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

Choe, Chol-Ung; Kim, Ryong-Son; Jang, Hyok; Hövel, Philipp; Schöll, Eckehard

doi:10.1007/s40435-013-0049-2

Cited by 17 publications

(9 citation statements)

References 59 publications

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“…There was no odd number limitation in our result. Differently from previous results [3,5,10,11], our result did not require any rotationally equivariant normal form of Stuart-Landau type, or any reduction to such normal forms as in [4]. The proper choices of feedback amplitudes 1/b and delay offsets ϑ, however, turned out to be a delicate matter, with only tiny Pyragas regions P ± of guaranteed success.…”

Section: The Line Of Neutral Hopf Tangenciescontrasting

confidence: 95%

“…For an exciting investigation of traveling wave stabilization by noninvasive delayed spatio-temporal feedback see [37]. Systematic studies of pattern selection at symmetry breaking Hopf bifurcation have been initiated in coupled oscillator settings; see for example [5,13,23,24,36,38,42,43] and the references there.…”

Section: X(t) = F(x(t)) + β(X(t) − X(t − τ ))mentioning

confidence: 99%

“…For some numerical observations in such settings see [3,5] for example. The only reason of our preference for the pure delay equation was the ease of a coherent treatment of all Hopf bifurcations, in Sect.…”

Section: The Line Of Neutral Hopf Tangenciesmentioning

confidence: 99%

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Delayed Feedback Control of a Delay Equation at Hopf Bifurcation

Fiedler

Oliva

2015

J Dyn Diff Equat

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We embark on a case study for the scalar delay equatioṅwith odd nonlinearity f , real nonzero parameters λ, b, and three positive time delays 1, ϑ, p/2. We assume supercritical Hopf bifurcation from x ≡ 0 in the well-understood single-delay case b = ∞. Normalizing f (0) = 1, branches of constant minimal period p k = 2π/ω k are known to bifurcate from eigenvalues iω k = i(k + 1 2 )π at λ k = (−1) k+1 ω k , for any nonnegative integer k. The unstable dimension is k, at the local branch k. We obtain stabilization of such branches, for arbitrarily large unstable dimension k. For p := p k the branch k of constant period p k persists as a solution, for any b = 0 and ϑ ≥ 0. Indeed the delayed feedback term controlled by b vanishes on branch k: the feedback control is noninvasive there. Following an idea of Pyragas, we seek parameter regions P of controls b = 0 and delays ϑ ≥ 0 such that the branch k becomes stable, locally at Hopf bifurcation. We determine rigorous expansions for P in the limit of large k. The only two regions P = P ± which we were able to detect, in this setting, required delays ϑ near 1, controls b near (−1) k · 2/ω k , and were of very small area of order k −4 . Our analysis is based on a 2-scale covering lift for the frequencies involved.

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Section: The Line Of Neutral Hopf Tangenciescontrasting

confidence: 95%

Section: X(t) = F(x(t)) + β(X(t) − X(t − τ ))mentioning

confidence: 99%

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Delayed Feedback Control of a Delay Equation at Hopf Bifurcation

Fiedler

Oliva

2015

J Dyn Diff Equat

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“…In [Choe et al, 2012], a rotation of the feedback term, which is closely related to a non-zero coupling phase, has been used to stabilize unstable periodic orbits and steady states in coupled Lorenz systems. Furthermore, a non-zero coupling phase can be used to obtain noninvasive control in delayed feedback schemes with arbitrary delays [Choe et al, 2014b].…”

