The Fuller index and global Hopf bifurcation

Using an index for periodic solutions of an autonomous equation defined by Fuller, we prove Alexander and Yorke's global Hopf bifurcation theorem. As the Fuller index can be defined for retarded functional differential equations, the global bifurcation theorem can also be proved in this case. These results imply the existence of periodic solutions for delay equations with several rationally related delays, for example, $\text{x}\text{?}\text{(t) = ?α[ax(t ? 1) + bx(t ? 2)]g(x(t))}$, with a and b non-negative and α greater than some computable quantity ξ(a, b) calculated from the linearized equation.