Some Researches On The Application Of Fractional Differential Equations

Fractional calculus is a 300-year-old mathematical topic. Although it had such along time research history, for many years, the studies were concentrated on a puretheoretical field. However, during the last ten years or so, with the development ofapplication of fractional calculus, such as the description of memory and hereditaryproperties of various materials, increasingly used to model problems in rheology and inmaterials and mechanical systems, signal processing and systems identification, ANN,fractal and chaos, and other areas of applications, the research on fractional differentialequations(FDEs) became more and more popular.This paper mainly discussed the application of FDEs in infectious disease modelsand chaotic systems. According to the memory and hereditary properties of spreadprocess of infectious virus, and fractional calculus have the feature of description ofmemory property, a fractional order model of HIV infection of CD4~+T-cells was establishedand the dynamic nature of HIV infection process was discussed. And for chaos inrecent years, the chaotic and hyperchaotic dynamics of fractional order systems beganto attract more and more attention. In this paper, the synchronization of a new fractionalorder hyperchaotic system and orbit control of general fractional order chaoticsystems were studied.This thesis consisted of four chapters, which could be divided into two parts. Thefirst part (Chapters 2) focused on the existence and uniqueness of positive solutions anddynamic nature of the HIV infection of CD4~+T-cells model. The second part(Chapters 3) concentrated on the synchronization and orbit control of fractional order chaoticsystem.In Chapters 2, a fractional order model of HIV infection of CD4~+ T-cells as followwas studied:where 0<γ≤1,D_*~γis the fractional derivative in the sense of Caputo. Accordingto the global existence theorem of FDEs as well as the property of fractional orderfunctions, the existence and uniqueness of positive solution of above model were proved.Then, the local stability of model's equilibrium points was discussed by application ofRouth-Hurwitz conditions of FDEs, and sufficient conditions for the local asymptoticstability of equilibrium points were obtained. Numerical simulations were presented toillustrate the results of the equilibrium points and the effectiveness of cure termρI.In Chapters 3, firstly, the following four-dimensional fractional order Chen-Leehyperchaotic system was considered:where 0＜q＜1. Numerical simulation showed that the fractional order Chen-Leesystem existed hyperchaotic attractor and its structure was more complex than thecorresponding hyperchaotic attractor of integer-order Chen-Lee hyperchaotic system.Then, based on the stability of fractional calculus theory, synchronization of abovesystem was achieved with adaptive feedback method and Laplace transform theorymethod. Numerical simulation showed that these two kinds of synchronization controlmethods were effective. Secondly, the orbit control of following fractional order chaotic system was analyzed:where A is a n×n constat matrix, and the nonlinear term g(t, x(t)) satisfies g(t, 0) = 0.By application of property and theory of fractional calculus, above chaotic system wasanalyzed and achieved: If a suitable linear state feed-back controller u(t) = Kx(t)was adopted to above nonlinear system, then the trajectory x(t) of the system couldforce into the target orbit x = 0.