3,4]. Of interest also are the recent studies of linear equations by Cai and Kou [5,6], Cheng et al [7] and Kou & Xia [8], of autonomous equations by Gasull, Llibre and Zhang [9,10], and the applications to neural networks by Li and his co-workers [11][12][13][14][15][16].…”