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In the spirit of the well-known odd-number limitation, we study failure ofPyragas control of periodic orbits and equilibria. Addressing the periodicorbits first, we derive a fundamental observation on the invariance of thegeometric multiplicity of the trivial Floquet multiplier. This observationleads to a clear and unifying understanding of the odd-number limitation, bothin the autonomous and the non-autonomous setting. Since the presence of thetrivial Floquet multiplier governs the possibility of successful stabilization,we refer to this multiplier as the determining center. The geometric invarianceof the determining center also leads to a necessary condition on the gainmatrix for the control to be successful. In particular, we exclude scalargains. Application of Pyragas control on equilibria does not only imply ageometric invariance of the determining center, but surprisingly also oncenters which resonate with the time delay. Consequently, we formulate odd- andany-number limitations both for real eigenvalues together with arbitrary timedelay as well as for complex conjugated eigenvalue pairs together with aresonating time delay. The very general nature of our results allows forvarious applications.

Geometric invariance of determining and resonating centers: Odd- and any-number limitations of Pyragas control