On the stability of delayed feedback control of chaos

Limitations in controlling chaos in continuous dynamical systems by delayed feedback are proved. The results are as follows: (1) If the linear variational equation about a hyperbolic unstable periodic orbit (UPO) has an odd number of real characteristic multipliers which are greater than unity, the UPO can never be stabilized with any value of feedback gain. (2) If all the characteristic exponents of the variational equation are different from each other and at least one of them is real and positive, then the UPO can never be stabilized with any feedback gain matrix of the form of diag(k, ..., k). These theorems are proved on the basis of Floquet theory. The result of the first theorem is also explained intuitively using bifurcation theory