Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control

Just, Wolfram; Fiedler, Bernold; Georgi, Marc; Flunkert, Valentín; Hövel, Philipp; Schöll, Eckehard

doi:10.1103/physreve.76.026210

Cited by 88 publications

(122 citation statements)

References 37 publications

(60 reference statements)

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“…It was commonly believed that unstable periodic orbits with an odd number of real Floquet multipliers larger than unity could never be stabilized by delayed-feedback control. Recently, this alleged odd-number theorem has been refuted by a counter example (Fiedler et al , 2008aJust et al 2007;Kehrt et al 2009); see §4 for a summary of these results.…”

Section: Z(t) = F (Z(t)) + B(z(t − τ ) − Z(t)) (12)mentioning

confidence: 99%

“…See also Fiedler et al (2007), Just et al (2007) and Schöll & Schuster (2008) for our previous analysis of this case.…”

Section: Beyond Odd-number Limitation For Planar Hopf Bifurcationmentioning

confidence: 99%

“…As usual, the unstable dimension denotes the number of Floquet multipliers strictly outside the complex unit circle, counting algebraic multipliers. See Diekmann et al (1995), Fiedler et al (2007), Just et al (2007) and Schöll & Schuster (2008) for the mathematical centre manifold machinery behind this result on exchange of stability at bifurcations. See also figure 1.…”

Section: Model Of Two Diffusively Coupled Oscillatorsmentioning

confidence: 99%

“…Their stabilization by non-invasive delayed feedback therefore refutes the so-called 'odd-number limitation' of Pyragas control. See Fiedler et al (2007) and Just et al (2007) for a detailed discussion of this aspect, including a rigorous mathematical analysis illustrated by numerical examples.…”

Section: Beyond Odd-number Limitation For Planar Hopf Bifurcationmentioning

confidence: 99%

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Delay stabilization of periodic orbits in coupled oscillator systems

Fiedler

Flunkert

Hövel

et al. 2010

Phil. Trans. R. Soc. A.

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We study diffusively coupled oscillators in Hopf normal form. By introducing a noninvasive delay coupling, we are able to stabilize the inherently unstable anti-phase orbits. For the super-and subcritical cases, we state a condition on the oscillator's nonlinearity that is necessary and sufficient to find coupling parameters for successful stabilization. We prove these conditions and review previous results on the stabilization of oddnumber orbits by time-delayed feedback. Finally, we illustrate the results with numerical simulations.

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Section: Z(t) = F (Z(t)) + B(z(t − τ ) − Z(t)) (12)mentioning

confidence: 99%

“…See also Fiedler et al (2007), Just et al (2007) and Schöll & Schuster (2008) for our previous analysis of this case.…”

Section: Beyond Odd-number Limitation For Planar Hopf Bifurcationmentioning

confidence: 99%

Section: Model Of Two Diffusively Coupled Oscillatorsmentioning

confidence: 99%

Section: Beyond Odd-number Limitation For Planar Hopf Bifurcationmentioning

confidence: 99%

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Delay stabilization of periodic orbits in coupled oscillator systems

Fiedler

Flunkert

Hövel

et al. 2010

Phil. Trans. R. Soc. A.

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“…[15,16]. The normal form of a system near a subcritical Hopf bifurcation also allows for an analytical treatment of the stability of the UPO, including the calculation of the Floquet exponents [17][18][19][20]. However, since the constraint is imposed that the delay time should be adjusted to the period of the UPO, strongly unstable periodic orbits are difficult to stabilize.…”

Section: Introductionmentioning

confidence: 99%

Delayed-feedback control: arbitrary and distributed delay-time and noninvasive control of synchrony in networks with heterogeneous delays

Choe¹,

Kim²,

Jang³

et al. 2014

Int. J. Dynam. Control

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We suggest a delayed feedback control scheme with arbitrary delay for stabilizing a periodic orbit, while maintaining the noninvasiveness of the controller. Since the constraint on the delay to be adjusted to the period of the unstable periodic orbit is not imposed, a richer structure of the dynamics can be observed: Not only weakly unstable, but also strongly unstable periodic orbits are stabilized and even stabilization of orbits with infinite period is achieved. The control mechanism is elucidated for the generic model of a subcritical Hopf bifurcation. A complete bifurcation analysis for the fixed point as well as the periodic orbit is presented and the stability domains are identified. Furthermore, we study the effects of distributed delayed feedback on the stabilization of periodic orbits, and show that larger variance of the delay distribution considerably enlarges the stabilization region in the parameter space. We extend the control scheme to a network of Hopf normal forms coupled with heterogeneous delays. By tuning the coupling parameters, different synchronization patterns, i.e., in-phase, splay, and clustering, can be selected. The characteristic equation for Floquet exponents of the heterogeneous delay network is derived in an analytical form, which reveals the coupling parameters for successful stabilization. The equation takes a C.-U. Choe · R.-S. Kim · H. Jang Center for Nonlinear Science, University of Science, Unjong-District, Pyongyang, DPR Korea P. Hövel · E. Schöll (B) Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstraße 36, 10623 Berlin, Germany e-mail: schoell@physik.tu-berlin.de P. Hövel Bernstein Center for Computational Neuroscience, Humboldt-Universität zu Berlin, Philippstraße 13, 10115 Berlin, Germany unified form for both subcritical and supercritical Hopf bifurcations regardless of the synchronization patterns. Analysis of Floquet exponents and direct numerical simulations show that the heterogeneity in the delays drastically facilitates stabilization and provides an enlarged parameter region for successful control. Finally, we consider the thermodynamic limit in the framework of a mean field approximation and show that heterogeneous delays offer an enhanced performance of control.

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Delay Stabilization of Rotating Waves without Odd Number Limitation

Fiedler

Flunkert

Georgi

et al. 2008

Reviews of Nonlinear Dynamics and Complexity

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A variety of methods have been developed in nonlinear science to stabilize unstable periodic orbits (UPOs) and control chaos [1], following the seminal work by Ott, Grebogi and Yorke [2], who employed a tiny control force to stabilize UPOs embedded in a chaotic attractor [3,4]. A particularly simple and efficient scheme is time-delayed feedback as suggested by Pyragas [5], which uses the difference z(t − τ) − z(t) of a signal z at a time t and a delayed time t − τ. It is an attempt to stabilize periodic orbits of (minimal) period T by a feedback control which involves a time delay τ = nT, for suitable positive integer n. A linear feedback example iṡ(2.1) whereż(t) = f (λ, z(t)) describes a d-dimensional nonlinear dynamical system with bifurcation parameter λ and an unstable orbit of (minimal) period T. The constant feedback control matrix B is chosen suitably. Typical choices are multiples of the identity or of rotations, or matrices of low rank. More general nonlinear feedbacks are conceivable, of course. The main point, however, is that the Pyragas choice τ P

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Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control

Cited by 88 publications

References 37 publications

Delay stabilization of periodic orbits in coupled oscillator systems

Delay stabilization of periodic orbits in coupled oscillator systems

Delayed-feedback control: arbitrary and distributed delay-time and noninvasive control of synchrony in networks with heterogeneous delays

Delay Stabilization of Rotating Waves without Odd Number Limitation

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