Global Exponential Attractive Sets Research Articles

ULTIMATE BOUND FOR AN ACTIVE LORENZ SYSTEM

Considering the Lorenz-like model [1] derived from Active Model H for scalar active matter introduced by Tiribocchi et al. [2], we investigate the ultimate bound and global attractive sets for the system, and obtain the ultimate bound characterized by an ellipsoid and a global exponential attractive set for the system.

Global attractive set for quaternion-valued neural networks with neutral items

This paper investigated the global attractive set for quaternion-valued neural networks (QVNNs) with leakage delay, time-varying delay, and neutral items. Based on various basic conditions of activation function, the global attractive set and global exponential attractive set of QVNNs are given combined with novel analytical techniques and Lyapunov theory. The QVNNs are studied by a direct method, without any decomposition. The time delay can be non-differential, which makes the results more pragmatic. Restrictions on the activation function of the neutral item are relaxed. The neutral activation function can be bounded or unbounded, which makes the results more practical. Two simulation examples are given to verify the validity of the theory results.

Dissipativity Analysis of Memristive Inertial Competitive Neural Networks with Mixed Delays

Without altering the inertial system into the two first-order differential systems, this paper primarily works over the global exponential dissipativity (GED) of memristive inertial competitive neural networks (MICNNs) with mixed delays. For this purpose, a novel differential inequality is primarily established around the discussed system. Then, by applying the founded inequality and constructing some novel Lyapunov functionals, the GED criteria in the algebraic form and the linear matrix inequality (LMI) form are given, respectively. Furthermore, the estimation of the global exponential attractive set (GEAS) is furnished. Finally, a specific illustrative example is analyzed to check the correctness and feasibility of the obtained findings.

Boundary estimation and cascade control for a Lorenz‐like system

In this paper, we devoted to investigate a hyperelliptic estimate of the ultimate bound and positively invariant set for the new system via the Lyapunov function method. Furthermore, we obtained the global exponential attractive set , which is an exponential decay rate. The rate of the trajectories of the system going from the exterior to the interior of the set . Numerical simulations are presented to show the effectiveness and the feasibility. The theoretical results obtained will find wide applications in cascade control and chaos synchronization.

Dynamical analysis and boundedness for a generalized chaotic Lorenz model

<abstract><p>The dynamical behavior of a 5-dimensional Lorenz model (5DLM) is investigated. Bifurcation diagrams address the chaotic and periodic behaviors associated with the bifurcation parameter. The Hamilton energy and its dependence on the stability of the dynamical system are presented. The global exponential attractive set (GEAS) is estimated in different 3-dimensional projection planes. A more conservative bound for the system is determined, that can be applied in synchronization and chaos control of dynamical systems. Finally, the finite time synchronization of the 5DLM, indicating the role of the ultimate bound for each variable, is studied. Simulations illustrate the effectiveness of the achieved theoretical results.</p></abstract>

Global dissipativity of fuzzy genetic regulatory networks with mixed delays

The synthetic genetic regulatory networks have proven to be a powerful tool in studying gene regulation processes in living organisms. In this article, the global dissipativity and corresponding attractive set for the fuzzy genetic regulatory networks with mixed delays are investigated. By utilising the Lyapunov functional method and the linear matrix inequalities (LMIs) techniques, new sufficient conditions ensuring the global dissipativity and the global exponential dissipativity of the suggested system are given. Moreover, the global attractive set and global exponential attractive set are obtained. The derived criteria are of the form of LMI, and hence they can be verified easily by the numerical software. Lastly, two numerical examples with its simulations are given to illustrate the effectiveness of the obtained results.

Global exponential stability of memristor based uncertain neural networks with time-varying delays via Lagrange sense

ABSTRACT This paper addresses the global exponential stability in Lagrange sense for memristor-based neural networks (MNNs) with time-varying delays. This paper attempts to derive the delay-dependent Lagrange stability conditions in terms of linear matrix inequalities by designing a suitable Lyapunov-Krasovskii functionaland used Wirtinger inequality, Jensen-based inequality for estimating the integral inequalities. The conditions which are derived confirms the globally exponential stability in Lagrange sense for the proposed MNNs and, the detailed estimation for global exponential attractive set is also given. To show the effectiveness and applicability of the proposed criteria, two numerical examples are also provided in this paper.

Chaotic Dynamics in Generalized Rabinovich System

The article is devoted to the study of the global behavior of the generalized Rabinovich system describing the process of interaction between waves in plasma. For this generalized system, we obtain the positive invariant set (ultimate bound) and globally exponential attractive set using the approach where we transform the initial problem of finding the corresponding set to the conditional extremum problem and solve this problem. Furthermore, the rate of the trajectories going from the exterior of the attractive set to the interior of the attractive set is also obtained. Numerical localization of attractor is presented. Meanwhile, the volumes of the ultimate bound set and the global exponential attractive set are obtained, respectively. The main innovation of this article lays in considering the generalized form of the Rabinovich system with [Formula: see text] and obtaining the results on the ultimate bound set and globally exponential attractive set for this general case.

