Advances in Synchronization of Coupled Fractional Order Systems : Fundamentals and Methods After a short introduction to the fundamentals this book provides a detailed account of major advances in applying fractional calculus to dynamical systems. Fractional order dynamical systems currently continue to gain further importance in many areas of science and engineering. As with many other approaches to mathematical modeling the first issue to be addressed is the need to couple a definition of the fractional differentiation or integration operator with the types of dynamical systems that are analyzed. As such for the fundamentals the focus is on basic aspects of fractional calculus in particular stability analysis which is required to tackle synchronization in coupled fractional order systems to understand the essence of estimators for related integer order systems and to keep track of the interplay between synchronization and parameter observation. This serves as the common basis for the more advanced topics and applications presented in the subsequent chapters which include an introduction to the Immersion and Invariance (I&I) methodology the masterslave synchronization scheme for partially known nonlinear fractional order systems Fractional Algebraic Observability (FAO) and Fractional Generalized quasi Synchronization (FGqS) to name but a few. This book is intended not only for applied mathematicians and theoretical physicists but also for anyone in applied science dealing with complex nonlinear systems Preface Introduction Basic Concepts and Preliminares Synchronization of Chaotic Systems by means of a nonlinear observer: An application to Secure Communications Synchronization for Chaotic system through an Observer using the Immersion and Invariance (I&I) Approach Synchronization of Nonlinear Fractional Order Systems by Means of PIra Reduced Order Observer Estimators for a class of commensurate fractional order systems with Caputo derivative Generalized Multi synchronization of Fractional Order Liouvillian Chaotic Systems using Fractional Dynamical Controller An Observer for a Class of Incommensurate Fractional Order Systems Fractional Generalized quasi synchronization of incommensurate fractional order oscillators Synchronization and Anti synchronization of fractional order chaotic systems by means of a fractional integral observer Appendix Index