Quaternion-valued Neural Networks Research Articles

Coexistence of locally multistable equilibrium points for [formula omitted]-neuron delayed quaternion-valued neural networks with continuous piecewise nonlinear activation functions

This article investigates the coexistence of multistable equilibrium points (EPs) for n-neuron delayed quaternion-valued neural networks (DQVNNs) in which the continuous and piecewise nonlinear functions are used as the activation functions. Using the state space decomposition technique, Brouwer’s fixed point theorem, and Lagrange’s mean value theorem, some novel sufficient conditions have been derived to ensure the coexistence of 54n EPs in which 34n of them are locally stable. Furthermore, in this scientific contribution, positively invariant sets have been estimated which actually turned out to be quite challenging for the activation functions of the designed DQVNNs. Finally, some comparisons and convincing simulations with the application to associative memory of DQVNNs are given to verify the theoretical results at the end of this article.

Synchronization of fractional-order quaternion-valued neural networks with image encryption via event-triggered impulsive control

The finite-time synchronization and impulsive synchronization of fractional-order quaternion-valued neural networks (FOQVNNs) with delay are investigated in this research. To begin, instead of employing a decomposition method, a Lyapunov direct approach is proposed to achieve the finite-time synchronization of FOQVNNs using event-triggered control based on state errors. The settling time is then estimated using a unique fractional differential inequality. The event-triggered impulsive controller, which updates only during triggering instants, is proposed to be more inexpensive and efficient. Then, utilizing fractional Lyapunov theory and the comparison principle, several novel quaternion-valued linear matrix inequalities criteria are defined to assure the impulsive synchronization of the delayed FOQVNNs. Furthermore, the positive lower bound of the inter-event time is obtained to exclude the Zeno behavior. As proof of the feasibility and validity of these theoretical findings obtained, numerical examples are provided. At last, the use of the event-triggered control technique for FOQVNNs in image encryption and decryption is demonstrated and validated.

Fixed/Preassigned-Time Synchronization of Fuzzy Memristive Fully Quaternion-Valued Neural Networks Based on Event-Triggered Control

In this paper, the fixed-time and preassigned-time synchronization issues of fully quaternion-valued fuzzy memristive neural networks are studied based on the dynamic event-triggered control mechanism. Initially, the fuzzy rules are defined within the quaternion domain and the relevant properties are established through rigorous analysis. Subsequently, to conserve resources and enhance the efficiency of the controller, a kind of dynamic event-triggered control mechanism is introduced for the fuzzy memristive neural networks. Based on the non-separation analysis, fixed-time and preassigned-time synchronization criteria are presented and the Zeno phenomenon under the event-triggered mechanism is excluded successfully. Finally, the effectiveness of the theoretical results is verified through numerical simulations.

Exponential synchronization of a class of quaternion-valued neural network with time-varying delays: A Matrix Measure Approach

This article investigates the exponential synchronization (ES) of quaternion-valued neural networks (QVNNs) with time-varying delays. Global ES has been studied using matrix measure techniques. Due to the incompatibility of quaternion multiplication, the QVNN system is first separated into four real-valued systems. Then ES is established in the master-response systems of QVNN by constructing a state feedback controller. The stability criteria for the error system of QVNN is then provided using the existing lemmas and the Halanay inequality. Additionally, it is no longer assumed that the activation functions are differentiable, which make the analytical process more concise. The viability of our findings are demonstrated using two numerical examples. The present theoretical findings have been confirmed in the first case, while the second example includes the use of QVNNs connected to associative memory to demonstrate their ability to restore real color image patterns properly.

