Realization of critical eigenvalues for scalar and symmetric linear delay-differential equations

TL;DR

This paper investigates the relationship between the number of delays and critical eigenvalues in scalar and symmetric delay-differential equations, providing conditions for their realization and applications to symmetric coupled systems.

Contribution

It establishes the existence of delay-differential equations with prescribed purely imaginary eigenvalues and generalizes this to systems with multiple factors and symmetry.

Findings

Existence of scalar delay equations with specified imaginary eigenvalues.

Extension to systems with multiple characteristic factors.

Application to delay-coupled symmetric cell systems.

Abstract

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.