Physics Letters A 372 (2008) 5110–5114 Contents lists available at ScienceDirect Physics Letters A www.elsevier.com/locate/pla Delayed feedback control of time-delayed chaotic systems: Analytical approach at Hopf bifurcation Nastaran Vasegh ∗ , Ali Khaki Sedigh Faculty of Electrical Engineering, K.N. Toosi University of Technology, PO Box 16315-1355, Tehran, Iran article info abstract Article history: Received 14 January 2008 Received in revised form 8 June 2008 Accepted 10 June 2008 Available online 17 June 2008 Communicated by A.P. Fordy PACS: 05.45.Gg 02.30.Yy 02.03.Ks Keywords: Chaos control Delayed feedback controller Delayed chaotic models This Letter is concerned with bifurcation and chaos control in scalar delayed differential equations with delay parameter τ . By linear stability analysis, the conditions under which a sequence of Hopf bifurcation occurs at the equilibrium points are obtained. The delayed feedback controller is used to stabilize unstable periodic orbits. To find the controller delay, it is chosen such that the Hopf bifurcation remains unchanged. Also, the controller feedback gain is determined such that the corresponding unstable periodic orbit becomes stable. Numerical simulations are used to verify the analytical results.  2008 Elsevier B.V. All rights reserved. 1. Introduction It was a well established fact for decades that time delay re- duces the efficiency of a control scheme. Therefore, it was quite a surprise when the delayed feedback controller (DFC) proposed to generate control force for stabilizing unstable periodic orbits (UPOs) [1]. Since DFC is an effective method for chaos control, it has been receiving considerable attention recently [2–7]. The basic idea of DFC is to realize a continuous control of a dynam- ical system by applying a feedback signal which is proportional to the difference between the dynamical variable x(t ) and its delayed value: u(t ) = k ( x(t − T ) − x(t ) ) , (1) where, T is the delay time and k is the feedback gain. If the delay time T coincides with the period of UPO, then the feedback van- ishes on this UPO. This means that the feedback in form (1) does not change the solution of the system. The DFC has the advantage of not requiring prior knowledge of anything but the period of the desired orbit T , so it has been successfully used in quite diverse experimental contexts includ- * Corresponding author. Tel.: +98 21 4421 3993; fax: +98 21 6643 2673. E-mail address: vasegh@eetd.kntu.ac.ir (N. Vasegh). ing electronic oscillators [2], lasers [6], and a magneto-elastic sys- tem [7]. The task of stability analysis of DFC is not easy. Nevertheless, a full analytical eigenmode expansion of the linear delayed sys- tems and a weakly nonlinear analysis has been given in [8]. In [9] local and global Hopf bifurcation of scalar delayed model is considered. In [10,11], the problem of subcritical Hopf bifurcation of a generic model under DFC is studied. Theses results attempt to fill the gaps between the abstract theoretical results and nu- merical simulations about time delay dynamics. DFC with multiple delays has also been considered in [12,13]. Also, [14] contains a large number of relevant articles. Since delayed chaotic systems are a large class of chaotic sys- tems [15] and references therein, the DFC has been used to control them, where the controller time delay can be different from the model time delay [16–18]. However, the delay has to be exactly adjusted to stabilize a true periodic orbit, despite the period of an unstable orbit not being known a priori. For this reason, several re- searchers [1–5,19,20] have suggested iterative techniques to adapt the delay. To the best of our knowledge, there is only one analyt- ical result in the current literature to adjust T for stabilizing UFPs [21] in time delayed chaotic models and no analytical result for stabilizing UPOs. In this Letter we use DFC to stabilize UPOs of time delayed chaotic systems. We determine the time delay T of DFC by bi- furcation analysis of free system. Then by using center manifold 0375-9601/$ – see front matter  2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2008.06.023