Bounds of the hyper-chaotic Lorenz–Stenflo system Pei Wang a, * , Damei Li a,1 , Qianli Hu b a School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China b Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China article info Article history: Received 13 March 2009 Received in revised form 12 September 2009 Accepted 12 September 2009 Available online 18 September 2009 Keywords: Hyper-chaotic system Ultimate bounds Positively invariant sets Optimization abstract To estimate the ultimate bound and positively invariant set for a dynamical system is an important but quite challenging task in general. This paper attempts to investigate the ulti- mate bounds and positively invariant sets of the hyper-chaotic Lorenz–Stenflo (L–S) sys- tem, which is based on the optimization method and the comparison principle. A family of ellipsoidal bounds for all the positive parameters values a, b, c, dand a cylindrical bound for a > 0, b > 1, c > 0, d > 0 are derived. Numerical results show the effectiveness and advan- tage of our methods. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction In 1996, Stenflo [1] derived a system to describe the evolution of finite amplitude acoustic gravity waves (AGW) in a rotating atmosphere. The equations are rather simple and reduce to the well-known Lorenz system [2] when the param- eter associated with the flow rotation is set to zero. AGW are of interest in atmospheric physics since they are ubiqui- tous and responsible for minor local weather changes as well as large scale phenomena such as storms and hurricanes. The governing equations for AGW can also be generic for many physical systems which exhibit universal phenomena such as intermittency, bifurcation, and deterministic chaos [3]. So, it is meaningful and valuable to investigate this system. Chaotic systems are bounded, the bounds have important application in chaos control and synchronization; But it is a difficult task to estimate the bounds of them; Some results have been derived only for the Lorenz system family [4–11]; Generally speaking, there are mainly three methods to estimate the bounds of chaotic systems in current literatures, that is the hyper-plane oriented method [4], Lyapunov stability theory combined with the comparison principle method [8] and the optimization method [7]. Among which the latter two methods are proved to be effec- tive and simpler. For hyper-chaotic system, to the best knowledge of the author, there is only one result about the Hyper-chaotic Lorenz–Haken system [12]. So it is meaningful and valuable to investigate the bounds of the hyper- chaotic L–S system. This paper investigates the bounds of the L–S system; we combine the above mentioned two methods to deeply inves- tigate the bound of this system. It is organized as follows: In section 2, the general method in this paper is proposed; in 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.09.015 * Corresponding author. E-mail addresses: wp0307@126.com (P. Wang), lidm@whu.edu.cn (D. Li), tsinghuaqlhu@yahoo.com.cn (Q. Hu). 1 Tel.: +86 027 62268575; fax: +86 027 68752256. Commun Nonlinear Sci Numer Simulat 15 (2010) 2514–2520 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns