Time-Delayed Feedback Control: From Simple Models to Lasers and Neural Systems

Schöll, Eckehard; Hövel, Philipp; Flunkert, Valentín; Dahlem, Markus

doi:10.1007/978-3-642-02329-3_4

Cited by 18 publications

(18 citation statements)

References 158 publications

(282 reference statements)

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“…Hence their period is not determined by the local properties of the fixed point, but by the global structure of the phase space: the period is dominated by the slow motion along the two stable branches of the activator nullcline. The dependence of the period of oscillations on the bifurcation parameter a can be estimated by analytical approximations, very similar to the approximations for the firing time in Schöll et al (2009b).…”

Section: (B) the Oscillatory Regimementioning

confidence: 98%

Interplay of time-delayed feedback control and temporally correlated noise in excitable systems

Brandstetter

Dahlem

Schöll

2010

Phil. Trans. R. Soc. A.

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The interplay of time-delayed feedback and temporally correlated coloured noise in a single and two coupled excitable systems is studied in the framework of the FitzHughNagumo (FHN) model. By using coloured noise instead of white noise, the noise correlation time is introduced as an additional time scale. We show that in a single FHN system the major time scale of oscillations is strongly influenced by the noise correlation time, which in turn affects the maxima of coherence with respect to the delay time. In two coupled FHN systems, coloured noise input to one subsystem influences coherence resonance and stochastic synchronization of both subsystems. Application of delayed feedback control to the coloured noise-driven subsystem is shown to change coherence and time scales of noise-induced oscillations in both systems, and to enhance or suppress stochastic synchronization under certain conditions.

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Section: (B) the Oscillatory Regimementioning

confidence: 98%

Interplay of time-delayed feedback control and temporally correlated noise in excitable systems

Brandstetter

Dahlem

Schöll

2010

Phil. Trans. R. Soc. A.

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“…Proof Let λ ∈ C be an eigenvalue of the time-T map M of (35)- (36). Let x 0 ,x 0 (both [−T , 0] → R n ) be the components of an eigenvector corresponding to λ, and let x 1 ,x 1 (also both [−T , 0] → R n ) be the corresponding components of M[x,x].…”

Section: A1 Floquet Multipliers For Extended Time-delayed Feedback mentioning

confidence: 99%

Generic stabilizability for time-delayed feedback control

Sieber

2016

Proc. R. Soc. A.

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Time-delayed feedback control is one of the most successful methods to discover dynamically unstable features of a dynamical system in an experiment. This approach feeds back only terms that depend on the difference between the current output and the output from a fixed time T ago. Thus, any periodic orbit of period T in the feedbackcontrolled system is also a periodic orbit of the uncontrolled system, independent of any modelling assumptions. It has been an open problem whether this approach can be successful in general, that is, under genericity conditions similar to those in linear control theory (controllability), or if there are fundamental restrictions to time-delayed feedback control. We show that, in principle, there are no restrictions. This paper proves the following: for every periodic orbit satisfying a genericity condition slightly stronger than classical linear controllability, one can find control gains that stabilize this orbit with extended time-delayed feedback control. While the paper's techniques are based on linear stability analysis, they exploit the specific properties of linearizations near autonomous periodic orbits in nonlinear systems, and are, thus, mostly relevant for the analysis of nonlinear experiments.

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“…Using the notation of Refs. [SCH08,SCH09a], the system is given for this configuration by the following equations…”

Section: Single Fitzhugh-nagumo System and Time-delayed Feedbackmentioning

confidence: 99%

“…This is done for better comparison with Refs. [SCH08,SCH09a] and the Master's thesis [HIL08]. This comparison can be found at the end of this Section.…”

Section: Single Fitzhugh-nagumo System and Time-delayed Feedbackmentioning

confidence: 99%