  In the spirit of the well-known odd-number limitation, we study failure of
Pyragas control of periodic orbits and equilibria. Addressing the periodic
orbits first, we derive a fundamental observation on the invariance of the
geometric multiplicity of the trivial Floquet multiplier. This observation
leads to a clear and unifying understanding of the odd-number limitation, both
in the autonomous and the non-autonomous setting. Since the presence of the
trivial Floquet multiplier governs the possibility of successful stabilization,
we refer to this multiplier as the organizing center. The geometric invariance
of the organizing center also leads to a necessary condition on the gain matrix
for the control to be successful. In particular, we exclude scalar gains.
Application of Pyragas control on equilibria does not only imply a geometric
invariance of the organizing center, but surprisingly also on centers which
resonate with the time delay. Consequently, we formulate odd- and any-number
limitations both for real eigenvalues together with arbitrary time delay as
well as for complex conjugated eigenvalue pairs together with a resonating time
delay. The very general nature of our results allows for various applications.
