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

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