Global 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.

Hamilton energy, competitive modes and ultimate bound estimation of a new 3D chaotic system, and its application in chaos synchronization

Abstract This paper discusses the dynamical behavior of a new 3D chaotic system of integer and fractional order. To get a comprehensive knowledge of the dynamics of the proposed system, we have studied competitive modes and Hamilton energy for different parameter values. In order to get the ultimate bound set for the proposed system, we employed the Lagrange coefficient approach to solve the optimization problem. We have also explored the use of the bound set in synchronization. Furthermore, we have examined the Hamilton energy, time series, bifurcation diagrams, and Lyapunov exponents for the fractional version of the proposed chaotic system. Finally, we looked at the Mittage-Leffler positive invariant sets and global attractive sets by merging the Lyapunov function approach with the Mittage-Leffler function. Numerical simulations have shown the obtained bound sets and other analytical outcomes.

Global Mittag-Leffler Attractive Sets, Boundedness, and Finite-Time Stabilization in Novel Chaotic 4D Supply Chain Models with Fractional Order Form

This research explores the complex dynamics of a Novel Four-Dimensional Fractional Supply Chain System (NFDFSCS) that integrates a quadratic interaction term involving the actual demand of customers and the inventory level of distributors. The introduction of the quadratic term results in significantly larger maximal Lyapunov exponents (MLE) compared to the original model, indicating increased system complexity. The existence, uniqueness, and Ulam–Hyers stability of the proposed system are verified. Additionally, we establish the global Mittag-Leffler attractive set (MLAS) and Mittag-Leffler positive invariant set (MLPIS) for the system. Numerical simulations and MATLAB phase portraits demonstrate the chaotic nature of the proposed system. Furthermore, a dynamical analysis achieves verification via the Lyapunov exponents, a bifurcation diagram, a 0–1 test, and a complexity analysis. A new numerical approximation method is proposed to solve non-linear fractional differential equations, utilizing fractional differentiation with a non-singular and non-local kernel. These numerical simulations illustrate the primary findings, showing that both external and internal factors can accelerate the process. Furthermore, a robust control scheme is designed to stabilize the system in finite time, effectively suppressing chaotic behaviors. The theoretical findings are supported by the numerical results, highlighting the effectiveness of the control strategy and its potential application in real-world supply chain management (SCM).

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.

Boundary analysis and energy feedback control of fractional-order extended Malkus–Robbins dynamo system

In this paper, the fractional-order extended Malkus-Robbins dynamo system (FOEMRD) is introduced. Additionally, the stability, complexity diagrams, Lyapunov dimension, Lyapunov exponents, and bifurcation diagrams of the system are studied. Also, we estimate the global Mittag-Leffler positive invariant sets and global Mittag-Leffler attractive sets of the proposed system. In addition, the Hamilton energy function of the FOEMRD system is computed by the Helmholtz theorem. Next, we design an energy feedback controller to control the chaos. Due to the relationship of Hamilton energy with the occurrence of chaos, the energy control method is very efficient and powerful. The applicability of the presented theories was proven with numerical simulations and related diagrams.

Quasi-invariant and attracting sets of competitive neural networks with time-varying and infinite distributed delays

This paper focuses on the quasi-invariant set (QIS), global attracting set (GAS) and global exponential attracting set (GEAS) of competitive neural networks (CNNs) with time-varying and infinite distributed delays. For these purposes, based on the characteristics of nonnegative matrix and M-matrix, a new bidirectional delay integral inequality and a novel integro-differential inequality are first established. From the founded integral inequality, the existence conditions of the QIS and the GAS of the discussed system are obtained. Besides, the existence conditions of the GEAS are also given by the proposed integro-differential inequality, which gets rid of the construction of complex Lyapunov functions and functionals. The frameworks of the QIS, GAS and GEAS are also given. A numerical example is analyzed to confirm the validity of the obtained results in the end.

The global dynamics of a new fractional-order chaotic system

This paper investigates the global dynamics of a new 3-dimensional fractional-order (FO) system that presents just cross-product nonlinearities. Firstly, the FO forced Lorenz-84 system is introduced and the stability of its equilibrium points, as well as the chaos control for their stabilization, are addressed. Secondly, dynamical behavior is further analyzed and bifurcation diagrams, phase portraits, and largest Lyapunov exponent (LE) are discussed. Then, the global Mittag-Leffler attractive sets (MLASs) and Mittag-Leffler positive invariant sets (MLPISs) of the FO forced Lorenz-84 system are presented. Finally, the Hamilton energy function (HEF) of the Lorenz-84 system is calculated by using the Helmholtz theorem. The calculation of the Hamilton energy has an essential role on the estimation of chaos in dynamical systems, the guidance of orbits, and stability. In fact, any control action on the dynamical system completely changes the HEF. Numerical simulations are presented for illustrating the theoretical findings.

Dissipativity for a class of quaternion‐valued memristor‐based neutral‐type neural networks with time‐varying delays

In this work, the global dissipativity and exponential dissipativity of quaternion‐valued memristor‐based neutral‐type neural networks (QVNTMNNs) with time‐varying delays are studied. It is worth noting that the QVNTMNN is explored as a single entity without decomposition. Considering the QVNTMNNs as a switching system and utilizing Lyapunov theorem as well as inequality techniques, several sufficient criteria depending on the state‐dependent switching parameters to guarantee global dissipativity and exponential dissipativity of QVNTMNNs with time‐varying delays are obtained in the form of quaternion‐valued linear matrix inequalities (QVLMIs) and corresponding complex‐valued liner matrix inequalities (CVLMIs). Moreover, the positive invariant set, global attractive set, and exponential attractive set of QVNTMNNs with time‐varying delays can also be derived. Finally, two numerical examples are given to substantiate the reliability and validity of our theoretical criteria presented in this work.