Stability and Synchronization of Delayed Quaternion-Valued Neural Networks under Multi-Disturbances

This paper discusses a type of mixed-delay quaternion-valued neural networks (QVNNs) under impulsive and stochastic disturbances. The considered QVNNs model are treated as a whole, rather than as complex-valued neural networks (NNs) or four real-valued NNs. Using the vector Lyapunov function method, some criteria are provided for securing the mean-square exponential stability of the mixed-delay QVNNs under impulsive and stochastic disturbances. Furthermore, a type of chaotic QVNNs under stochastic and impulsive disturbances is considered using a previously established stability analysis method. After the completion of designing the linear feedback control law, some sufficient conditions are obtained using the vector Lyapunov function method for determining the mean-square exponential synchronization of drive–response systems. Finally, two examples are provided to demonstrate the correctness and feasibility of the main findings and one example is provided to validate the use of QVNNs for image associative memory.

Projective Synchronization of Inertial Quaternion-Valued Neural Networks via Non-reduced Order Approach

In this article, the issue on projective synchronization of delayed inertial quaternion-valued neural networks (IQVNNs) is investigated. Different from most existing literature, we adopt the non-reduced order approach to deal with IQVNNs described by second order differential equations. By introducing a novel Lyapunov functional, several sufficient criteria are presented in component form to ensure the projective synchronization between master–slave systems. A numerical experiment demonstrates the feasibility of control strategy as well as the correctness of theoretical results.

Fixed-Time Pinning Common Synchronization and Adaptive Synchronization for Delayed Quaternion-Valued Neural Networks.

This article focuses on the fixed-time pinning common synchronization and adaptive synchronization for quaternion-valued neural networks with time-varying delays. First, to reduce transmission burdens and limit convergence time, a pinning controller which only controls partial nodes directly rather than the entire nodes is proposed based on fixed-time control theory. Then, by Lyapunov function approach and some inequalities techniques, fixed-time common synchronization criterion is established. Second, further to realize the self-regulation function of pinning controller, an adaptive pinning controller which can adjust automatically the control gains is developed, the desired fixed-time adaptive synchronization is achieved for the considered system, and the corresponding criterion is also derived. Finally, the availability of these results is tested by simulation example.

Synchronization analysis of delayed quaternion-valued memristor-based neural networks by a direct analytical approach

<abstract><p>This issue discusses the asymptotic synchronization and the exponential synchronization for memristor-based quaternion-valued neural networks under the time-varying delays. Some criteria for synchronization of the memristor-based quaternion-valued neural networks are given by exploiting the set-valued theory, the differential inclusion theory, some analytic techniques, as well as constructing novel controllers, It is worth noting that the synchronization problem about the memristor-based quaternion-valued neural networks were studied by the direct analysis method in this paper. Finally, the main theoretical results were verified by numerical simulations.</p></abstract>

Adaptive impulsive synchronization for fractional-order quaternion-valued neural networks with different structures and unknown parameters

The synchronization issue between fractional-order quaternion-valued neural networks with different structures is more common than that with the same structure in the real world. This paper addresses the Mittag-Leffler stability and adaptive impulsive synchronization between two fractional-order quaternion-valued neural networks with different structures. We also propose two assumptions to analyze the parameters of the nonlinear activation functions. Based on the Mittag-Leffler stability and fractional-order Lyapunov direct method, an impulsive controller scheme is designed to ensure the global stability of fractional-order time-delay quaternion-valued neural networks. We also analyzed the situation when the self-feedback matrix and connection matrix of quaternion-valued fractional-order neural networks with different structures are unknown and designed adaptive impulsive controllers to achieve synchronization. Finally, we verified the feasibility and effectiveness of the proposed theorem through numerical simulation experiments.

Lagrange stability criteria for hypercomplex neural networks with time varying delays

This article deals with the Lagrange stability (LS) of hypercomplex neural networks (HCNNs) with time-varying delays. To overcome the non-commutativity and non-associativity of HCNNs, the HCNNs are separated into equivalent n+1 real-valued neural networks (RVNNs). Then, by Lyapunov theory, LS and the globally attractive exponential set are obtained. Since there are no assumptions about the number of equilibria, these results can also be used to evaluate monostable, multistable, and higher neural networks. Since, complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and octonion-valued neural networks (OVNNs), etc., are the particular cases of the HCNNs, therefore our results are more general. Finally, three numerical examples for CVNNs, QVNNs, and OVNNs are provided and analyzed to validate the obtained results.

