Refuting the odd number limitation of time-delayed feedback control

Details Refuting the odd number limitation of time-delayed feedback control Refuting the odd number limitation of time-delayed feedback control

2006-09-22

arxiv.org/abs/nlin/0609056v2

Phys. Rev. Lett. 98, 114101 (2007)

Nonlinear Sciences

Chaotic Dynamics

4 pages, 3 figures, in print at Phys Rev Lett

Scientific paper

10.1103/PhysRevLett.98.114101

We refute an often invoked theorem which claims that a periodic orbit with an odd number of real Floquet multipliers greater than unity can never be stabilized by time-delayed feedback control in the form proposed by Pyragas. Using a generic normal form, we demonstrate that the unstable periodic orbit generated by a subcritical Hopf bifurcation, which has a single real unstable Floquet multiplier, can in fact be stabilized. We derive explicit analytical conditions for the control matrix in terms of the amplitude and the phase of the feedback control gain, and present a numerical example. Our results are of relevance for a wide range of systems in physics, chemistry, technology,and life sciences, where subcritical Hopf bifurcations occur.

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