  The renormalization method which is a type of perturbation method is extended
to a tool to study weakly nonlinear time-delay systems. For systems with
order-one delay, we show that the renormalization method leads to reduced
systems without delay. For systems with order-one and large-delay, we propose
an extended renormalization method which leads to reduced systems with delay.
In some examples, the validities of our perturbative results are confirmed
analytically and numerically. We also compare our reduced equations with
reduced ones obtained by another perturbation method.
