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Study on the Stability and Blowing-up Solutions of Fractional order Dynamic System

Author: HuHua
Tutor: YuJiMin
School: Guangxi Normal University
Course: Applied Mathematics
Keywords: multi-variables fractional-order nonlinear dynamic systems Lyapunov direct method generalized Mittag-Lefer stability blow-up solutions fractional comparison principle
CLC: O19
Type: Master's thesis
Year: 2012
Downloads: 53
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Abstract


Fractional order nonlinear dynamic system is fractional non-autonomous sys-tem on the basis of fractional diferential equations. It is widely used in biologi-cal model,viscoelastic materials, various material of genetic and memory, etc. In2011, Liyan.etc, who propose the definition of generalized Mittag-Lefer stability,study the generalized Mittag-Lefer stability of fractional order nonlinear dynam-ic system by fractional Lyapunov direct method. However, they lack the prac-tical application of relevant theory, and cannot solve the case of multi-variablesfractional order nonlinear dynamic system. On that Base, we (my advisors, co-operators and I) study the generalized Mittag-Lefer stability of multi-variablesfractional order nonlinear dynamic system, and give a meaningful fractional orderLotka-Volterra predator-prey model to explain the stability and efectiveness ofthe system by the fractional Lyapunov direct method.In this paper, the research of another part is the existence of blowing-up solu-tions of fractional order nonlinear dynamic system. The existence of blowing-upsolutions of integer order dynamic system has been researched by scholars. How-ever, the existence of blowing-up solutions of fractional order nonlinear dynamicsystem, which is more superior than integer order system, has been in blank. Westudy the existence of blowing-up solutions of fractional order nonlinear dynamicsystem by fractional comparison principle and fractional diferential inequality,and illustrate the existence of blowing-up solutions of the system by two examples.The thesis is divided into four Chapters.In the first chapter, we introduce the theoretical background of fractionalcalculus, in which field the application has advantages. We introduce the originof fractional calculus, the forming process of fractional calculus, the study resultsat this stage and the present situation of development.In the second chapter, we introduce the definition and basic properties ofRiemann-Liouville and Caputo fractional calculus. In addition, we introduce thedefinition and basic properties of Mittag-Lefer functions, and give the solutionof fractional diferential equations with Mittag-Lefer function. In the third chapter, we analysis the present method and its shortcomings ofthe stability analysis of fractional order nonlinear dynamic system. We prove thegeneralized Mittag-Lefer stability of multi-variables fractional order nonlineardynamic system, by the definition of generalized fractional order Mittag-Leferstability and generalized fractional Lyapunov direct method. We apply a sim-ple method to find the candidate Lyapunov function to fractional order Lotka-Volterra predator-prey system, and plot by using Matlab/Simulink software.In the fourth chapter, we introduce the development background and theobtained results about the existence of blowing-up solutions of integer order non-linear dynamic system. We study the existence of blowing-up solutions of frac-tional order nonlinear dynamic system, by the basic properties and theorem offractional order nonlinear dynamic system.We also study the stability and stabilization of multi-variables fractional-order linear retarded systems with nonlinear uncertain parameters. For the con-sideration of the continuity, this part do not appear in this paper.

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CLC: > Mathematical sciences and chemical > Mathematics > Dynamical systems theory
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