A comparison of numerical solvers for the delay eigenvalue problem of coupled oscillators

In this work, we conduct a comparison for various delay differential equations' solvers to determine the strength and weakness points of each one. The comparison is clarified through a numerical example in which we consider the coupled harmonic oscillators. We obtain the eigenvalues of the delayed coupled harmonic oscillators system by solving the governing equations by all considered methods. However, the methods of concern in this paper are as follows: the Taylor series expansion, Rekasius's substitution, self-consistent approach, and the Krylov method with Chebyshev interpolation. The main aspects that we investigate in this paper are the stability, robustness, and numerical efficiency. We found that the Krylov method with Chebyshev interpolation performs better than all the other considered ones.