International Journal of Bifurcation and Chaos, Vol. 21, No. 9 (2011) 2679–2694 c  World Scientific Publishing Company DOI: 10.1142/S0218127411030027 ULTIMATE BOUND ESTIMATION OF A CLASS OF HIGH DIMENSIONAL QUADRATIC AUTONOMOUS DYNAMICAL SYSTEMS PEI WANG ∗ , DAMEI LI † and XIAOQUN WU ‡ School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P. R. China ∗ wp0307@126.com † lidm@whu.edu.cn ‡ wuxiaoqunjj@yahoo.com.cn JINHU L ¨ U §,¶ and XINGHUO YU ‖ School of Electrical and Computer Engineering, RMIT University, Melbourne VIC 3001, Australia § LSC, Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China ¶ jhlu@iss.ac.cn ‖ x.yu@rmit.edu.au Received August 22, 2010; Revised October 13, 2010 This paper aims to propose a unified approach for the ultimate bound estimation of a class of High Dimensional Quadratic Autonomous Dynamical Systems (HDQADS). Using the proposed method and the optimization idea, a sufficient condition is then given for estimating the ultimate bounds of a class of HDQADS. To validate the above sufficient condition, this paper further investigates the ultimate bound estimation of a hyperchaotic system, a 6D and a 9D chaotic system, separately. Moreover, the ultimate bounds for a general Lorenz system, a low-order atmospheric circulation model, and a new 3D chaotic system are also discussed in detail. In particular, it should be pointed out that a unified and accurate ultimate bound estimation is attained for the generalized Lorenz system and it includes several well-known results as its special cases. Some numerical simulations are also given to verify and visualize the corresponding theoretical results. Keywords : Ultimate bound estimation; high dimensional quadratic autonomous systems; chaos; optimization. 1. Introduction Chaos, as an interesting nonlinear phenomenon, has been intensively investigated over the last few decades in various disciplines, such as mathemat- ics, physics, biology, engineering and social sciences [Leonov, 2001; Lorenz, 1963; Pogromsky et al., 2003; Zhou et al., 2003]. Moreover, chaotic behavior has been observed and verified in various real-world systems, including lasers, oscillating chemical reac- tions, electrical circuits, fluid dynamics, mechanical 2679