Periodic Normal Forms for Bifurcations of Limit Cycles in DDEs

A recent work arXiv:2207.02480 by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates derivation of periodic normal forms. In this paper, we prove the existence of a special coordinate system on the center manifold that will allow us to describe the local dynamics on the center manifold in terms of these periodic normal forms. Furthermore, we characterize the center eigenspace by proving the existence of time periodic smooth Jordan chains for the original and the adjoint system.