  We present an analytical approach to deal with nonlinear delay differential
equations close to instabilities of time periodic reference states. To this end
we start with approximately determining such reference states by extending the
Poincar'e Lindstedt and the Shohat expansions which were originally developed
for ordinary differential equations. Then we systematically elaborate a linear
stability analysis around a time periodic reference state. This allows to
approximately calculate the Floquet eigenvalues and their corresponding
eigensolutions by using matrix valued continued fractions.
