Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2013, Article ID 863659, 2 pages http://dx.doi.org/10.1155/2013/863659 Editorial Recent Advances in Hybrid Dynamical Systems Xinzhi Liu, 1 Piyapong Niamsup, 2 Qiru Wang, 3 and Yi Zhang 4 1 Department of Applied Mathematics, University of Waterloo, Waterloo, Canada N2L 3G1 2 Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, Tailand 3 School of Mathematics and Computational Science, Sun Yat-Sen University, Guangzhou 510275, China 4 School of Science, China University of Petroleum, Beijing 102249, China Correspondence should be addressed to Xinzhi Liu; xzliu@uwaterloo.ca Received 29 October 2013; Accepted 29 October 2013 Copyright © 2013 Xinzhi Liu et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Tere has been extensive research in hybrid dynamical systems in recent years due to their important applica- tions in various industrial and technological areas such as communication, complex networks, biotechnology, artifcial intelligence, switching circuits in power electronics, space- crafs control, and ecosystems management. Tis special issue consists of eight excellent papers that represent new and important developments in the feld of hybrid dynamical systems. Hopefully, this special issue will stimulate further research on this topic and collaboration to the world of scientifc community. Te paper “A new series of three-dimensional chaotic sys- tems with cross-product nonlinearities and their switching ” by X. Zhao, F. Jiang, Z. Zhang, and J. Hu introduces a new series of three-dimensional chaotic systems. Based on some conditions, it analyzes the globally (conditional) exponen- tially attractive set and positive invariant set of these chaotic systems. Moreover, it gives some examples to show that the results and the exponential estimate are explicitly derived. It also constructs some three-dimensional chaotic systems with cross product nonlinearities and studies the switching system between them. In the paper “Exponential stabilizability of switched sys- tems with polytopic uncertainties” by X. Zhang, Z.-Q. Xia, and Y. Gao, the exponential stabilizability of switched non- linear systems with polytopic uncertainties are explored by employing the methods of nonsmooth analysis and the min- imum quadratic Lyapunov function. Te switchings among subsystems are dependent on the directional derivative along the vertex directions of subsystems. Especially, a sufficient condition for exponential stabilizability of the switched linear systems is established considering the sliding modes and the directional derivatives along sliding modes. Furthermore, the matrix conditions of exponential stabilizability are derived for the case of switched linear system and some numerical examples are given to show the validity of the synthesis results. Te paper “Stability in terms of two measures for nonlinear impulsive systems on time scales” by K. Zhang and X. Liu investigates the stability problems of nonlinear impulsive systems with fxed moments of impulses in terms of two measures on time scales. Sufficient conditions for (uniform) stability, (uniform) asymptotic stability, and instability in terms of two measures are derived by using the method of Lyapunov functions. Te obtained results include the existing results as the time scale reduces to the set of real numbers. Particularly, the results provide stability criteria for impulsive discrete systems in terms of two measures, which have not been investigated extensively. Two examples are presented to illustrate the efficiency of the proposed results. In the paper “-Stability and -stabilizability of stochastic nonlinear and bilinear hybrid systems under stabilizing switch- ing rules” by E. Seroka and L. Socha, the problem of th mean exponential stability and stabilizability of a class of stochastic nonlinear and bilinear hybrid systems with unstable and stable subsystems is considered. Sufficient conditions for the th mean exponential stability and stabilizability under a feedback control and stabilizing switching rules are derived. A method for the construction of stabilizing switching rules based on the Lyapunov technique and the knowledge of the regions of decreasing of Lyapunov functions for subsystems is given. Two cases, including a single Lyapunov function and