Matrix Measure Approach Research Articles

Asymptotic synchronisation for coupled heterogeneous complex networks via L' Hoptial' rule

This paper considers the global asymptotic synchronisation (GAS) for the drive-response coupled heterogeneous complex networks (DRCHCNS). By using the different inequality techniques (DIT), L' Hoptial' rule and limit algorithms, three criteria to assure the GAS for the DRCHCNS are derived. Up to now, the results on GAS for DRCHCNS have been obtained usually by applying linear matrix inequality (LMI) and matrix measure approach (MMA). But, in this paper, without applying above approaches, we establish the GAS criteria for the DRCHCNS by using L' Hoptial' rule and limit algorithms. So, our study approach for the complex networks (CNS) extends the study fields of synchronisation of the CNS.

Synchronization of fractional order time delayed neural networks using matrix measure approach

Synchronization of fractional order time delayed neural networks using matrix measure approach

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.

Projective quasi-synchronization of complex-valued recurrent neural networks with proportional delay and mismatched parameters via matrix measure approach

This article is concerned with the projective quasi-synchronization of non-identical complex-valued recurrent neural networks (CVRNNs) with proportional delays and mismatched parameters. The nonlinear Lipschitz activation functions under Lyapunov stability criteria and matrix measure approach have been employed. By designing a suitable controller, a sufficient condition for projective quasi-synchronization criteria of the non-identical CVRNNs model has been derived through the proper description of the matrix measure approach. A significant result for the CVRNNs with mismatched parameters and proportional delays is provided. Finally, a numerical simulation result is given to validate the usefulness and persistence of the theoretical results. The results for different particular cases are displayed graphically.

Quasi-projective synchronization of inertial complex-valued recurrent neural networks with mixed time-varying delay and mismatched parameters

This article investigates the quasi-projective synchronization of inertial complex-valued recurrent neural networks (ICVRNNs) with mixed time-varying delay and mismatched parameters. By using an appropriate variable transformation, the order of the ICVRNN differential system is reduced, and then, by applying the real decomposition method, it is separated into real and imaginary components. The matrix measure approach with the nonlinear Lipschitz activation functions is employed in the ICVRNN model. Through the proper description of the matrix measure approach, some sufficient conditions have been derived for the quasi-projective synchronization criteria of the considered model through designing a suitable controller. Here, some significant results have been provided for the ICVRNNs with mismatched parameters and mixed time-varying delay. Finally, two numerical simulations are discussed to validate the feasibility and persistence of our obtained results with few conditions.

Stability of inertial delayed neural networks with stochastic delayed impulses via matrix measure method

This paper is mainly concerned with the stability of inertial delayed neural networks (IDNNs) with stochastic delayed impulses. In contrast with previous related works, the stochastic impulsive intensity and density, and impulsive delays are considered simultaneously. Firstly, by appropriate variable substitution, the considered IDNNs model is converted to a first order differential equation with stochastic delayed impulsive effect. Secondly, by using the matrix measure approach and some stochastic analysis techniques, the stability criteria of IDNNs with stochastic delayed impulses are presented. Finally, numerical examples are provided to illustrate the obtained theoretical results.

Matrix Measure Approach for Stability and Synchronization of Complex-Valued Neural Networks with Deviating Argument

This paper concentrates on global exponential stability and synchronization for complex-valued neural networks (CVNNs) with deviating argument by matrix measure approach. The Lyapunov function is no longer required, and some sufficient conditions are firstly obtained to ascertain the addressed system to be exponentially stable under different activation functions. Moreover, after designing a suitable controller, the synchronization of two complex-valued coupled neural networks is realized, and the derived condition is easy to be confirmed. Finally, some numerical examples are given to demonstrate the superiority and feasibility of the presented theoretical analysis and results.

Stability of delayed inertial neural networks on time scales: A unified matrix-measure approach

This note introduces a unified matrix-measure concept to study the stability of a class of inertial neural networks with bounded time delays on time scales. The novel matrix-measure concept unifies the classic matrix-measure and the generalized matrix-measure concept. One sufficient global exponential stability criterion is obtained based on this key matrix-measure and no Lyapunov function is required. To make the stability performance better, another stability criterion in which more detailed information is involved has been acquired. The theoretical results in this note contain and extend some existing continuous-time and discrete-time works. A numerical example is given to show the validity of the results.

