A functional analytic approach to validated numerics for eigenvalues of delay equations

Abstract

This work develops validated numerical methods for linear stability analysis at an equilibrium solution of a system of delay differential equations (DDEs). In addition to providing mathematically rigorous bounds on the locations of eigenvalues, our method leads to validated counts. For example we obtain the computer assisted theorems about Morse indices (number of unstable eigenvalues). The case of a single constant delay is considered. The method downplays the role of the scalar transcendental characteristic equation in favor of a functional analytic approach exploiting the strengths of numerical linear algebra/techniques of scientific computing. The idea is to consider an equivalent implicitly defined discrete time dynamical system which is projected onto a countable basis of Chebyshev series coefficients. The projected problem reduces to questions about certain sparse infinite matrices, which are well approximated by \begin{document}$ N \times N $\end{document} matrices for large enough \begin{document}$ N $\end{document} . We develop the appropriate truncation error bounds for the infinite matrices, provide a general numerical implementation which works for any system with one delay, and discuss computer-assisted theorems in a number of example problems.