Stability Analysis of Quaternion-Valued Neural Networks: Decomposition and Direct Approaches

Abstract

In this paper, we investigate the global stability of quaternion-valued neural networks (QVNNs) with time-varying delays. On one hand, in order to avoid the noncommutativity of quaternion multiplication, the QVNN is decomposed into four real-valued systems based on Hamilton rules: $ij=-ji=k,~jk=-kj=i$ , $ki=-ik=j$ , $i^{2}=j^{2}=k^{2}=ijk=-1$ . With the Lyapunov function method, some criteria are, respectively, presented to ensure the global $\mu $ -stability and power stability of the delayed QVNN. On the other hand, by considering the noncommutativity of quaternion multiplication and time-varying delays, the QVNN is investigated directly by the techniques of the Lyapunov-Krasovskii functional and the linear matrix inequality (LMI) where quaternion self-conjugate matrices and quaternion positive definite matrices are used. Some new sufficient conditions in the form of quaternion-valued LMI are, respectively, established for the global $\mu $ -stability and exponential stability of the considered QVNN. Besides, some assumptions are presented for the two different methods, which can help to choose quaternion-valued activation functions. Finally, two numerical examples are given to show the feasibility and the effectiveness of the main results.

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

“…Remark In Theorem , we do not require the activation functions to be bounded. But, in recent works,() the activation functions are assumed to be bounded. If we assume that the activation functions are bounded, then we have the following verifiable results.…”

“…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.…”

“…Our results are new even when system is degenerated into a real‐valued system, and our method is different from those used in other studies. ()…”

Parameter-range-dependent robust stability conditions for quaternion-valued neural networks with time delays

“…Compared to the general systems, the dynamics of neural networks with delays becomes more complicated and it may lead to volatility, instability and even chaos. In fact, some authors also have considered the existence, uniqueness and global stability of the equilibrium point for delayed quaternion-valued neural networks [14,[26][27][28][29][30][31][32][33][34].…”

Stability Analysis of Quaternion-Valued Neutral-Type Neural Networks with Time-Varying Delay

“…Until now, quaternion algebra has been successfully applied to communications problems and signal processing, such as color image processing [24] and wind forecasting [25]. Since then, numerous scholars have produced many excellent results in the field of QVNNs (see, e.g., [26][27][28][29] and literature referenced therein). QVNN was changed into two complex-valued systems by using a simple transformation, and [26] reduced the complexity of computation generated by the non-commutativity of quaternion multiplication.…”