| 摘 要： | The Hopf bifurcation Theorem requires that one at first knows all eigenvalues of the linearized system;and a diffieulty in applying the Hopf Theorem is that in the general case we are unable to solve an algebraic equation of degree n (n≥3) with parameter coefficients.This paper firstly proves Theorem 1 which enables us to know the existence of the Hopf bifuecation by direct use of the coefficients of the characteristic equ ation of the linearized system for the case where the characteristic equation of the linearized system has degree 2,3 or 4.Then by using Theorem 1,we gave out the condition for existing a periodic traveling-wave solution of Brusselator in diffusion by Hopf bifurcat on.Finally,we discuss the problem of bifurcation direction and the problem of stability of the bifurcation.
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