Global Exponential Dissipativity of Impulsive Recurrent Neural Networks with Multi-proportional Delays

This paper addresses the global exponential dissipativity (GED) of impulsive recurrent neural networks (IRNNs) with proportional delays. By introducing some adjustable parameters, skillfully designing several Lyapunov functionals and utilizing matrix norm properties, serval delay-dependent GED criteria are developed, and global attractive sets (GAS) and global exponential attractive sets (GEAS) of the proposed system are given. These adjustable parameters are related to the exponential decay rate and contribute greatly to expand the attractive sets of this paper. Here the criteria proposed improve and extend the earlier global dissipativity criteria. Several numerical examples are used to verify the obtained results and show that the obtained results are independent of each other.

DYNAMICS OF A GENERALIZED LORENZ-LIKE CHAOS DYNAMICAL SYSTEMS

<abstract> In this work, a new seven-parameter Lorenz-like chaotic system is presented and discussed by combining nonlinear dynamical systems theory with computer simulation. The existence of the ultimate bound set and global exponential attractive set of this chaotic system is proved by using Lyapunov's direct method. A family of analytic mathematical expression of the ultimate bound sets and global exponential attractive sets involving two parameters are obtained, respectively. Meanwhile, the volumes of the ultimate bound set and global exponential attractive set are obtained, respectively. Numerical simulations are conducted which validates the correctness of the proposed theoretical analysis. </abstract>

Global dissipativity of Clifford-valued multidirectional associative memory neural networks with mixed delays

The main goal of this article is to study the global dissipativity problem of Clifford-Valued Multidirectional Associative Memory Neural Networks (CVMAMNNs) with time-varying delays and distributed delays. Based on Lyapunov functionals and Linear Matrix Inequalities (LMIs) approach, new sufficient conditions are derived to ensure the global dissipativity and global exponential dissipativity of the considered network model. Moreover, the global attractive set and global exponential attractive set are obtained which are positive invariant ones. Finally, two numerical examples with simulations are given to illustrate the effectiveness of the analytical findings.

Global Robust Exponential Dissipativity of Uncertain Second-Order BAM Neural Networks With Mixed Time-Varying Delays.

This article focuses on the global robust exponential dissipativity (GRED) of uncertain second-order BAM neural networks with mixed time-varying delays. First, a new differential inequality for the concerned second-order system is established. Second, by constructing some new Lyapunov-Krasovskii functionals (LKFs) and applying this new inequality and some other inequalities, some new GRED criteria in the form of linear matrix inequalities are presented. The global exponential attractive sets are also provided simultaneously. Different from the existing reduced-order methods, this article considers some new LKFs to directly analyze the dynamics of the addressed system via a nonreduced-order strategy. Finally, the correctness of the theoretical results is verified by simulation experiments.

Global Lagrange stability for neutral-type inertial neural networks with discrete and distributed time delays

This paper focus on the problem of global Lagrange stability for neutral-type inertial neural networks with discrete and distributed time delays. By choosing a proper variable substitution, an inertial neural network consisting of second-order differential equations can be converted into a first-order differential model. The sufficient conditions of the inertial neural network with neutral delay are derived by constructing suitable Lyapunov-Krasovskii functional candidates, introducing new free weighting matrices, utilizing inequality techniques and analytical method. Through the LMI condition, we analyze the global exponential stability of the delayed inertial neural networks in Lagrange sense. Meanwhile, the global exponential attractive set is also given. Finally, some example is given to illustrate our theoretical results.

Global Lagrange Stability of Inertial Neutral Type Neural Networks with Mixed Time-Varying Delays

This paper deals with the Lagrange stability of inertial neutral type neural networks with mixed time-varying delays. Two different types of activation functions are considered, including bounded and general unbounded activation functions. Under a proper variable transformation, the original inertial system is converted to a first order differential network. Based on Lyapunov method and applying inequality techniques and analytical method, some sufficient criteria are derived to ensure the global Lagrange exponential stability of the addressed neural networks. Moreover, the global exponential attractive sets are established. These results here generalize and improve the earlier publications on inertial neural networks. Finally, some numerical examples with simulations are given to demonstrate the effectiveness of our theoretical results.

