In this paper, we explore the complete synchronization and quasi-projective synchronization in a class of stochastic delayed quaternion-valued neural networks, utilizing a state-feedback control scheme. The studied neural networks into real-valued networks are short of known decomposing, by designing a very general nonlinear controller, according to the quaternion form It? formula with a number of inequality techniques in the configuration of quaternion domain, we obtained a quasi-projective synchronization criterion for drive-response networks. Moreover, we estimate the error margin for quasi-projective synchronization. At last, the theoretical results are confirmed by a numerical simulation.

This paper examines the issue of almost periodic quasi-projective synchronization of delayed fractional-order quaternion-valued neural networks. First, using a direct method rather than decomposing the fractional quaternion-valued system into four equivalent fractional real-valued systems, using Banach's fixed point theorem, according to the basic properties of fractional calculus and some inequality methods, we obtain that there is a unique almost periodic solution for this class of neural network with some sufficient conditions. Next, by constructing a suitable Lyapunov functional, using the characteristic of the Mittag-Leffler function and the scaling idea of the inequality, the adequate conditions for the quasi-projective synchronization of the established model are derived, and the upper bound of the systematic error is estimated. Finally, further use Matlab is used to carry out two numerical simulations to prove the results of theoretical analysis.

This paper mainly utilizes the non-decomposition method to examine a problem of polynomial synchronization for the quaternion-valued fuzzy cellular neural networks with proportional delays. Firstly, we prove a lemma explaining the solution form of a quaternion-valued differential system. Furthermore, under the Banach fixed point theory and some scaling techniques, we prove that this system’s unique almost periodic solution exists in the specific closed convex subset. Secondly, depending on the infinite norm of the quaternion-valued vector, the Lyapunov functional is formulated without time delay. Besides, the global polynomial synchronization of the whole quaternion-valued system is investigated by constructing a suitable nonlinear controller and exploiting the properties of quaternions and related inequalities. Ultimately, an illustrative numerical example is included to substantiate the conclusion. © The Author(s), under exclusive licence to Springer Nature B.V. 2024.

This paper deals with the quasi-projective synchronization problem of delayed stochastic quaternion fuzzy cellular neural networks with mismatch parameters. Although the parameter mismatch of the drive-response system increases the computational complexity of the article, it is of practical significance to consider the existence of deviations between the two systems. The method of this article is to design an appropriate controller and construct Lyapunov functional and stochastic analysis theory based on the Itô formula in the quaternion domain. We adopt the non-decomposable method of quaternion FCNN, which preserves the original data and reduces computational effort. We obtain sufficient conditions for quasi-projective synchronization of the considered random quaternion numerical FCNNs with mismatched parameters. Additionally, we estimate the error bounds of quasi-projective synchronization and then carry out a numerical example to verify their validity. Our results are novel even if the considered neural networks degenerate into real-valued or complex-valued neural networks. This article provides a good research idea for studying the quasi-projective synchronization problem of random quaternion numerical FCNN with time delay and has obtained good results. The method in this article can also be used to study the quasi-projective synchronization of a Clifford-valued neural network. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.