Applications Of Some Algebraic Methods In Researching Dynamical Quality Of Delayed Differential Systems

By the development of science and technology, the cross among diferent subjects or inner-cross in mathematics has played more important roles in all kinds of the science. The cross between diferential equations and algebras is an inner-cross subject. Infact, the algebraic theory is applied in diferential equations in all of its history. Especially, the matrix theory and the linear space have been applied in diferential equations for many years.As one branch of diferential equations, the theory of delay-diferential equations is a hot field. For recent years, by the evolution of the computer technique, the theory on delay-diferential equations has been researched by manypeople. Algebraic methods will be applied to research the stable theory on some diferent delay-diferential equations in this paper. The algebraic methods in this paper mainly conclude matrix pencil, gen-eral eigenvalues, spectrum, Kronecker product, linear operations and symmetric groups. By those algebraic methods, the dynamic behavior on several diferent delay-diferential systems will be studied. Meantime the algebraic conditions on the distribution of the eigenvalues and the stability are discussed in the paper. The main issues are organized as the following:(1) This paper will study the bifurcation theory, which mainly contains general Hopf bifurcation theoryand symmetric bifurcation theory. Especially,by researchinga classof coupled delay-diferential equation, symmetric group and its presentation theory are used to elaborate on the spatial behavior, such as D3invariance and symmetric bifurcations.(2) This paper will research the dynamic property of general linear delay-diferential systems by some algebraic methods. The mainly researched cases of the general delay-diferential equations in this paper are singular or neutral systems with single delay or multiple delays. Especially,for the systems with multiple delays, the commensurate cases are discussed in this paper. As we all know, the distribution of the eigenvalues plays an important role in studying the stability and Hopf bifurcations on delay-diferential systems.Particularlythe appearanceof imaginaryeigenvaluesis usuallythe critical status between stability and instability. So the distribution of imaginary eigenvalues for distinct delay-diferential systems will be researched. Besides, the conditions on stability and bifurcations for those delay-diferential systems are discussed in this paper. (3) By algebraic methods, such as matrix pencil, Kronecker product, linear opera-tor and so on, a delayed reaction-difusion system with single delay will be investigated, which is also called semi-linear partial diferential equation. The distribution of eigen-values in the complex plane determines the stability of delayed reaction-difusion system. For the time parameterτ and the Neumann boundary condition on spatial domain, this paper will try to find the critical condition of the stable switch on the parameter τ.