Research On Synchronization Of One Class Of Nonlinear Fractional Dynamical Chaotic System

The chaos phenomenon is very important in the field of nonlinear science, and has been widely used in the fields of biology, engineering, communication and chemistry. Reviewing the history, experts and scholars have always considered that the phenomenon of chaos is a harmful behavior, and avoided in many practical fields. Until the end of twentieth century, the concepts of chaos synchronization and chaos control have been proposed. People have a comprehensive understanding of the nonlinear chaotic systems, and have a great research interest.The history of fractional calculus has a history about 300 years, and has been developing rapidly in the twentieth century. The study shows that lots of chaotic systems can be chaotic in the fractional derivative. At present, the study of fractional order chaotic system has become an important part of the nonlinear dynamics, however, it is not mature, and so it has great development space. The main contents of the paper are follows:Chapter One: The development of nonlinear science, the definition, characteristics, and research status of fractional order chaotic systems are briefly described.Chapter Two: The basic knowledge of fractional calculus is discussed, including the definition of fractional calculus, the operation property, the method of solving, and the stability theorems of fractional order chaotic systems.Chapter Three: In view of one parameter uncertain fractional Lorenz-Stenflo(LS) chaotic system. Firstly, the chaotic attractors of different phase plane are given. Then, based on the fractional-order stability theory, suitable adaptive synchronization controllers are designed. The method is proved by using Lyapunov stability theory. Numerical simulations are presented to verify the effectiveness of the control scheme.Chapter Four: The projective synchronization of different fractional order chaotic systems(R-F system and N-L system) with non-identical orders and parameters identification is studied. The adaptive projective synchronization controllers and identification parameter laws are developed. At last, based on the J function criterion, strict mathematic proof is given, numerical simulation demonstrates the effective and correctness of the method.Chapter Five: a summary of the full text is given. And the further research direction is pointed out.