Bifurcation Study Of Birkhoff System Based On Gradient Method

Based on the methoddogy of the gradient system,this paper investigated the bifurcation of the generalized Birkhoff system,which includes the stability of equilibrium,types of singular points,bifurcation of singular points,static bifurcation as well as the nonexistence of limit cycles.Chapter one presents an brief overview of previous and present studies concerning the qualitative theory of Birkhoff system.Chapter two discusses the stability of equilibrium of the autonomously-constrained generalized Birkhoff system and the conditions for the transformation of the autonomously-constrained generalized Birkhoff system to the gradient system,whose qualities provide a foundation for the discussion.Chapter three analyzes the types of singular points which belong to the second-order autonomous generalized Brikhoff system and puts forward the conditions for the transformation of the corresponding linearized system to the gradient system,whose qualities serves as a basis for the analysis of the singular points of the corresponding linearized system.Types of singular points of the corresponding non-linearized system,on the other hand,are also explained on the basis of Perron theorem.Chapter four studies the bifurcation of singular points of the second-order autonomous generalized Birkhoff system,which presents necessary condition for the bifurcation in the first place and analyzes bifurcation by adopting Lypunov-Schmidt methoddogy.Chapter five,probes into the static bifurcation of a type of non-autonomous generalized Birkhoff system by further explaining its gradient representation as well as the stability of the whole system.Chapter six elaborates on the nonexistence of limit cycles of the second-order autonomous generalized Birkhoff system by the means of using the characteristic of the gradient system.At last,we make a conclusion of the whole paper and show some expectations for future researches.