Proof (transyersality condition for Hopf-bifurcation). If then the transyersality condition holds: clA
(3.20)
at r = To Proof. By using implicit function theorem and differentiating the with respect to the
In this section, system parameter k is regarded as a bifurcation parameter to analyze the problem of bifurcation of NN model (1). The conditions of occurrence of Hopf bifurcation for NN model (1) will be obtained.
The linearized system of NN model (1) at the origin gives x3(t) = -kx1(t)+.11.1(t – r)+ anx2(t – r), (c,DPx2(t) = -kx2(t)+.21.1(t – r)+ .22.2(t — r), where ou = f’ (0)(i, j = 1, 2).
The associated characteristic equation of Eq. (2) can be obtained as Xl(s)+ X2(s)e-” + x3(s)e’s’ = 0, (3) where Xl(s)= s'”+*2 + + s*) + , Xz(s) = +.11s'”‘ + +000)1, X3(s)= .0.22 – .12.21. Multiplying e•-” on both sides of Eq. (3), then we have ar(s)e” + X2(s)+ X3(s)e-” = 0. (4) It is assumed that s = zu(cos i+i sin 1)(m = 0) is a purely imaginary root of Eq. (4), then one has IP0 costor + r212 sin tor = p., p2-, cos tor + pn sin tor = p25,
(2)
where
(5)
P Sear.