Bifurcations, ultimate boundedness and singular orbits in a unified hyperchaotic Lorenz-type system

In this note, by using the theory of bifurcation and Lyapunov function, one performs a qualitative analysis on a novel four-dimensional unified hyperchaotic Lorenz-type system (UHLTS), including stability, pitchfork bifurcation, Hopf bifurcation, singularly degenerate heteroclinic cycle, ultimate bound estimation, global exponential attractive set, heteroclinic orbit and so on. Numerical simulations not only are consistent with the results of theoretical analysis, but also illustrate singularly degenerate heteroclinic cycles with distinct geometrical structures and nearby hyperchaotic attractors in the case of small \begin{document}$ b > 0 $\end{document} , i.e. conjugate hyperchaotic Lorenz-type attractors (CHCLTA) and nearby a short-duration transient of singularly degenerate heteroclinic cycles approaching infinity or singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci and stable node-foci, etc. In particular, by a linear scaling, a possibly new forming mechanism behind the creation of well-known hyperchaotic attractor with \begin{document}$ (a_{1}, a_{2}, c, d, e, f, b, p, q) = (-12, 12, 23,-1, -1, 1, 2.1, -6, -0.2) $\end{document} , consisting of occurrence of degenerate pitchfork bifurcation at \begin{document}$ S_{z} $\end{document} , the change in the stability index of the saddle at the origin as \begin{document}$ b $\end{document} crosses the null value, explosion of normally hyperbolic stable node-foci, collapse of singularly degenerate heteroclinic cycles consisting of normally hyperbolic saddle-foci or saddle-nodes, and stable node-foci, is revealed. The findings and results of this paper may provide theoretical support in some future applications, since they improve and complement the known ones.

On Singular Orbits and Global Exponential Attractive Set of a Lorenz-Type System

This paper deals with some unsolved problems of the global dynamics of a three-dimensional (3D) Lorenz-type system: [Formula: see text], [Formula: see text], [Formula: see text] by constructing a series of Lyapunov functions. The main contribution of the present work is that one not only proves the existence of singularly degenerate heteroclinic cycles, existence and nonexistence of homoclinic orbits for a certain range of the parameters according to some known results and LaSalle theorem but also gives a family of mathematical expressions of global exponential attractive sets for that system with respect to its parameters, which is available only in very few papers as far as one knows. In addition, numerical simulations illustrate the consistence with the theoretical conclusions. The results together not only improve and complement the known ones, but also provide support in some future applications.

Dissipativity analysis of neutral-type memristive neural network with two additive time-varying and leakage delays

In this paper, we offer an approach about the dissipativity of neutral-type memristive neural networks (MNNs) with leakage, additive time, and distributed delays. By applying a suitable Lyapunov–Krasovskii functional (LKF), some integral inequality techniques, linear matrix inequalities (LMIs) and free-weighting matrix method, some new sufficient conditions are derived to ensure the dissipativity of the aforementioned MNNs. Furthermore, the global exponential attractive and positive invariant sets are also presented. Finally, a numerical simulation is given to illustrate the effectiveness of our results.

Analysis of a Lorenz‐Like Chaotic System by Lyapunov Functions

In this paper, we investigate the ultimate bound set and positively invariant set of a 3D Lorenz‐like chaotic system, which is different from the well‐known Lorenz system, Rössler system, Chen system, Lü system, and even Lorenz system family. Furthermore, we investigate the global exponential attractive set of this system via the Lyapunov function method. The rate of the trajectories going from the exterior of the globally exponential attractive set to the interior of the globally exponential attractive set is also obtained for all the positive parameters values a, b, c. The innovation of this paper is that our approach to construct the ultimate bounded and globally exponential attractivity sets assumes that the corresponding sets depend on some artificial parameters (λ and m); that is, for the fixed parameters of the system, we have a series of sets depending on λ and m. The results contain the known result as a special case for the fixed λ and m. The efficiency of the scheme is shown numerically. The theoretical results may find wide applications in chaos control and chaos synchronization.

Exponential stability in Lagrange sense for inertial neural networks with time-varying delays

This paper addresses the global exponential stability in Lagrange sense for inertial neural networks (INNs) with time-varying discrete delays and distributed delays. By utilizing a proper variable substitution to transform the original system into a first-order differential system, choosing an appropriate Lyapunov–Krasovskii functional (LKF), applying Jensen inequality, Jensen-based inequality, Wirtinger-based integral inequality and improved reciprocally convex inequality to estimate the derivative of the LKF, several sufficient conditions, which guarantee the global exponential stability in Lagrange sense for the INNs, are newly obtained in terms of linear matrix inequalities. Meanwhile, the detailed estimation for global exponential attractive set is also given. And two numerical examples are provided to validate the effectiveness of the proposed results.

Qualitative Study of a 4D Chaos Financial System

Some dynamics of a new 4D chaotic system describing the dynamical behavior of the finance are considered. Ultimate boundedness and global attraction domain are obtained according to Lyapunov stability theory. These results are useful in estimating the Lyapunov dimension of attractors, Hausdorff dimension of attractors, chaos control, and chaos synchronization. We will also present some simulation results. Furthermore, the volumes of the ultimate bound set and the global exponential attractive set are obtained.