Global dynamical analysis of the integer and fractional 4D hyperchaotic Rabinovich system

In this paper, the dynamical behavior of integer and fractional 4D hyperchaotic Rabinovich system is studied. We use the Lagrange coefficient method to solve an optimization problem analytically so that find an accurate ultimate bound set (UBS) for the 4D hyperchaotic Rabinovich system. Furthermore, bifurcation diagrams, Lyapunov exponents, global attractive sets (GASs), and positive invariant sets (PISs) of the fractional-order system are studied. Finally, using the Mittag-Leffler function and Lyapunov function method, the Mittag-Leffler GAS and Mittag-Leffler PIS of the proposed system are estimated. Our investigations indicate that there is a close relationship between changing the parameters and the dynamical behavior of the system, Hamilton energy consumption and changing the bound of the variables. The corresponding boundedness is numerically verified to show the effectiveness of the theoretical analysis.

Global attractive set of neural networks with neutral item

This paper investigates the global attractive set of neural networks with neutral item. To better deal with the neutral terms, different types of activation functions are considered. Based on matrix measures, inequality techniques, and Lyapunov theory, three new types of Lyapunov functions are designed to find the global attractive set of the system. We give out a simulation example to verify the validity of theory results. The result is very inclusive, whether the system has equilibrium or not. As long as the system is stable, we can find its global attractive set.

Ultimate Boundedness and Finite Time Stability for a High Dimensional Fractional-Order Lorenz Model

In this paper, the global attractive set (GAS) and positive invariant set (PIS) of the five-dimensional Lorenz model with the fractional order derivative are studied. Using the Mittag-Leffler function and Lyapunov function method, the ultimate boundedness of the proposed system are estimated. An effective control strategy is also designed to achieve the finite time stability of this fractional chaotic system. The corresponding boundedness and control scheme are numerically verified to show the effectiveness of the theoretical analysis.

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.

The global attractive sets and synchronization of a fractional-order complex dynamical system

<abstract><p>This paper presents a chaotic complex system with a fractional-order derivative. The dynamical behaviors of the proposed system such as phase portraits, bifurcation diagrams, and the Lyapunov exponents are investigated. The main contribution of this effort is an implementation of Mittag-Leffler boundedness. The global attractive sets (GASs) and positive invariant sets (PISs) for the fractional chaotic complex system are derived based on the Lyapunov stability theory and the Mittag-Leffler function. Furthermore, an effective control strategy is also designed to achieve the global synchronization of two fractional chaotic systems. The corresponding boundedness is numerically verified to show the effectiveness of the theoretical analysis.</p></abstract>

Global Attracting Sets of Neutral Stochastic Functional Differential Equations Driven by Poisson Jumps

Global Attracting Sets of Neutral Stochastic Functional Differential Equations Driven by Poisson Jumps

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.

Asymptotic properties of solutions for impulsive neutral stochastic functional integro-differential equations

In this work, we are concerned with the asymptotic properties of solutions for an impulsive neutral stochastic functional integro-differential equation. By applying the theory of resolvent operators, the Banach fixed point principle theorem, and results on stochastic analysis, we study respectively the existence, uniqueness, and global attracting and quasi-invariant sets of mild solutions for the considered equation. We also derive some sufficient conditions of pth moment exponential stability and almost surely exponential stability of the mild solutions. An example is provided in the end to illustrate the applications of the obtained results.

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.

Polynomial dissipativity of proportional delayed BAM neural networks

This paper pays close attention to the global polynomial dissipativity (GPD) for proportional delayed BAM neural networks (PDBAMNNs). The global exponential dissipativity (GED) and the global dissipativity (GD) are also talked about. Under the help of novel Lyapunov functionals and a generalized Halanay inequality, a set of dissipative criteria for such systems are led out, together with the global polynomial attracting set (GPAS) and the global attracting set (GAS). Further, the relationship among GPD, GED and GD is unveiled. Finally, a proposed theoretical condition is validated through a simulation experiment.

Global dissipativity of delayed discrete-time inertial neural networks

This paper addresses global dissipativity for a class of delayed discrete-time inertial neural networks (DINN). With a newly developed discrete-time Halanay inequality, a novel criterion of the global dissipativity for the DINN is established. Moreover, the global robust dissipativity of DINN with bounded parameter uncertainties is also investigated. Meanwhile, some specific estimates of global attractive sets and positive invariant sets are derived. Finally, two simulation examples validate the efficacy of the proposed results.

Well-posedness and dynamics of impulsive fractional stochastic evolution equations with unbounded delay

This paper is concerned with the well-posedness and dynamics of delay impulsive fractional stochastic evolution equations with time fractional differential operator α ∈ (0, 1). After establishing the well-posedness of the problem, and a result ensuring the existence and uniqueness of mild solutions globally defined in future, the existence of a minimal global attracting set is investigated in the mean-square topology, under general assumptions not ensuing the uniqueness of solutions. Furthermore, in the case of uniqueness, it is possible to provide more information about the geometrical structure of such global attracting set. In particular, it is proved that the minimal compact globally attracting set for the solutions of the problem becomes a singleton. It is remarkable that the attraction property is proved in the usual forward sense, unlike the pullback concept used in the context of random dynamical systems, but the main point is that the model under study has not been proved to generate a random dynamical system.