Control Problems Of Fractional-order Chaotic System

Non-linear science acts as one of the most hot topics in the world,which involves every aspects of science and nature.Chaos is a new topic,and it is one of the most important topics in non-linear science,whose development is also concerned by more and more scholars.Besides,triangle system is an important system form,which control design has some good advantages,such as the controller designed based on triangle system form is not only easy but also extended,which can solve a lot of control problems for both linear and non-linear systems.Therefore,this paper applies the theorem results of triangle system to integer-order chaotic systems and fractional-order chaotic systems,and has many significant results.In this paper,we mainly investigate the control problems based on triangle system form both for integer-order chaotic systems and fractional-order chaotic systems,such as stabilization,synchronization,anti-synchronization,simultaneous synchronization and anti-synchronization,projective synchronization.Firstly,based on triangle system form,we investigate the control problems of integer-order chaotic systems in three cases: 1)integer-order chaotic systems are triangle system form itself;2)there exist a non-singular transformation which can convert the integer-order chaotic systems into triangle system form;3)there exist a non-singular transformation which can convert the integer-order chaotic systems into triangle system form partly.Based on the theorem results of triangle system and the control design results of non-linear systems,we have some new results of control design of the above integer-order chaotic systems.Similarly,we use the similar methods investigate the control problems of fractional-order chaotic systems based on triangle system form,and we have some new results.The first contribution of this paper is that based on the results of triangle system,we investigate the control problems of integer-order chaotic systems and fractionalorder chaotic systems.The second contribution of this paper is that we have some new results of triangle system based on adaptive feedback method.The controller designed by this method is not only easy but also has popularity.The third contribution of this paper is that we apply back-stepping method to triangle system and we have some new results.The controller designed by this method is not only the single controller but also has popularity.The first chapter of this paper is the introduction,we introduce the fractional-order chaotic systems,and whose current situation,background and significance in detail.The second chapter of this paper is preliminaries,we present the stability theorem of integerorder chaotic systems and fractional-order chaotic systems respectively.At the same time,we introduce the definition of stabilization,synchronization and antisynchronization,simultaneous synchronization and anti-synchronization,projective synchronization,which are the basic theorems for the next research.The third chapter of this paper is the control problems of integer-order chaotic systems.In this chapter,firstly,based on the theorem results of integer-order triangle system,we make the transformation for integer-order chaotic systems in three cases.After that,based on back-stepping method and adaptive feedback control method,we investigate the control problems of integer-order chaotic systems.Finally,illustrative examples with numerical simulations are used to demonstrate the validity and effectiveness of the above results.The fourth chapter of this paper is the control problems of fractional-order chaotic systems.In this chapter,firstly,based on the theorem results of fractional-order triangle system,we make the transformation for fractional-order chaotic systems in three cases.After that,based on back-stepping method and linear feedback control method,we investigate the control problems of fractional-order chaotic systems.Finally,illustrative examples with numerical simulations are used to demonstrate the validity and effectiveness of the above results.The fifth chapter includes the prospect and summary,which gives the innovation points,deficiencies and future research.