Here x is a vector in R”, the parameter c( is an element of a finite dimensional real Banach space d, and F and G are mappings from d x BUC(R; R”) into R”. Moreover, F(a, 0) = G(a, 0) = 0, so x E 0 is a solution of both (1.1) and (1.2). In addition we suppose that the linearizations of (1.1) and (1.2) have a one-parameter family of nontrivial periodic solutions at a critical value a0 of the parameter. Our aim is to show that the nonlinear equations have one-parameter families of nontrivial periodic solutions for values of a close to ao. Most of the existing Hopf bifurcation results for (1.1) and (1.2) require F and G to be nonanticipatory functionals with a finite delay, i.e., the terms F(a, x,) and G(a, x,) are allowed to depend only on the values of x(s) for t-r<s<t, where r is some fixed finite number (see, e.g., [l, 4, 5, 13, 14,261). Some resent results do exists (see [6] and [30]) where F is allowed to include an infinite delay, but then it is still assumed that F and G have an “exponentially fading” memory, i.e., that F(a, x,) and G(a, x,) are well defined for functions x which grow exponentially at minus infinity. In [9] a Hopf bifurcation result is proved for a certain quadratic 114 OO22-0396187 $3.00