Real-valued Systems Research Articles

Finite-time synchronization of a class of nonlinear complex-valued networks with time-varying delays

This paper focuses on the problem of finite-time synchronization for a class of nonlinear complex-valued networks with time-varying delays. A novel model is proposed to generalize the previous works. The nonlinear couplings, time-varying node delay and multiple time-varying coupling delays are considered; all the variables and parameters in this model are considered as complex values. By decomposing the network model to real and imaginary parts, an equivalent real-valued system is built. Then, by utilizing the linear feedback controllers, synchronization is achieved in a given finite time rather than merely asymptotically synchronize. Based on the finite-time stability theory, Lyapunov functional method and some mathematical analysis techniques, sufficient conditions are strictly established to ensure the finite-time synchronization, the main results can be also extended to the networks without node delay or coupling delays. At last, examples are provided in details to demonstrate our results.

Delay independent robust stability analysis of delayed fractional quaternion-valued leaky integrator echo state neural networks with QUAD condition

This paper studies the stability of fractional order (FO) continuous-time quaternion-valued Leaky Integrator Echo State Neural Networks (NN). The delay independent robust stability of NN with QUAD vector field activation functions under time varying delays is derived. The analysis follows the FO Lyapunov theorem while decomposing the NN into four real-valued systems. The existence and uniqueness of the equilibrium point are also verified by means of a contraction map on the decomposed NN. The new approach is tested in the stability analysis of the FO quaternion-valued Echo State NN whitout time delays, complex-valued Echo State and quaternion-/complex-valued Hopfield NN with/without time delays. Two numerical examples demonstrate the feasibility of the proposed method.

Anti-periodic Synchronization of Quaternion-valued Generalized Cellular Neural Networks with Time-varying Delays and Impulsive Effects

In this paper, a class of quaternion-valued generalized cellular neural networks (QVGCNNs) with time-varying delays and impulsive effects is considered. Firstly, by constructing an appropriate Lyapunov function and applying inequality techniques, some sufficient conditions on the existence of anti-periodic solutions are established. Then, the global exponential anti-periodic synchronization of delayed QVGCNNs with impulsive effects and anti-periodic coefficients is investigated by designing a novel nonlinear state-feedback controller and constructing suitable Lyapunov functions. Our results are completely new even if the considered quaternion-valued systems degenerated into real-valued or complex-valued systems. Finally, two numerical examples are given to illustrate the feasibility and effectiveness of the obtained results.

Periodic solutions for quaternion-valued fuzzy cellular neural networks with time-varying delays

In this paper, quaternion-valued fuzzy cellular neural networks (QVFCNNs) with time-varying delays are considered. First, we decompose QVFCNNs into their equivalent real-valued systems according to Hamilton’s multiplication rules. Then, we establish the existence and global exponential stability of periodic solutions of QVFCNNs by using the Schauder fixed point theorem and by constructing an appropriate Lyapunov function. Our results are completely new and supplementary to the known results. Finally, we give a numerical example to illustrate the effectiveness of our results.

Existence and Exponential Stability of Anti-Periodic Solutions for Inertial Quaternion-Valued High-Order Hopfield Neural Networks With State-Dependent Delays

This paper deals with a class of inertial quaternion-valued high-order Hopfield neural networks with state-dependent delays. Without decomposing the considered neural networks into real-valued systems, based on a continuation theorem of coincidence degree theory and the Wirtinger inequality, the existence of anti-periodic solutions of the networks is established. By constructing a suitable Lyapunov function, the global exponential stability of anti-periodic solutions of the networks is obtained. Finally, a numerical example is given to show the feasibility of our results.

Existence and Global Exponential Stability of Pseudo Almost Periodic Solution for Clifford- Valued Neutral High-Order Hopfield Neural Networks With Leakage Delays

In this paper, we are concerned with a class of Clifford-valued neutral high-order Hopfield neural networks with leakage delays. Although the multiplication of Clifford numbers does not satisfy the commutativity, which brings great difficulties to the study of Clifford-valued systems, we have found a method that does not decompose Clifford-valued systems into real-valued systems to study the existence and global exponential stability of pseudo-almost periodic solutions of this class of neural networks. Our results are completely new. Finally, two examples are given to illustrate the effectiveness and feasibility of our main results.

