Available online at www.sciencedirect.com Physica A 327 (2003) 124–128 www.elsevier.com/locate/physa Coherence of noisy oscillators with delayed feedback D. Goldobin a; b , M. Rosenblum a , A. Pikovsky a ; ∗ a Department of Physics, University of Potsdam, Postfach 601553, D-14415 Potsdam, Germany b Department of Theoretical Physics, Perm State University, 15 Bukireva str., 614990 Perm, Russia Abstract A theory of eect of delayed feedback on the coherence of the noisy self-sustained oscillations is developed. In the Gaussian approximation a closed system of equations is derived for the phase diusion constant and the mean frequency. A comparison with numerics shows that the theory works well for weak feedback and strong noise. c  2003 Elsevier B.V. All rights reserved. PACS: 05.40.-a; 02.50.Ey Keywords: Noise; Delay; Diusion; Coherence Coherence is one of the main characteristics of self-oscillating systems. For periodic oscillators it determines their quality as clocks, and usually the improvement of the coherence is one of the major goals in the construction of such oscillators. In terms of the phase dynamics, the coherence is quantitatively measured by the phase diusion constant, it is proportional to the width of the spectral peak of oscillations. Many chaotic oscillators can be also represented via the phase dynamics, thus allowing one to characterize their coherence by virtue of the phase diusion constant as well [1]. In this paper we study how the coherence of oscillations is inuenced by exter- nal delayed feedback. Applying a delayed feedback is widely use to control dierent properties of the dynamical systems: to make chaotic oscillators operate periodically (Pyragas’ control method [2]) and to suppress space–time chaos [3,4]. In our consid- eration below we are not considering such delay-induced qualitative changes in the * Corresponding author. Fax: +49-331-977-1142. E-mail address: pikovsky@stat.physik.uni-potsdam.de (A. Pikovsky). 0378-4371/03/$ - see front matter c  2003 Elsevier B.V. All rights reserved. doi:10.1016/S0378-4371(03)00463-1