Exponential stability of inertial BAM neural network with time-varying impulses and mixed time-varying delays via matrix measure approach

This article is concerned with the effects of time-varying impulses on exponential stability to a unique equilibrium point of inertial Bidirectional Associative Memories (BAM) neural network with mixed time-varying delays. A suitable variable transformation is chosen to transform the original system into a system of first order differential equations. The concept of homeomorphism has been implemented to find a distributed delay-dependent sufficient condition which assures that the system has a unique equilibrium point. In order to study the impulsive effects on stability problems, a time-varying impulses, including stabilizing and destabilizing impulses, are considered with the transformed system. Based on the matrix measure approach and an extended impulsive differential inequality for a time-varying delayed system, we have derived sufficient criteria in matrix measure form which ensure the exponential stability of the system towards an equilibrium point for two classes of activation functions. Further, different convergence rates of the system’s trajectory have been discussed for the cases of time-varying stabilizing and destabilizing impulses using the concept of an average impulsive interval. Finally, the efficiency of the theoretical results has been illustrated by providing two numerical examples.

Weak, modified and function projective synchronization of Cohen–Grossberg neural networks with mixed time-varying delays and parameter mismatch via matrix measure approach

This paper is concerned with the modified function projective synchronization of Cohen–Grossberg neural networks systems with parameter mismatch and mixed time-varying delays. Due to the existence of parameter mismatch between the drive and slave systems, complete modified function projective synchronization is not possible to achieve. So a new concept, viz., weak modified function projective synchronization, is discussed up to a small error bound. Several generic criteria are derived to show weak modified function projective synchronization between the systems. The estimation of error bound is done using matrix measure and Halanay inequality. Simulation results are proposed graphically for different particular cases to show the synchronization between parameter-mismatched systems, which validate the effectiveness of our proposed theoretical results.

Stability and synchronization for impulsive Markovian switching CVNNs: matrix measure approach

This paper devotes to the global exponential stability and synchronization problem for impulsive Markovian switching complex-valued neural networks (CVNNs) with time-varying delays. Based on the matrix measure approach and the impulsive differential inequality, some sufficient conditions are firstly derived to guarantee the impulsive network to be exponentially stable, where the exponential convergence rate is explicitly estimated. After that, the synchronization problem is investigated for the coupled impulsive complex-valued networks, and the obtained criteria are easy to be verified and implemented in practice. Finally, two examples are presented to illustrate effectiveness of the proposed theoretical results.

Global Power Stability of Neural Networks with Impulses and Proportional Delays

By establishing two new impulsive delay differential inequalities from impulsive perturbation and impulsive control point of view, respectively, constructing some Lyapunov functionals, and employing the matrix measure approach, some novel and sufficient conditions are obtained to guarantee global power stability of neural networks with impulses and proportional delays. The obtained stability criteria are dependent on impulses and the proportional delay factor so that it is convenient to derive some feasible impulsive control laws according to the proportional delay factor allowed by such neural networks. It is shown that impulses can act as stabilizers to globally power stabilize an unstable neural network with proportional delay based on suitable impulsive control laws. Moreover, the power convergence rate can be estimated and obtained by simple computation. Three numerical examples are given to illustrate the effectiveness and advantages of the results obtained.

Matrix measure approach to Lyapunov-type inequalities for linear Hamiltonian systems with impulse effect

We present new Lyapunov-type inequalities for Hamiltonian systems, consisting of 2n-first-order linear impulsive differential equations, by making use of matrix measure approach. The matrix measure estimates of fundamental matrices of linear impulsive systems are crucial in obtaining sharp inequalities. To illustrate usefulness of the inequalities we have derived new disconjugacy criteria for Hamiltonian systems under impulse effect and obtained new lower bound estimates for eigenvalues of impulsive eigenvalue problems.

Stability of synchronous state in networks of chaotic maps by matrix measure approach

Stability of synchronous state in networks of chaotic maps by matrix measure approach

Stability of bidirectional associative memory neural networks with Markov switching via ergodic method and the law of large numbers

This paper devotes to stability analysis of continuous time and discrete time bidirectional associative memory (BAM) neural networks whose parameters are randomly varying in a finite state Markov chain sense. Based on the ergodic theory of continuous time Markov chain, the matrix measure approach and Lyapunov theory, almost sure stability and exponential stability in the mean square for continuous time BAM neural networks are derived. We also present some new stability results for discrete time BAM neural networks with the help of the law of large numbers. Meanwhile, some examples with numerical simulations are given to show that the Markov chain plays an important role in stability of neural networks.

