A Hamiltonian with periodic orbits having several delays

In 1974 Kaplan and Yorke introduced a certain delay equation with an arbitrary number of delays, x′(t)=-f(x(t-1))-f(x(t-2))-⋯-f(x(t-n)), conjecturing that it has periodic solutions when f is an odd homeomorphism of the reals which is differentiable at the origin and infinity. By coupling the delay equation to a vector field on Rn+1 they were able to prove, when certain conditions on the derivative of f at 0 and ∞ are satisfied, one delay and two delay versions. We find that the closed orbits of the coupled vector field occur at the points of intersection of two hyperplane fields which are invariant under the flow and invariant under deformation to a linear vector field. By analyzing properties of the linear vector field we are able to give an elementary construction of periodic solutions to the general delay equation when conditions naturally extending those of Kaplan and Yorke are satisfied.