The goal of these notes is to give a reasonahly com- plete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to spe- cific problems, including stability calculations. Historical- ly, the subject had its origins in the works of Poincare [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930. Hopf's basic paper [1] appeared in 1942...