Controlling Stochastic Oscillations Close to a Hopf Bifurcation by Time-Delayed Feedback

Abstract

We study the effect of a time-delayed feedback upon a Van der Pol oscillator under the influence of white noise in the regime below the Hopf bifurcation where the deterministic system has a stable fixed point. We show that both the coherence and the frequency of the noise-induced oscillations can be controlled by varying the delay time and the strength of the control force. Approximate analytical expressions for the power spectral density and the coherence properties of the stochastic delay differential equation are developed, and are in good agreement with our numerical simulations. Our analytical results elucidate how the correlation time of the controlled stochastic oscillations can be maximized as a function of delay and feedback strength.

Time-delayed feedback in neurosystems

“…One could also consider the application of the feedback scheme to both subsystems and the effects of different values of the control parameters for each subsystem, but these investigations are beyond the scope of this work. Previously, time-delayed feedback has also been used to influence noise-induced oscillations of a single excitable system Janson et al 2004;Prager et al 2007), of systems below a Hopf bifurcation (Pomplun et al 2005;Schöll et al 2005;Flunkert & Schöll 2007;Pototsky & Janson 2007) or below a global bifurcation (Hizanidis et al 2006;Hizanidis & Schöll 2008) and of spatially extended reactiondiffusion systems Stegemann et al 2006;Dahlem et al 2008). Extensions to multiple time-delay control schemes have also been considered (Hövel et al 2007, submitted;Pomplun et al 2007;).…”

Control of Complex Nonlinear Systems with Delay

“…Time-delayed feedback was also considered for the Vander-Pol system in Refs. [POM05a,SCH04b] and in Ref. [POM07] the extended form of the control method [SOC94] was applied.…”

Noise-Induced Oscillations and Their Control in Semiconductor Superlattices

“…Previous works mainly concentrate on the control of stochastic oscillations in low-dimensional simple models Christini & Collins, 1995;Janson et al, 2004;Landa et al, 1997;Masoller, 2002;Schöll et al, 2005], or self-oscillations in the presence of noise [Goldobin et al, 2003], while control of noise-induced dynamics in spatially extended systems seems still to be an open problem.…”

“…However, nowadays for a large class of extended systems of reaction-diffusion type it has been shown that noise can play a constructive role inducing quite coherent dynamical space-time patterns [García-Ojalvo et al, 1993]. Recently, such noise-induced patterns were also found in semiconductor nanostructures described by a reaction-diffusion model for the current density distribution [Stegemann et al, 2005] It was recently shown for two general classes of simple nonlinear systems with temporal degrees of freedom only, that the coherence properties and the time scales of noise-induced oscillations can be changed by applying a time-delayed feedback Janson et al, 2004;Schöll et al, 2005] in the form which was introduced earlier by Pyragas [Pyragas, 1992] for chaos control of deterministic dynamics. In that previous work purely temporal noise-induced dynamics was considered within the example of a Van-der-Pol oscillator, i.e.…”