in this paper, the synchronization of complex dynamical networks (CDNs) under 'successful' but recoverable attacks is studied by utilizing the framework of switching systems. 'Successful' attacks refer to a class of attacks that break a network into a group of isolated clusters. Recoverable attacks refer to a class of attacks that the network can recover from after a period of time. To facilitate the investigation of the synchronization of a CDN under 'successful' but recoverable attacks, the attack frequency and the average recovering time are introduced. By using a piecewise Lyapunov function, the upper bounds of the attack frequency and the average recovering time are obtained to ensure that the whole network can achieve global synchronization under attacks. (C) 2009 Elsevier Ltd. All rights reserved.