The Hopf Bifurcation And Periodic Solutions Of Price Reyleigh Equation With Delay

This paper studies the Hopf bifurcation diagram and the sufficient criteria that a nonconstantperiodic solutions exist of the price Reyleigh equation with delay. First we introduce the process ofbuilding up a Reyleigh ordinary differential equation and its basic results before we discuss theequation with delay. Then we add delay to the ordinary differential equation and give the Hopfbifurcation diagram in the r-γparameter plane of the equation with delay by the method paper [1]provides and the properties of function of functions and the monotonicity of functions. It'smeaningful in the field of distribution partition of the characteristic roots of RFDE. Furthermore, we point out the direction of the Hopf bifurcation on the first bifurcation curve Cand the stability of the periodic solutions according to paper [2]. Finally, we give the sufficientcriteria for the existence of a nonconstant periodic solutions of Reyleigh equation with delay byusing the theorem in paper [6]. Finally we compare the results of ordinary differential equations and retarded functionaldifferential equations, some new phenomena occur in ecology after we add delay to the equation.The results are meaningful both in theories and in practices.