Global Exponential Attractive Sets Research Articles

Dynamics of a New 5D Hyperchaotic System of Lorenz Type

Ultimate boundedness of chaotic dynamical systems is one of the fundamental concepts in dynamical systems, which plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors and the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, chaos synchronization. However, it is often difficult to obtain the bounds of the hyperchaotic systems due to the complex algebraic structure of the hyperchaotic systems. This paper has investigated the boundedness of solutions of a nonlinear hyperchaotic system. We have obtained the global exponential attractive set and the ultimate bound set for this system. To obtain the ellipsoidal ultimate bound, the ultimate bound of the proposed system is theoretically estimated using Lagrange multiplier method, Lyapunov stability theory and optimization theory. To show the ultimate bound region, numerical simulations are provided.

Complex dynamics of a new 3D Lorenz-type autonomous chaotic system

This paper investigates a new three-dimensional continuous quadratic autonomous chaotic system which is not topologically equivalent to the Lorenz system. The dynamical behaviours of this system are further investigated in detail, including the ultimate boundedness, the invariant sets and the global attraction domain according to Lyapunov stability theory of dynamical systems. The innovation of the paper lies in the fact that this paper not only proves this chaotic system is globally bounded for the parameters of this system but also gives a family of mathematical expressions of global exponential attractive sets with respect to the parameters of this system. To validate the ultimate bound estimation, numerical simulations are also investigated. Numerical simulations verify the effectiveness and feasibility of the theoretical scheme.

Qualitative behaviors of the high-order Lorenz-Stenflo chaotic system arising in mathematical physics describing the atmospheric acoustic-gravity waves

The boundedness of chaotic systems plays an important role in investigating the stability of the equilibrium, estimating the Lyapunov dimension of attractors, the Hausdorff dimension of attractors, the existence of periodic solutions, chaos control, and chaos synchronization. However, as far as the authors know, there are only a few papers dealing with bounds of high-order chaotic systems due to their complex algebraic structure. To sort this out, in this paper, we study the bounds of a high-order Lorenz-Stenflo system arising in mathematical physics. Based on Lyapunov stability theory, we show that there exists a globally exponential attractive set for this system. The innovation of the paper is that we not only prove that this system is globally bounded for all the parameters, but also give a family of mathematical expressions of global exponential attractive sets of this system with respect to its parameters. We also study some other dynamical characteristics of this chaotic system such as invariant sets and chaotic behaviors. To justify the theoretical analysis, we carry out detailed numerical simulations.

Constructive proof of Lagrange stability and sufficient – Necessary conditions of Lyapunov stability for Yang–Chen chaotic system

This paper studies the stability problem of Yang–Chen system. By introducing different radial unbounded Lyapunov functions in different regions, global exponential attractive set of Yang–Chen chaotic system is constructed with geometrical and algebraic methods. Then, simple algebraic sufficient and necessary conditions of global exponential stability, global asymptotic stability, and exponential instability of equilibrium are proposed. And the relevant expression of corresponding parameters for local exponential stability, local asymptotic stability, exponential instability of equilibria are obtained. Furthermore, the branch problem of the system is discussed, some branch expressions are given for the parameters of the system.

Qualitative behaviors of the continuous-time chaotic dynamical systems describing the interaction of waves in plasma

In this paper, a new Lorenz-like chaotic system describing the interaction of three resonantly coupled waves in plasma is studied. Explicit ultimate boundedness and global attraction domain are derived according to stability theory of dynamical systems. The innovation of the paper is that this paper not only proves this chaotic system is globally bounded for the parameters of this system but also gives a family of mathematical expressions of global exponential attractive sets for this system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained. Finally, numerical localization of attractor is presented.

Global Lagrange stability for inertial neural networks with mixed time-varying delays

This paper concerns with the global Lagrange stability of inertial neural networks with discrete and distributed time-varying delays. By choosing a proper variable substitution, the inertial neural networks can be rewritten as a first-order differential system. Based on the Lyapunov functional method, inequality techniques and analytical method, several sufficient conditions are derived to guarantee the global exponential stability of the inertial neural networks in Lagrange sense. Meanwhile, the global exponential attractive set is also given. Simulation results demonstrate the effectiveness of the theoretical results.

On global boundedness of the Chen system

This paper deals with the open problem of the global boundedness of the Chen system based on Lyapunov stability theory, which was proposed by Qin and Chen (2007). The innovation of the paper is that this paper not only proves the Chen system is global bounded for a certain range of the parameters according to stability theory of dynamical systems but also gives a family of mathematical expressions of global exponential attractive sets for the Chen system with respect to the parameters of this system. Furthermore, the exponential rate of the trajectories is also obtained.

