Physics Letters A 366 (2007) 52–60 www.elsevier.com/locate/pla Impulsive synchronization and parameter mismatch of the three-variable autocatalator model Yang Li a,∗ , Xiaofeng Liao a , Chuandong Li a , Tingwen Huang b , Degang Yang c a Department of Computer Science and Engineering, Chongqing University, Chongqing 400044, China b Texas A&M University at Qatar, c/o Qatar Foundation, PO Box 5825, Doha, Qatar c College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China Received 12 May 2006; received in revised form 10 December 2006; accepted 18 December 2006 Available online 9 January 2007 Communicated by A.P. Fordy Abstract The synchronization problems of the three-variable autocatalator model via impulsive control approach are investigated; several theorems on the stability of impulsive control systems are also investigated. These theorems are then used to find the conditions under which the three-variable autocatalator model can be asymptotically controlled to the equilibrium point. This Letter derives some sufficient conditions for the stabilization and synchronization of a three-variable autocatalator model via impulsive control with varying impulsive intervals. Furthermore, we address the chaos quasi-synchronization in the presence of single-parameter mismatch. To illustrate the effectiveness of the new scheme, several numerical examples are given.  2006 Elsevier B.V. All rights reserved. Keywords: Impulsive stabilization; Chaos; Impulsive synchronization; Parameter mismatch 1. Introduction Since 1990s, the issues on chaotic control and synchro- nization have attracted the interest of many researchers. Many schemes have been proposed to achieve chaotic control and synchronization [1–3]. Linear control techniques for low- dimensional chaotic systems, such as the Ott–Grebogi–Yorke method [4], have been successfully applied in a variety of phys- ical, chemical, and biological settings [5]. Because impulsive control allows the stabilization and synchronization of chaotic systems using only small control impulses [6], even though the chaotic behavior may follow unpredictable patterns, it has been widely used to stabilize and synchronize chaotic systems [7–10]. Impulsive synchronization of chaotic systems [11,12], lag synchronization of hyperchaos with application to secure communications [13] and a unified approach for impulsive lag synchronization of chaotic systems with time delay [14] have * Corresponding author. E-mail address: liyang991411@sohu.com (Y. Li). been also discussed. In practice, there exist many examples of impulsive control systems [15,16]. For instance, T. Yang et al. [8] achieved the synchronization of two identical chaotic sys- tems, i.e., Chua circuit, using the state variable at the fixed instant time as the impulsive signal. The importance of impul- sive control is that in many cases impulsive control may give an efficient method to deal with systems, which cannot endure continuous disturbance. Chaos displayed by phenomenological equations describing macroscopic dissipative processes such as chemical reactions, hydrodynamics and electrical circuits, its underlying micro- scopic dynamics has attracted considerable attention in the past few years. The chaotic dynamics governing the three-variable autocatalytic model is studied in [17]. The three-variable auto- catalytic system exhibits chaotic dynamics in an open system configuration. In [18], it uses a reactive lattice-gas automaton to provide a macroscopic description of the reaction dynamics, investigates a specific mass action chemical scheme, and to in- vestigate the three variable autocatalator, which shows a period- doubling cascade to a chaotic attractor. The three-variable au- tocatalator exhibits chaotic dynamics in an open system config- 0375-9601/$ – see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.12.073