  This paper considers the effect of additive white noise on the normal form
for the supercritical Hopf bifurcation in 2 dimensions. The main results
involve the asymptotic behavior of the top Lyapunov exponent lambda associated
with this random dynamical system as one or more of the parameters in the
system tend to 0 or infinity. This enables the construction of a bifurcation
diagram in parameter space showing stable regions where lambda is negative
(implying synchronization) and unstable regions where lambda is positive
(implying chaotic behavior). The value of lambda depends strongly on the
shearing effect of the twist factor of the deterministic Hopf bifurcation. If
the twist factor is sufficiently small then lambda is negative regardless of
all the other parameters in the system. But when all the parameters except the
shear coefficient are fixed then lambda tends to infinity as the shear
coefficient tends to infinity.