Section: Phase Of the Complex Coupling Strengthmentioning

confidence: 99%

Controlling Synchronization Patterns in Complex Networks

Lehnert¹

2016

Springer Theses

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In this thesis, I consider the control of synchronization in delay-coupled complex networks. As one main focus, several applications to neural networks will be discussed. In the first part, I focus on the stability of synchronization in complex networks meaning that the control is realized by considering the stability of synchrony in dependence on the parameters. In the second part, adaptive control of synchronization is studied. To this end, adaptive control algorithms are developed that tune the system parameters such that the desired control goal is reached.Besides zero-lag synchrony -a state where all nodes follow the same dynamics without a phase lag -groups and cluster states are considered, i.e., states where the network consists of several groups where the nodes within one group are in zero-lag synchrony and, in the case of cluster synchrony, with a constant phase lag between the clusters.The stability of synchronization can be accessed via the master stability function [Pecora and Carroll, 1998]. This convenient tool allows for treating the node dynamics and the network topology in two separate steps, and, thus, allows for a quite general treatment of different network topologies. In this thesis, I will discuss the generalization of the master stability function to group and cluster states and to non-smooth systems.The master stability function can be used to investigate synchrony in neural networks. Neurons are excitable systems where type-I and type-II excitability can be distinguished. Here, the stability of synchrony for both types in complex networks with excitatory and inhibitory links is investigated on two generic models, namely the normal form of the saddle-node infinite period bifurcation and the FitzHugh-Nagumo system. Furthermore, synchronization in systems with heterogeneous delays or node parameters is studied.In situations where parameters are unknown or drift, adaptive control methods are useful since they allow for an automatically realized adaption of the control parameters. A convenient adaptive method is the speed-gradient method that minimizes a predefined goal function [Fradkov, 1979[Fradkov, , 2007. I first apply this method in the control of an unstable focus and an unstable periodic orbit embedded in the Rössler attractor. Furthermore, I show that clusters states in networks of delayed coupled Stuart-Landau oscillators can be controlled by adaptively tuning the phase of the complex coupling strength or by adapting the topology. The first method is particularly simple because only one parameter has to be adapted, while the second method is more reliable in the sense that its success is widely independent of the choice of the control parameters and the initial conditions. ZUSAMMENFASSUNGIn dieser Arbeit wird die Kontrolle von Synchronisation in komplexen Netzwerken mit retardierten Kopplungen untersucht. Dabei werden verschiedene Anwendungen auf neuronale Netzwerke diskutiert. Der erste Teil der Arbeit beschäftigt sich mit der Stabilität von Synchronisation in komplexen Ne...

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“…The master stability function has been successfully applied to study synchronization in various delay-coupled networks of different node dynamics [Dah12,Leh11,Kin09,Cho09,Flu10b,Hei11,Kea12,Wil13,Lad13,Bla13,Sch13,Cho14].…”

Section: Dynamics Of Complex Networkmentioning

confidence: 99%

Dynamics of Complex Autonomous Boolean Networks

Rosin

2015

Springer Theses

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Network science provides a powerful framework for analyzing complex systems found in physics, biology, and social sciences. One way of studying the dynamics of networks is to engineer and measure them in the laboratory, which is particularly difficult with established approaches. In this thesis, I approach this problem using a hardware device with time-delay elements executing Boolean functions that can be connected to autonomous Boolean networks with chaotic, periodic, or excitable dynamics. I am able to make scientific discoveries for networks with each of these three different node dynamics, driven by the large flexibility and the non-ideal effects of the experiment complemented by analytical and numerical investigations.Using network realizations with periodic Boolean oscillators, I study socalled chimera states and find that they can disappear and reappear-the resurgence of chimera states. I measure the transient times of chimera states and find a power-law relationship between the average transient time and the phase space volume with an exponent of κ = −0.28 ± 0.10.I also study cluster synchronization in networks of coupled excitable systems. In these artificial neural networks, I find a breakdown of an established theoretical tool when the heterogeneity of the link time delays is greater than the neural refractory period. This phenomenon is used to derive a control scheme for spiking patterns generated by neural networks.Experimental implementations of these systems take advantage of the fast timescale of electronic logic gates, large scalability, and low price. These properties make the system attractive for technological applications, as I demonstrate by realizing a physical random number generator that has an ultra-high bitrate of 12.8 Gbit/s and a silicon neuron that is a thousand times faster than the fastest preceding silicon neuron. For the study of coupled oscillator networks, I develop a phase-locked loop allowing for multiple drivers that may be advantageous for clock synchronization. Instead of the common topologies with one driver per oscillator, it allows for heavily connected clock networks to increase robustness against failure.iii Z U S A M M E N FA S S U N G • D. P. Rosin, D. Rontani, and D. J. Gauthier. Ultrafast physical generation of random numbers using hybrid Boolean networks. Phys. Rev. E 87, 040902(R) (2013).• D. P. Rosin, D. Rontani, D. J. Gauthier, and E. Schöll. Excitability in autonomous Boolean networks. Europhys. Lett. 100, 30003 (2012).• D. P. Rosin, D. Rontani, D. J. Gauthier, and E. Schöll. Experiments on autonomous Boolean networks. Chaos 23, 025102 (2013).• D. P. Rosin, D. Rontani, and D. J. Gauthier. Synchronization of coupled Boolean phase oscillators. Phys. Rev. E 89, 042907 (2014).• D. P. Rosin, D. Rontani, E. Schöll, and D. J. Gauthier. Transient scaling and resurgence of chimera states in coupled Boolean phase oscillators.

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

Cited by 17 publications

References 59 publications

Delayed Feedback Control of a Delay Equation at Hopf Bifurcation

Delayed Feedback Control of a Delay Equation at Hopf Bifurcation

Controlling Synchronization Patterns in Complex Networks

Dynamics of Complex Autonomous Boolean Networks

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