Abstract: This paper deals with the dynamical properties of the Hopf bifurcation diagram of the price Reyleigh equation model with delay, and the stability of equilibrium point and the existence of Hopf bifurcation by the method of τ-Dpartitioning approach of exponential polynomial. Furthermore, we gained the computational forumla of the direction of the Hopf bifurcation and the stability of the periodic solutions with the Hopf bifurcation formula of Liénard equation with finite delay and completely gave the Hopf bifurcat ion diagram in the parameterplane of r-γ. Finally we found an important conclusion that delay reflects that price has an hysteresis on supply , by which we can explain the shake phenomenon of price in economy life.

Key words: price Reyleigh equation, delay, Hopf bifurcation, direction of bifurcation, stability, image of bifurcation