Dynamics of a low-order atmospheric circulation chaotic model

In this paper, the dynamics of the Lorenz model of general circulation of the atmosphere is investigated. The global exponential attractive set and positively invariant set are obtained based on well-known Lyapunov stability theory and the function extremum theory for this low-order atmospheric circulation chaotic model. Furthermore, the exponential rate of the trajectories going from the exterior of the attractive set to the interior of the attractive set is also obtained. At last, numerical studies are provided to illustrate the effectiveness of the presented scheme.

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

In this paper, the global exponential stability in Lagrange sense related to inertial neural networks with time-varying delay is investigated. Firstly, by constructing a proper variable substitution, the original system is transformed into the first order differential system. Next, some succinct criteria for the ultimate boundedness and global exponential attractive set are derived via the Lyapunov function method, inequality techniques and analytical method. Meanwhile, the detailed estimations for the global exponential attractive set are established. Finally, the effectiveness of theoretical results has been illustrated via two numerical examples.

Global Lagrange stability for neutral-type Cohen–Grossberg BAM neural networks with mixed time-varying delays

The paper discusses global Lagrange stability for a class of Cohen–Grossberg BAM neural networks of neutral-type with multiple time-varying and finite distributed delays by constructing appropriate Lyapunov-like functions. To this end, we first establish a new differential integral inequality for non-autonomous Cohen–Grossberg BAM neural networks with finite distributed delays. By using the new inequality and employing some other inequality techniques, we analyze two different types of activation functions which include both bounded and unbounded activation functions. Some easily verifiable criteria are obtained for global exponential attractive sets in which all trajectories converge. These results can also be applied to analyze monostable as well as multistable and more extensive neural networks due to making no assumptions on the number of equilibria. Meanwhile, the results obtained in this paper are more general and challenging than that of the existing references. Finally, some examples with numerical simulations are given and analyzed to verify our results.

New results of the ultimate bound on the trajectories of the family of the Lorenz systems

In this paper, the global exponential attractive sets of a class of continuous-time dynamical systems defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^3},$ are studied. The elements of main diagonal of matrix $A$ are both negative numbers and zero, where matrix $A$ is the Jacobian matrix $\frac{{df}}{{dx}}$ of a continuous-time dynamical system defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^3},$ evaluated at the origin ${x_0} = \left( {0,0,0} \right).$ The former equations [1-6] that we are searching for a global bounded region have a common characteristic: The elements of main diagonal of matrix $A$ are all negative, where matrix $A$ is the Jacobian matrix $\frac{{df}}{{dx}}$ of a continuous-time dynamical system defined by $\dot x = f\left( x \right),{\kern 1pt} {\kern 1pt} {\kern 1pt} x \in {R^n},$ evaluated at the origin ${x_0} = {\left( {0,0, \cdots ,0} \right)_{1 \times n}}.$ For the reason that the elements of main diagonal of matrix $A$ are both negative numbers and zero for this class of dynamical systems , the method for constructing the Lyapunov functions that applied to the former dynamical systems does not work for this class of dynamical systems. We overcome this difficulty by adding a cross term $xy$ to the Lyapunov functions of this class of dynamical systems and get a perfect result through many integral inequalities and the generalized Lyapunov functions.

Global stability in Lagrange sense for BAM-type Cohen–Grossberg neural networks with time-varying delays

In this paper, we investigate the positive invariant sets and global exponential attractive sets for a class of bidirectional associative memory (BAM)-type Cohen–Grossberg neural networks with multiple time-varying delays. By applying inequality techniques, some easily verifiable delay-independent criteria for the ultimate boundedness and global exponential attractive sets of BAM-type Cohen–Grossberg neural networks are obtained by constructing appropriate Lyapunov functions. Finally, one example with numerical simulations is given to illustrate the results obtained in this paper.

