Stability Analysis And Normal Form Computation Of Fractional Differential Equations

Fractional calculus (fractional derivative and fractional integral) has been pre-sented for more than three centuries. It has attracted more and more attention by applied scientists and engineers. Although in the earlier, many schloars did not understand it clearly, it has been found that fractional calculus has many poten-tial applications, such as in soft matter, control engineering, anomalous diffusion, rheology and others have made great progress.In brevity, this paper mainly contains three parts:The first part consists of chaptersⅠandⅡ, this section introduces the difinations and the properties of fractional calculus and the comparison theorem for fractional differential system in depth. The second part contains chaptersⅢ,ⅣandⅤ, which mainly deal with the stability of fractional differential systems, including the differential systems with Riemann-Liouville fractional derivative and with Miller-Ross sequential derivative. The third part of this dissertation investigates the basic bifurcations of the frac-tional differential systems, such as the transcritical bifurcation, fold bifurcation and pitchfork bifurcation.In details, the first chapter introduces the definitions of fractional calculus and its basic properties. The second chapter studies the comparison theorem of the fractional differential equations. The existence and uniqueness of the solutions of the fractional differential equations are also presented.In chapterⅢ, we discuss the stability of the fractional differential systems with Riemann-Liouville derivative. In this chapter, we investigate the stability of the linear system and the perturbed systems. According to the Jacobian matrix eigenvalue of the fractional differential systems, we discuss the stabilty of the systems with different cases of the eigenvalues, including the critical eigenvalue|arg(λ)|=απ/2, and zero eigenvalue.Chapter IV deals with the fractional differential systems with Miller-Ross se-quential derivative in the sense of the Riemann-Liouville derivative. We study the stability of the fractional differential system with the Miller-Ross sequential deriva-tive, including homogeneous system and non-homogeneous one. In chapter V, we consider the fractional differential systems with the Miller-Ross sequential derivative in the sense of the Caputo derivative. We mainly investigate the stability of the systems, where two cases are discussed, the homogeneous system and the non-homogeneous one.In the last chapter, we investigate the fundamental bifurcations of the fractional differential systems with the Caputo derivative, which include the transcritical bi-furcation, the fold bifurcation and the pitchfork bifurcation. By using the Taylor expansion and the Implicit Function Theorem, we obtain the normal forms of the fractional differential systems, which include the transcritical bifurcation, fold bifur-cation and the pitchfork bifurcation.