Realization of critical eigenvalues for scalar and symmetric linear   delay-differential equations
  This paper studies the link between the number of critical eigenvalues and the number of delays in certain classes of delay-differential equations. There are two main results. The first states that for k purely imaginary numbers which are linearly independent over the rationals, there exists a scalar delay-differential equation depending on k fixed delays whose spectrum contains those k purely imaginary numbers. The second result is a generalization of the first result for delay-differential equations which admit a characteristic equation consisting of a product of s factors of scalar type. In the second result, the k eigenvalues can be distributed amongst the different factors. Since the characteristic equation of scalar equations contain only exponential terms, the proof exploits a toroidal structure which comes from the arguments of the exponential terms in the characteristic equation. Our second result is applied to delay coupled D_n-symmetric cell systems with one-dimensional cells. In particular, we provide a general characterization of delay coupled D_n-symmetric systems with arbitrary number of delays and cell dimension. 
Pietro-Luciano Buono and Victor G. LeBlanc