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.

Provides a self-contained account on recent advances in fractional order dynamical systems Contains a lot of introductory and background material, as well as worked examples for newcomers to the field Offers a special chapter on secure communications as newly emerging topic for fractional order systems modeling

Autorentext

Rafael Martínez-Guerra obtained the B.Sc. degree in Physics and Mathematics from the National Polytechnic Institute (IPN), Mexico City (Mexico) and the M.Sc. degree in Automatic Control from CINVESTAV-IPN (Center for Research and Advanced Studies of IPN), Mexico City. He also got a M.Sc. degree in Mathematics and Automatic from Ecole des Mines, Paris (France) and earned his Ph.D. Degree from the Universidad Autónoma Metropolitana in 1996. Currently, he is a Researcher at the Automatic Control Department of the CINVESTAV-IPN and a Member of the National System Researchers since 1992 (Level 3, now). He is author and co-author of more than 90 papers in international journals and more than 140 contributions in international conference proceedings. He is co-author of 9 books. His main research interests are in the areas of nonlinear systems, differential geometric and differential algebraic methods, fractional nonlinear observers, fractional fault detection and diagnosis, secure communications, fractional synchronization and chaos. Juan Pablo Flores-Flores obtained the B.Sc. degree as Aerospace Engineer from the National Polytechnic Institute (IPN) and the M.Sc. degree in Control Theory from the Automatic Control Department of CINVESTAV-IPN (Center for Research and Advanced Studies of IPN). He earned his Ph.D. from the same institution in 2022. He is co-author of the book Encryption and Decryption Algorithms for Plain Text and Images using Fractional Calculus (Springer, 2023) and has published research papers in international journals and conference proceedings. His main research interests are nonlinear systems, differential algebra methods, partial differential equations systems and integer and fractional synchronization of nonlinear systems.

Inhalt
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.