On analytical properties of delayed feedback control of chaos

Two theorems on limitations in controlling chaos by delayed feedback control are proved. The results are as follows: (1) If the linear variational equation about the target hyperbolic unstable periodic orbit (UFO) has an odd number of real characteristic multipliers which is greater than unity, the UFO can never be stabilized with any value of the 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 UFO can never be stabilized with any feedback gain matrix of the form diag(k,..., k). Both theorems are proved on the basis of Floquet theory. The result of the first theorem is also explained intuitively using bifurcation theory. (C) 1997 Elsevier Science B.V.