  We introduce a time-delay function in bulk viscosity cosmology. Even for bulk
viscosity functions where closed-form solutions are known, because of the
time-delay term the exact solutions are lost. Therefore in order to study the
cosmological evolution of the resulting models we perform a detail analysis of
the stability of the critical points, which describe de Sitter solutions, by
using Lindstedt's method. We find that for the stability of the critical points
it depends also on the time-delay parameter, where a critical time-delay value
is found which play the role of a bifurcation point. For time-delay values near
to the critical value, the cosmological evolution has a periodic evolution,
this oscillating behaviour is because of the time-delay function. We find a new
behaviour near the exponential expansion point, which can be seen also as an
alternative way to exit the exponential inflation.