Mean-square exponential input-to-state stability for stochastic neutral-type quaternion-valued neural networks via Itô’s formula of quaternion version

The input-to-state stability of stochastic quaternion-valued neural networks with neutral delays is explored in this study. Unlike previous researches, this study treats the neural network as a unified entity, rather than isolating and examining the real and imaginary components separately. Through the construction of a Lyapunov functional and the use of the Itô’s formula of quaternion version, a sufficient criterion for achieving mean-square exponential input-to-state stability is obtained for stochastic quaternion-valued neural networks with neutral delays. Three numerical instances are presented to validate the reliability of the obtained conditions.

Results on finite time passivity of fractional-order quaternion-valued neural networks with time delay via linear matrix inequalities

Results on finite time passivity of fractional-order quaternion-valued neural networks with time delay via linear matrix inequalities

Fixed/Preassigned-Time Synchronization of Fully Quaternion-Valued Cohen–Grossberg Neural Networks with Generalized Time Delay

This article is concerned with fixed-time synchronization and preassigned-time synchronization of Cohen–Grossberg quaternion-valued neural networks with discontinuous activation functions and generalized time-varying delays. Firstly, a dynamic model of Cohen–Grossberg neural networks is introduced in the quaternion field, where the time delay successfully integrates discrete-time delay and proportional delay. Secondly, two types of discontinuous controllers employing the quaternion-valued signum function are designed. Without utilizing the conventional separation technique, by developing a direct analytical approach and using the theory of non-smooth analysis, several adequate criteria are derived to achieve fixed-time synchronization of Cohen–Grossberg neural networks and some more precise convergence times are estimated. To cater to practical requirements, preassigned-time synchronization is also addressed, which shows that the drive-slave networks reach synchronization within a specified time. Finally, two numerical simulations are presented to validate the effectiveness of the designed controllers and criteria.

Asymptotic and Mittag–Leffler Synchronization of Fractional-Order Octonion-Valued Neural Networks with Neutral-Type and Mixed Delays

Very recently, a different generalization of real-valued neural networks (RVNNs) to multidimensional domains beside the complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and Clifford-valued neural networks (ClVNNs) has appeared, namely octonion-valued neural networks (OVNNs), which are not a subset of ClVNNs. They are defined on the octonion algebra, which is an 8D algebra over the reals, and is also the only other normed division algebra that can be defined over the reals beside the complex and quaternion algebras. On the other hand, fractional-order neural networks (FONNs) have also been very intensively researched in the recent past. Thus, the present work combines FONNs and OVNNs and puts forward a fractional-order octonion-valued neural network (FOOVNN) with neutral-type, time-varying, and distributed delays, a very general model not yet discussed in the literature, to our awareness. Sufficient criteria expressed as linear matrix inequalities (LMIs) and algebraic inequalities are deduced, which ensure the asymptotic and Mittag–Leffler synchronization properties of the proposed model by decomposing the OVNN system of equations into a real-valued one, in order to avoid the non-associativity problem of the octonion algebra. To accomplish synchronization, we use two different state feedback controllers, two different types of Lyapunov-like functionals in conjunction with two Halanay-type lemmas for FONNs, the free-weighting matrix method, a classical lemma, and Young’s inequality. The four theorems presented in the paper are each illustrated by a numerical example.

Asymptotically Almost Automorphic Solutions for Impulsive Quaternion-Valued Neural Networks with Mixed Delays

Asymptotically Almost Automorphic Solutions for Impulsive Quaternion-Valued Neural Networks with Mixed Delays

Almost periodic quasi-projective synchronization of delayed fractional-order quaternion-valued neural networks

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.

