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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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

It is often difficult to obtain the bounds of the hyperchaotic systems due to very complex algebraic structure of the hyperchaotic systems. After an exhaustive research on a new 4D Lorenz-type hyperchaotic system and a coupled dynamo chaotic system, we obtain the bounds of the new 4D Lorenz-type hyperchaotic system and the globally attractive set of the coupled dynamo chaotic system. To validate the ultimate bound estimation, numerical simulations are also investigated. The innovation of this article lies in that the method of constructing Lyapunov-like functions applied to the Lorenz system is not applicable to this 4D Lorenz-type hyperchaotic system; moreover, one Lyapunov-like function cannot estimate the bounds of this 4D Lorenz-type hyperchaos system. To sort this out, we construct three Lyapunov-like functions step by step to estimate the bounds of this new 4D Lorenz-type hyperchaotic system successfully. [ABSTRACT FROM AUTHOR]

In this paper, the ultimate bound for a new chaotic system is derived based on stability theory of dynamical systems. The meaningful contribution of this article is that the results presented in this paper contain the existing results as special cases. Finally, numerical simulations are given to verify the effectiveness and correctness of the obtained results. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This article deals with the ultimate bound on the trajectories of the hyperchaos Lorenz system based on Lyapunov stability theory. The innovation of this article lies in that the method of constructing Lyapunov functions applied to the former chaotic systems is not applicable to this hyperchaos system, and moreover, one Lyapunov function can not estimate the bounds of this hyperchaos Lorenz system. We successfully estimate the bounds of this hyperchaos system by constructing three generalized Lyapunov functions step by step. Some computer simulations are also given to show the effectiveness of the proposed scheme. © 2016 Wiley Periodicals, Inc. Complexity 21: 440-445, 2016 [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this article, the dynamical behaviors of two classes of chaotic systems are considered based on generalized Lyapunov function theorem with integral inequalities. Explicit estimations of the ultimate bounds are derived. The results presented in this article contain the existing results as special cases. Computer simulation results show that the proposed method is effective. © 2014 Wiley Periodicals, Inc. Complexity 21: 363-369, 2015 [ABSTRACT FROM AUTHOR]

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 Φ to the interior of the attractive set Φ is an exponential rate. The rate of the trajectories is also obtained. Numerical simulations are presented to show the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]