Several Problems Of Fractional Dynamical Systems

Fractional calculus (fractional diferential and fractional integral) emerged on1695.It did not attracted extensive attention until1970s. Especially, in recent years, a largenumber of fractional systems in soft matter, control engineering, anomalous difusion,rheology and other areas are derived, the theories and applications of fractional diferen-tial systems are more studied. Compared to the classic dynamic systems generated byordinary diferential equations, the fractional diferential systems are more complex, andmathematical theories and methods are far from mature. Therefore, further studying onfractional dynamical systems is essential in both theoretical aspect and practical aspect.This dissertation mainly includes three parts: The linearization theorem and sta-bility analysis of fractional dynamical systems. The existence of solution to fractionaldiferential equations and the asymptotics of solution to fractional anomalous difusionequations. The main research works are as follows.The first chapter briefly introduces the development of fractional calculus and thedefinitions and properties of fractional calculus.The second chapter studies the linearization theorem of fractional dynamical sys-tems with Caputo derivative (namely, the fractional Hartman theorem). First, we give thedefinition of fractional dynamical systems with Caputo derivative. Then, using the prop-erties of fractional dynamical systems and the definition of topological equivalence, weprove the linearization theorem, and two examples are provided to support the theoreticalanalysis. At last, we investigate the stability of nonlinear fractional diferential systemswith Caputo derivative, and get the asymptotically stable theorem (namely, Audounet-Matignon-Montseny conjecture).Chapter III discusses the existence conditions of solution to the fractional diferen-tial equation. First, using the fixed point theorem, we get the existence conditions ofsolution to fractional diferential equation in Riemann-Liouville sense. And then, wegive the conditions when the fractional diferential equation with Caputo derivative hasperiodic solution.In the last chapter, by Laplace transform and Fourier transform, we get the asymp-totics of solution to anomalous difusion equations (the sub-difusion equation, the order α∈(0,1) and the super-difusion equation, the order α∈(1,2)).