Pseudo almost automorphic solutions of quaternion-valued neural networks with infinitely distributed delays via a non-decomposing method

Abstract

In this paper, we consider the existence and global exponential stability of pseudo almost automorphic solutions to quaternion-valued cellular neural networks with infinitely distributed delays. Unlike most previous studies of quaternion-valued cellular neural networks, we do not decompose the systems under consideration into real-valued or complex-valued systems, but rather directly study quaternion-valued systems. Our method and the results of this paper are new. An example is given to show the feasibility of our main results.

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

“…Fortunately, over the past 20 years, especially in algebra area, quaternion has been a topic for the effective applications in the real world. Also, a new class of differential equations named quaternion differential equations has been already applied successfully to the fields, such as quantum mechanics [2,3], robotic manipulation [4], fluid mechanics [5], differential geometry [6], communication problems and signal processing [7][8][9], and neural networks [10][11][12][13]. Many scholars tried to shed some light on the information about solutions of quaternion differential equations.…”

“…Curves of x R p (t) = (x R 1 (t), x R 2 (t)) T and x I p (t) = (x I 1 (t), x I 2 (t)) T of system(13) with the initial values (x R 1 (0), x R 2 (0)) T = (0.01, -0.02) T , (0.03, -0.04) T , (0.05, -0.05) T and (x I 1 (0), x I 2 (0)) T = (-0.01, 0.03) T , (0.02, 0.05) T , (-0.05, -0.02) T τ 11 (t) τ 12 (t) τ 21 (t) τ 22 (f q (x) ≤ 0.0715, g q (x) ≤ 0.0561, f q (x) ≤ 1 20 xy , g q (x) ≤ 1 25 xy , a -1 = 1.4, a -2 = 1.6, a + 1 = 1.41, a + 2 = 1.62, b + 11 ≤ 0.0224, b + 12 ≤ 0.0332, b + 21 ≤ 0.0548, b + 22 ≤ 0.0245, c + 11 ≤ 0.0224, c + 12 ≤ 0.0332, c + 21 ≤ 0.0548, c + 22 ≤ 0.0245, Q + 1 ≤ 0.1334, Q + 2 ≤ 0.1546. So (H 1 )-(H 4 ) are satisfied.…”

Pseudo-almost-periodic solutions of quaternion-valued RNNs with mixed delays via a direct method

“…Therefore, the research on quaternion-valued neural networks has become a hot topic in the theory and applications of neural networks. However, due to the noncommutativity of quaternion multiplication, the results of quaternion-valued neural network dynamics are very few [32][33][34][35][36][37][38][39][40][41]. Especially, the results obtained by a method of not decomposing quaternionvalued systems into real-or complex-valued systems are even rarer.…”

On Pseudo Almost Automorphic Solutions to Quaternion-Valued Cellular Neural Networks With Delays

“…Generally speaking, almost automorphicity can be regarded as the extension of almost periodicity and pseudo automorphicity can be regarded as the generalization of almost automorphicity [34]. In the objective world, the pseudo automorphicity is more common than periodicity, almost periodicity and almost automorphicity.…”

“…In recent years, some excellent works on pseudo almost automorphic solutions on neural networks have been displayed. For example, Xiang and Li [34] considered the pesudo almost automorphic solutions of delayed quaternion-valued neural networks; M hamdiin [35] discussed the pseudo almost automorphic solutions to delayed BAM neural networks; Cieutat and Ezzinbi [36] investigated the pseudo almost automorphic solutions for some dissipative differential equations in Banach spaces; Zhao et al [37] handled the weighted pseudo-almost automorphic solutions for high-order Hopfield neural networks; Aouiti and Dridi [38] made a detailed analysis on piecewise asymptotically almost automorphic solutions of high-order Hopfield neural networks. Zhu et al [39] studied the existence and exponential stability of pseudo almost automorphic solutions to delayed Cohen-Grossberg neural networks.…”