Global Exponential Attractive Sets for Recurrent Neural Networks with Infinite Distributed Delays

This paper is concerned with the problem on global exponential attractive sets of recurrent neural networks (RNNs) with general activation functions and infinite distributed delays. By employing a new differential inequality and abandoning the limitation on activation functions, several sufficient conditions for detailed estimations of the global exponential attractive sets are derived in terms of linear matrix inequalities (LMIs) which can be easily checked by LMI Control Toolbox in Matlab. Compared with previous methods, the results obtained are independent of the time-varying delays and do not require the differentiability of delay functions. They extend and improve the earlier publications. Finally, a numerical example is provided to demonstrate the potential effectiveness of the proposed results. KeywordsRecurrent Neural Network; Global Exponential Attractive Sets; Infinite Distributed Delays; Linear Matrix Inequalities

Dynamical behaviors of a new hyperchaotic system

Currently, chaotic systems and chaos‐based applications are commonly used in the engineering fields. One of the main structures used in these applications is chaotic control and synchronization. In this paper, the dynamical behaviors of a new hyperchaotic system are considered. Based on Lyapunov Theorem with differential and integral inequalities, the global exponential attractive sets and positively invariant sets are obtained. Furthermore, the rate of the trajectories is also obtained. The global exponential attractive sets of the system obtained in this paper also offer theoretical support to study chaotic control, chaotic synchronization for this system. Computer simulation results show that the proposed method is effective. Copyright © 2014 John Wiley & Sons, Ltd.

On the New Results of Global Exponential Attractive Sets of the T Chaotic System

In this paper, the global exponential attractive sets of the T chaotic system are studied. The method for constructing Lyapunov functions that are applied to the former chaotic systems [Liao, 2004; Liao et al., 2008; Wang et al., 2011; Zhang et al., 2011a; Zhang et al., 2011b; Yu & Liao, 2008; Li et al., 2009; Yu et al., 2009] does not work for the T chaotic system. We overcome this difficulty by introducing a cross term xy. We get a perfect result that contains the case for the parameter values considered covering the most interesting case of the chaotic attractor system.

Boundedness solutions of the complex Lorenz chaotic system

This paper is concerned with the boundedness of solutions of the complex Lorenz system. We have obtained the global exponential attractive set Ψλ,m and the ultimate bound Ωλ,m for this system. Furthermore, we confirm that the rate of the trajectories of the system going from the exterior of the set Ψλ,m to the interior of the set Ψλ,m is an exponential rate. The rate of the trajectories is also obtained. Numerical simulations are presented to show the effectiveness of the proposed scheme.

Positive invariant sets and global exponential attractive sets of BAM neural networks with time-varying and infinite distributed delays

In this paper, the problem on positive invariant sets and global exponential attractive sets for a class of BAM neural networks with time-varying and infinite distributed delays is discussed. By virtue of appropriate Lyapunov-like functions and some inequality techniques, some algebraic criterions in linear matrix inequality form for the existence of positive invariant sets and global exponential attractive sets are obtained. Meanwhile, the estimations of the global exponential attractive sets are given out. The results derived here generalize and improve the earlier publications by removing the usual assumptions for the boundedness, monotonicity and differentiability on the activation functions. Finally, one numerical example is given and analyzed to demonstrate our results.

Global attractive sets of a novel bounded chaotic system

This paper is concerned with the boundedness of solutions of a new chaotic system. For this system, the global exponential attractive set and positively invariant set are derived based on generalized Lyapunov function theory and the extremum principle of function. Furthermore, we can conclude that the rate of the trajectories of the system going from the exterior of the attractive set ? ? to the interior of the attractive set ? ? is an exponential rate. The rate of the trajectories is also obtained. Numerical simulations are presented to show the effectiveness of the proposed scheme.

Estimations for ultimate boundary of a new hyperchaotic system and its simulation

This paper has investigated the boundedness of a new hyperchaotic Rabinovich system. We have obtained the global exponential attractive set and the ultimate bound Ωλ for this system. Furthermore, we can conclude that the rate of the trajectories of the system going from the exterior of the set Ωλ,2 to the interior of the set Ωλ,2 is an exponential rate. The estimate of the trajectories rate is also obtained. Numerical simulations are presented to show the effectiveness of the proposed scheme.

On the new results of global exponential attractive set

Abstract In this paper, the global exponential attractive sets of a class of continuous-time dynamical systems defined by x = f ( x ) , x ∈ R n are studied. The elements of main diagonal of matrix A are both negative numbers and zero, where matrix A is the Jacobian matrix d f d x of a continuous-time dynamical system defined by x = f ( x ) , x ∈ R n evaluated at the origin x 0 = ( 0 , 0 , … , 0 ) 1 × n . However, note that the former equations that we are searching for a global bounded region have a common characteristic: the elements of main diagonal of matrix A are all negative. As far as we know, very few papers have addressed this problem.