The Research On Theory And Applications On Function Projective Synchronization Of Fractional Order Chaotic Systems

Chaos phenomenon which is widely found in nature is a complex mathemati-cal form. There are many features of chaotic signals, such as extreme sensitivity toinitial value, noise and long-term unpredictable characteristics. So it is particularlyapplicable to the feld of information security and has been concerned about thedomestic and foreign scholars. Indeed integer order chaotic system is idealized realchaotic systems, and the use of fractional diferential operator can be more accurate-ly to describe the actual dynamic behavior.Compared to the integer order chaoticsystem, fractional order chaotic system has a more complex dynamic behavior, soit has a very broad application prospects in information security, image processingand other engineering felds.So far, the development of chaos theory integer order is relatively complete. Bycontrast, fractional chaos theory is still in the development stage, a lot of questionsneed further study. For example, fractional order chaotic system stability man-agement is not perfect, and some chaotic synchronization methods for the integerorder systems do not apply to fractional order chaotic system. In addition, peopleapply fractional chaos theory to communications, cryptography, and we achievesome initial results. Therefore, the study fractional synchronization has importanttheoretical signifcance and application value.Chaos synchronization refers to two or more of chaotic systems under theaction of external forces to show the common characteristic behavior. For now,synchronization of chaotic systems is often based on a scale factor, and there are few studies on the function of the scale factor. Because the functions can increasethe proportion of the security of confdential communications, the study of the func-tion projective synchronization problem has theoretical signifcance and applicationvalue.This paper mainly studies the function projection synchronization problemof fractional order chaotic systems and gives the corresponding synchronizationcontroller design method. The main content is divided into four chapters.1. In the frst chapter, a brief introduction to the origin and development ofchaos, the defnition of chaos and the basic characteristics of chaos is given andstatement of the concept of chaotic system synchronization and control methods ispresented. We summarize the defnition of fractional calculus and get the Lyapunovstability theorem of stability theorems and fractional nonlinear systems. This laysthe foundation for theoretical analysis in behind.2. In the second chapter, we introduce problems of the fractional order unifedsystem functions in single variable coupling projection condition synchronization,and we get the appropriate projective synchronization design of adaptive controlfunction.3. In the third chapter, a brief introduction of fractional Lorenz system isbeen given, and fractional Lorenz system synchronization is given as a functionof projected single-variable coupling state. The parameter identifcation rules ofLorenz system with unknown parameters is realized and we get the adaptive controldesign of the nonlinear controller.4. In the fourth chapter, we describe the function projective synchronizationof fractional Rabinovich-Fabrikant system.The content of the thesis is the conclusion including:(1) The design is given that is based on a single variable fractional stabilitytheory of fractional unifed chaotic system.Theorem2.1Let ek1=k1-k10, ek2=k2-k20, and k10,k20be constants, the parameter adaptive rate satisfy Let controller κ10and κ20satisfy where m is the bounder of the system.Then,for(x1(0),x2(0);y1(0),y2(0)；κ1(0),κ2(0)), the system will be synchronized,that is limtâ†'+∞|ei|=0(i=1,2).(2)By using fractional stability theorem,an adaptive controller is designed to achieve a projection function of fractional Lorenz synchronization under different conditions.Theorem3.1Let and eκ1=κ1一κ10,eκ2=κ2一κ20,the parameter adaptive rate satisfy dqκ1/dtq=e21, dqκ2/dtq=e22.Ifκ10and κ20satisfy in which m is the border of the chaos system,then the system(3.2.1)will be syn-chronized,that is limtâ†'＋∞|ei|=0(i=1,2,3).Theorem3.2Let controller where m is the border of the chaos system.the system will be synchronized,that is lim tâ†'＋∞|ei|=0(i=1,2,3)(3)By using the Lyapunov stability theorem,we get the function proj ective synchronization of fractional Rabinovich-Fabrikant system.Theorem4.1Let controllers I(?) eκ1=κ1—κ10,eκ2=κ2—κ20,eκ3=κ3—κ30,and κ10,κ20,κ30are all constants, the parameter adaptive rate satisfy and if in which m is the border of the chaos system,then the system will be synchro-nized,that is limtâ†'＋∞|ei|=0(i=1,2,3) In this paper, for each theorem are proved in detail.