Globally Asymptotic Almost Automorphic Synchronization of Clifford-Valued RNNs With Delays

In this paper, we consider a class of Clifford-valued recurrent neural networks (RNNs) with discrete and infinitely distributed delays. In order to overcome the non-commutativity of the multiplication of Clifford numbers, we first transform the Clifford-valued RNNs with discrete and infinitely distributed delays into real-valued systems based on the multiplication rule and properties of Clifford numbers. Then, we establish the existence and uniqueness of almost automorphic solutions for the neural networks under consideration by applying the contraction mapping principle, and we obtain the globally asymptotic almost automorphic synchronization of the neural networks under consideration by designing a novel state feedback controller and constructing an appropriate Lyapunov function. Finally, we present a numerical example to illustrate the feasibility of the results of this paper.

Finite-Time Synchronization of Complex-Valued Neural Networks with Multiple Time-Varying Delays and Infinite Distributed Delays

This paper investigates finite-time synchronization of complexed-valued neural networks with multiple time-varying delays and infinite distributed delays. By separating the complex-valued neural networks into the real and the imaginary parts, the corresponding equivalent real-valued systems are obtained. Some sufficient conditions are derived for finite-time synchronization of the drive-response system based on the new Lyapunov–Krasovskii function and the new analysis techniques. Numerical examples demonstrate the effectiveness of the theoretical results.

Stability analysis of fractional Quaternion-Valued Leaky Integrator Echo State Neural Networks with multiple time-varying delays

This paper studies the stability problem of fractional order continuous-time Quaternion-Valued Leaky Integrator Echo State Neural Networks (NN) with multiple time-varying delays. The NN is decomposed into four real-valued systems without considering the non-commutativity of quaternion multiplication resulting in Hamilton rules. Then, the delay-independent stability analysis of the NN equilibrium point with Lipschitz continuous activation function is derived in the real space. The existence and uniqueness of the equilibrium point are also given using the contraction map in the real space. Simulations results show the feasibility of the method.

Multistability analysis of delayed quaternion-valued neural networks with nonmonotonic piecewise nonlinear activation functions

This paper deals with the multistability problem of the quaternion-valued neural networks (QVNNs) with nonmonotonic piecewise nonlinear activation functions and unbounded time-varying delays. By virtue of the non-commutativity of quaternion multiplication resulting from Hamilton rules, the QVNNs can be separated into four real-valued systems. By using the fixed point theorem and other analytical tools, some novel algebraic criteria are established to guarantee that the QVNNs can have 54n equilibrium points, 34n of which are locally μ-stable. Some criteria that guarantee the multiple exponential stability, multiple power stability, multiple log-stability, multiple log–log-stability are also derived as special cases. The obtained results reveal that the introduced QVNNs in this paper can have larger storage capacity than the complex-valued ones. Finally, one numerical example is presented to clarify the validity of the theoretical results.

Stability and Hopf bifurcation analysis of a complex-valued Wilson–Cowan neural network with time delay

Stability and Hopf bifurcation of a class of delayed complex-valued neural networks are investigated in this paper. First, using proper translations and coordinate transformations, we decompose the activation functions and connection weights into their real and imaginary parts, so as to construct an equivalent real-valued system. Then, the sufficient conditions for Hopf bifurcation and its directions are provided through normal form theory and central manifold theorem. In the end, some numerical simulations are given to illustrate the correctness of the results.

Almost periodic solutions for quaternion-valued shunting inhibitory cellular neural networks of neutral type with time delays in the leakage term

ABSTRACTIn this paper, we propose a class of quaternion-valued shunting inhibitory cellular neural networks of neutral type with time delays in the leakage term. In order to overcome the difficulty of the non-commutativity of quaternion multiplication, we first decompose the system under consideration into four real-valued systems. Then by using the exponential dichotomy theory of linear differential equations and Banach's fixed point theorem, we establish sufficient conditions to ensure the existence and global exponential stability of almost periodic solutions for this class of neural networks. Finally, a numerical example is given to demonstrate the effectiveness of the obtained result.

Stability analysis for delayed quaternion-valued neural networks via nonlinear measure approach

In this paper, the existence and stability analysis of the quaternion-valued neural networks (QVNNs) with time delay are considered. Firstly, the QVNNs are equivalently transformed into four real-valued systems. Then, based on the Lyapunov theory, nonlinear measure approach, and inequality technique, some sufficient criteria are derived to ensure the existence and uniqueness of the equilibrium point as well as global stability of delayed QVNNs. In addition, the provided criteria are presented in the form of linear matrix inequality (LMI), which can be easily checked by LMI toolbox in MATLAB. Finally, two simulation examples are demonstrated to verify the effectiveness of obtained results. Moreover, the less conservatism of the obtained results is also showed by two comparison examples.

Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks with mixed delays via state-feedback control.

This paper studies the drive-response synchronization for quaternion-valued shunting inhibitory cellular neural networks (QVSICNNs) with mixed delays. First, QVSICNN is decomposed into an equivalent real-valued system in order to avoid the non-commutativity of the multiplicity. Then, the existence of almost periodic solutions is obtained based on the Banach fixed point theorem. An novel state-feedback controller is designed to ensure the global exponential almost periodic synchronization. At the end of the paper, an example is given to illustrate the effectiveness of the obtained results.

Anti-periodic Solutions for Quaternion-Valued High-Order Hopfield Neural Networks with Time-Varying Delays

In this paper, quaternion-valued high-order Hopfield neural networks (QVHHNNs) with time-varying delays are considered. Theoretically, a QVHHNN can be separated into four real-valued systems, forming an equivalent real-valued system. By using a novel continuation theorem of coincidence degree theory and constructing an appropriate Lyapunov function, some sufficient conditions are derived to guarantee the existence and global exponential stability of anti-periodic solutions for QVHHNN, which are new and complement previously known results.

Almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks with time-varying and distributed delays

AbstractIn this paper, we consider the problem of the almost automorphic synchronization of quaternion-valued high-order Hopfield neural networks (QVHHNNs) with time-varying and distributed delays. Firstly, to avoid the non-commutativity of quaternion multiplication, we decompose QVHHNNs into an equivalent real-valued system. Secondly, we use the Banach fixed point theorem to obtain the existence of almost automorphic solutions of QVHHNNs. Thirdly, by designing a novel state-feedback controller and constructing suitable Lyapunov functions, we obtain that the drive-response structure of QVHHNNs with almost automorphic coefficients can realize the exponential synchronization. Our results are completely new. Finally, a numerical example is given to illustrate the feasibility of our results.

Quasi-projective synchronization of fractional-order complex-valued recurrent neural networks

In this paper, without separating the complex-valued neural networks into two real-valued systems, the quasi-projective synchronization of fractional-order complex-valued neural networks is investigated. First, two new fractional-order inequalities are established by using the theory of complex functions, Laplace transform and Mittag-Leffler functions, which generalize traditional inequalities with the first-order derivative in the real domain. Additionally, different from hybrid control schemes given in the previous work concerning the projective synchronization, a simple and linear control strategy is designed in this paper and several criteria are derived to ensure quasi-projective synchronization of the complex-valued neural networks with fractional-order based on the established fractional-order inequalities and the theory of complex functions. Moreover, the error bounds of quasi-projective synchronization are estimated. Especially, some conditions are also presented for the Mittag-Leffler synchronization of the addressed neural networks. Finally, some numerical examples with simulations are provided to show the effectiveness of the derived theoretical results.

Delay-dependent dynamical analysis of complex-valued memristive neural networks: Continuous-time and discrete-time cases

This paper considers the delay-dependent stability of memristive complex-valued neural networks (MCVNNs). A novel linear mapping function is presented to transform the complex-valued system into the real-valued system. Under such mapping function, both continuous-time and discrete-time MCVNNs are analyzed in this paper. Firstly, when activation functions are continuous but not Lipschitz continuous, an extended matrix inequality is proved to ensure the stability of continuous-time MCVNNs. Furthermore, if activation functions are discontinuous, a discontinuous adaptive controller is designed to acquire its stability by applying Lyapunov–Krasovskii functionals. Secondly, compared with techniques in continuous-time MCVNNs, the Halanay-type inequality and comparison principle are firstly used to exploit the dynamical behaviors of discrete-time MCVNNs. Finally, the effectiveness of theoretical results is illustrated through numerical examples.

Finite-Time Synchronization for a Class of Fully Complex-Valued Networks With Coupling Delay

In this paper, the problem of synchronization for a class of fully complex-valued networks with coupling delay is investigated by the linear feedback control in a finite time. The state variables, system function, inner coupling matrix, and outer coupling matrix in this model are set as complex values. By decomposing the complex-valued network into real and imaginary parts, an equivalent real-valued system is established. Based on the finite-time stability theory and Lyapunov functional method, linear feedback controllers are constructed to guarantee the synchronization for the complex-valued networks with or without coupling delay in a desired finite time. Then, some sufficient conditions for assurance of finite-time synchronization are given by strict proof. Finally, simulation examples are provided to verify our results.

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

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.