In our adjacent work, we developed a spectral comparison principle for
compound cocycles generated by delay equations. In particular, this principle
allows to derive frequency conditions (inequalities) for the uniform
exponential stability of such cocycles by means of their comparison with
stationary problems. Such inequalities are hard to verify analytically since
they contain resolvents of additive compound operators and to compute the
resolvents it is required solving a first-order PDEs with boundary conditions
involving both partial derivatives and delays.
In this work, we develop approximation schemes to verify some of the arising
frequency inequalities. Beside some general results, we mainly stick to the
case of scalar equations. By means of the Suarez-Schopf delayed oscillator and
the Mackey-Glass equations, we demonstrate applications of the theory to reveal
regions in the space of parameters where the absence of closed invariant
contours can be guaranteed. Since our conditions are robust, so close systems
also satisfy them, we expect them to actually imply the global stability, as in
known finite-dimensional results utilizing variants of the Closing Lemma which
is still awaiting developments in infinite dimensions