Synchronization for complex networks with Markov switching via matrix measure approach

This paper devotes to almost sure synchronization and almost sure quasi-synchronization of complex networks with Markov switching. Some sufficient conditions are derived in terms of the ergodic theory of continuous time Markov chain and the matrix measure approach, which can guarantee that the dynamical networks almost surely synchronize or quasi-synchronize to a given manifold. According to the property of Markov chain and the exponential distribution of switching time sequence, we also estimate the probability distribution of the quasi-synchronization error for a two-state Markov chain and then generalize them to a finite state space Markov chain. Meanwhile, some examples with numerical simulations are given to show that the Markov chain plays an important role in synchronization of networks.

Prevalence and co-occurrence of addictive behaviors among former alternative high school youth

Background and Aims: Recent work has studied multiple addictions using a matrix measure, which taps multiple addictions through single responses for each type. Methods: The present study investigated use of a matrix measure approach among former alternative high school youth (average age = 19.8 years) at risk for addictions. Lifetime and last 30-day prevalence of one or more of 11 addictions reviewed in other work (Sussman, Lisha & Griffiths, 2011) was the primary focus (i.e., cigarettes, alcohol, other/hard drugs, eating, gambling, Internet, shopping, love, sex, exercise, and work). Also, the co-occurrence of two or more of these 11 addictive behaviors was investigated. Finally, the latent class structure of these addictions, and their associations with other measures, was examined. Results: We found that ever and last 30-day prevalence of one or more of these addictions was 79.2% and 61.5%, respectively. Ever and last 30-day co-occurrence of two or more of these addictions was 61.5% and 37.7%, respectively. Latent Class Analysis suggested two groups: a generally Non-addicted Group (67.2% of the sample) and a “Work Hard, Play Hard”-addicted Group that was particularly invested in addiction to love, sex, exercise, the Internet, and work. Supplementary analyses suggested that the single-response type self-reports may be measuring the addictions they intend to measure. Discussion and Conclusions: We suggest implications of these results for future studies and the development of prevention and treatment programs, though much more validation research is needed on the use of this type of measure.

Robust regional eigenvalue‐clustering analysis for grey discrete‐time systems with/without state delay

The regional eigenvalue-clustering robustness problem of grey discrete-time systems with/without state delay is considered. Based on both spectral radius approach and matrix measure approach, three new sufficient conditions are proposed to preserve the regional eigenvalue-clustering property of grey discrete-time systems with/without state delay. If all eigenvalues are only required to be located within the unit circle, the proposed criteria become stability criteria. The proposed sufficient conditions are mathematically proven to be less conservative than those reported in the literature. Three examples are given to demonstrate that the proposed sufficient conditions are applicable and obtain less conservative results compared with those reported recently in the literature.

Functional differential inclusions and dynamic behaviors for memristor-based BAM neural networks with time-varying delays

In this paper, we formulate and investigate a class of memristor-based BAM neural networks with time-varying delays. Under the framework of Filippov solutions, the viability and dissipativity of solutions for functional differential inclusions and memristive BAM neural networks can be guaranteed by the matrix measure approach and generalized Halanay inequalities. Then, a new method involving the application of set-valued version of Krasnoselskii’ fixed point theorem in a cone is successfully employed to derive the existence of the positive periodic solution. The dynamic analysis in this paper utilizes the theory of set-valued maps and functional differential equations with discontinuous right-hand sides of Filippov type. The obtained results extend and improve some previous works on conventional BAM neural networks. Finally, numerical examples are given to demonstrate the theoretical results via computer simulations.

Filippov systems and quasi-synchronization control for switched networks

This paper is concerned with the quasi-synchronization issue of linearly coupled networks with discontinuous nonlinear functions in each isolated node. Under the framework of Filippov systems, the existence and boundedness of solutions for such complex networks can be guaranteed by the matrix measure approach. A design method is presented for the synchronization controllers of coupled networks with non-identical discontinuous systems. Moreover, a sufficient condition is derived to ensure the quasi-synchronization of switched coupled complex networks with discontinuous isolated nodes, which could be controlled by some designed linear controllers. The obtained results extend the previous work on the synchronization issue of coupled complex networks with Lipschitz continuous conditions. Numerical simulations on the coupled chaotic systems are given to demonstrate the effectiveness of the theoretical results.