Nonseparation Approach to General-Decay Synchronization of Quaternion-Valued Neural Networks with Mixed Time Delays

In this paper, the general-decay synchronization issue of a class of quaternion-valued neural networks with mixed time delays is investigated. Firstly, unlike some previous works where the quaternion-valued model is separated into four real-valued networks or two complex-valued networks, we consider the mixed-delayed quaternion-valued neural network model as a whole and introduce a novel nonlinear feedback controller for the corresponding response system. Then, by introducing a suitable Lyapunov–Krasovskii functional and employing a novel inequality technique, some easily verifiable sufficient conditions are obtained to ensure the general-decay synchronization for the considered drive-response networks. Finally, the feasibility of the established theoretical results is verified by carrying out Matlab numerical simulations.

Anti-synchronization of inertial neural networks with quaternion-valued and unbounded delays: Non-reduction and non-separation approach

This article investigates the anti-synchronization (AS) problem for quaternion-valued inertial neural networks (QVINNs) with unbounded time delays. Using two different types of control strategies (feedback and adaptive controllers) and Lyapunov theory, various conditions are established to guarantee the AS of QVINNs. Most of the results available for QVINNs are based on the variable substitution approach, which reduces the order of the original second-order system into the first-order system. The quaternion-valued networks can be separated into four equivalent real-valued neural networks (RVNNs), which have made the process more complicated. However, in the present study, the authors deal with the non-reduction order method and non-separation approach for QVINNs, making the analysis approach more concise and easier to deal with inertial neural networks (INNs). Here, the QVINNs have unbounded time-varying delays since neurons’ actions are related to their previous states. Hence the approach adopted here is more realistic and easy to deal with the QVINNs with unbounded time delays, which has not yet been considered by any researcher. Finally, two numerical examples are provided to demonstrate the efficiency and effectiveness of the proposed approach. In the first example, the present theoretical results have been validated, whereas the second example contains the application of quaternion-valued neural networks (QVNNs) related to associative memory to demonstrate their capacity to restore true color image patterns accurately.

Adaptive Global Synchronization for a Class of Quaternion-Valued Cohen-Grossberg Neural Networks with Known or Unknown Parameters

In this paper, the adaptive synchronization problem of quaternion-valued Cohen–Grossberg neural networks (QVCGNNs), with and without known parameters, is investigated. On the basis of constructing an appropriate Lyapunov function, and utilizing parameter identification theory and decomposition methods, two effective adaptive feedback schemes are proposed, to guarantee the realization of global synchronization of CGQVNNs. The control gain of the above schemes can be obtained using the Matlab LMI toolbox. The theoretical results presented in this work enrich the literature exploring the adaptive synchronization problem of quaternion-valued neural networks (QVNNs). Finally, the reliability of the theoretical schemes derived in this work is shown in two interesting numerical examples.

On the Existence of the Exact Solution of Quaternion-Valued Neural Networks Based on a Sequence of Approximate Solutions.

In many practical applications, it is difficult or impossible to obtain the exact solution of the mathematical model due to the limitations of solving methods and the complexity of the neural network itself. A natural problem is given as follows: does the exact solution of quaternion-valued neural networks (QVNNs) exist when successively improved approximate solutions can be obtained? Fortunately, the Hyers-Ulam stability happens to be one of the important means to deal with this problem. In this article, the issue of Hyers-Ulam stability of QVNNs with time-varying delays is addressed. First, inspired by the Hyers-Ulam stability of general functional equations, the concept of the Hyers-Ulam stability of QVNNs is proposed along with the QVNNs model. Then, by utilizing the successive approximation method, both delay-dependent and delay-independent Hyers-Ulam stability criteria are obtained to ensure the Hyers-Ulam stability of the QVNNs considered. Finally, a simulation example is given to verify the effectiveness of the derived results.