Physica D 203 (2005) 209–223 Reconstruction of time-delayed feedback systems from time series M.D. Prokhorov a,∗ , V.I. Ponomarenko a,b , A.S. Karavaev b , B.P. Bezruchko a,b a Saratov Department of the Institute of RadioEngineering and Electronics, Russian Academy of Sciences, Zelyonaya Street 38, Saratov 410019, Russia b Department of Nonlinear Processes, Saratov State University, Astrakhanskaya Street 83, Saratov 410012, Russia Received 27 April 2004; received in revised form 13 February 2005; accepted 28 March 2005 Available online 13 April 2005 Communicated by M. Ding Abstract For various classes of time-delay systems we propose the methods of their model delay-differential equation reconstruction from time series. The methods are based on the characteristic location of extrema in the time series of time-delay systems and the projection of infinite-dimensional phase space of these systems to suitably chosen low-dimensional subspaces. We verify our methods by using them for the recovery of time-delay differential equations from their chaotic solutions and for modelling experimental systems with delay-induced dynamics from their chaotic time series. © 2005 Elsevier B.V. All rights reserved. PACS: 05.45.-a; 05.45.Tp Keywords: Parameter estimation; Delay-differential equations; Time series analysis; Nonlinear delayed feedback system 1. Introduction Systems, whose dynamics is affected not only by the current state, but also by past states, are wide spread in nature [1]. Usually these systems are modelled by delay-differential equations. Such models are success- fully used in many scientific disciplines, like physics, ∗ Corresponding author. Tel.: +7 8452 511 180; fax: +7 8452 261 156. E-mail address: sbire@sgu.ru (M.D. Prokhorov). physiology, biology, economics and cognitive sciences. Typical examples include population dynamics [2], where individuals participate in the reproduction of a species only after maturation, or spatially extended sys- tems, where signals have to cover distances with finite velocities [3]. Within this rather broad class of sys- tems, one can find the Ikeda equation [4] modelling the passive optical resonator system, the Lang–Kobayashi equations [5] describing semiconductor lasers with op- tical feedback, the Mackey–Glass equation [6] mod- elling the production of red blood cells, and many other 0167-2789/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2005.03.013