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We study the stability of unstable steady states in a class of retarded time-delayed systems, the systems of Mackey-Glass, Ikeda and Sprott, subjected to a variable-delay feedback control. The important aspect of such a control problem is that time-delayed systems are already infinite-dimensional before the delayed feedback control is turned on. The approach consists of relating the stability properties of the resulting variable-delay system with those of an analogous distributed delay system when the frequency of the modulation is large compared to the system's dynamics. Under such conditions we show that control domains are significantly larger than those in the usual time-delayed feedback control, and that the complexity of the domain structure depends on the form of the delay modulation. The conditions for stability of the fixed points of the linearized systems are obtained. In addition, in analogy to the odd-number limitation in finite-dimensional systems, we found that the control method fails to achieve controllability for certain unstable steady states for any value of the feedback control parameters.
REFERENCES: [1] D. Mackey and L. Glass, Science 197, 28 (1977). [2] K. Ikeda, Opt. Commun. 39, 257 (1979); K. Ikeda and K. Matsumoto, Physica D 29, 223 (1987). [3] J. C. Sprott, Phys. Lett. A 366, 397 (2007). [4] P. Hoevel and E. Schoell, Phys. Rev. E 72, 046203 (2005). [5] A. Gjurchinovski and V. Urumov, Europhysics Letters 84, 40013 (2008); A. Gjurchinovski and V. Urumov, God. Zb. Inst. Mat. 41, 57 (2008). |
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