
A functional
analytic approach to validated numerics for eigenvalues of
delay equations
JeanPhilippe Lessard
and Jason Mireles James
This work develops validated numerical
methods for linear stability analysis at an equilibrium
solution of a system of delay differential equations
(DDEs). 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 N x N matrices
for large enough N. 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 computerassisted theorems in a
number of example problems.
