In this paper, chaos synchronization in coupled discrete-time dynamical systems is studied. Computer results display the interesting synchronization behaviors in the mutually coupled systems. As possible applications of chaos synchronization, parameter estimations and secure communications are proposed. Furthermore, a modified OGY method is given, which converts a chaotic motion into a periodic motion.

The singular point analysis, such as the Painlev test and Yoshida's test, is a computational method and has been implemented in a symbolic computational manner. But, in applying the singular point analysis to high dimensional and/or "complex" dynamical systems, we face with some computational difficulties. To cope with these difficulties, we propose a new numerical technique of the singular point analysis with the aid of the self-validating numerics. Using this technique, the singular point analysis can now be applicable to a wide class of high dimensional and/or "complex" dynamical systems, and in many cases dynamical properties such as the algebraic non-integrability can be proven for such systems.