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Time-delayed feedback control: qualitative promise and quantitative constraints

We discuss stabilization of planar rotating waves via feedback which is delayed by an integer multiple of the minimal period. We refute the so-called "odd number limitation" which claims that periodic orbits cannot be stabilized, if they possess an odd number of positive real unstable Floquet multipliers. On the other hand, we present a large, but sharp, bound on such a multiplier, beyond which delayed feedback control cannot be achieved.
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