Delayed feedback control of chaos: Bifurcation analysis

Balanov, A. G.; Janson, Natalia B.; Schöll, Eckehard

doi:10.1103/physreve.71.016222

Cited by 112 publications

(48 citation statements)

References 34 publications

(37 reference statements)

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“…Thus, the control scheme does not rely on a reference system and has only a small number of control parameters, i.e., the feedback gain K and time delay τ . It has been shown that TDAS can stabilize both unstable periodic orbits, e.g., embedded in a strange attractor [4,5], and unstable steady states [6,7,8]. In the first case, TDAS is most efficient if τ corresponds to an integer multiple of the minimal period of the orbit.…”

Section: Introductionmentioning

confidence: 99%

Control of unstable steady states by extended time-delayed feedback

2007

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Time-delayed feedback methods can be used to control unstable periodic orbits as well as unstable steady states. We present an application of extended time delay autosynchronization introduced by Socolar et al. to an unstable focus. This system represents a generic model of an unstable steady state which can be found for instance in a Hopf bifurcation. In addition to the original controller design, we investigate effects of control loop latency and a bandpass filter on the domain of control. Furthermore, we consider coupling of the control force to the system via a rotational coupling matrix parametrized by a variable phase. We present an analysis of the domain of control and support our results by numerical calculations.

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Section: Introductionmentioning

confidence: 99%

Control of unstable steady states by extended time-delayed feedback

2007

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“…The idea that time-delayed feedback may not only be used for controlling a system but also for creating new dynamics is not new. Nevertheless, the investigation of delay-induced bifurcations and multistability is still a growing field [Balanov et al, 2005;Xu & Yu, 2004] with applications to the logistic map as well as to laser equations [Martínez-Zérega & Pisarchik, 2005;Martínez-Zérega et al, 2003].…”

Section: Introductionmentioning

confidence: 99%

Delay-Induced Multistability Near a Global Bifurcation

Hizanidis

Aust

Schöll

2008

Int. J. Bifurcation Chaos

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We study the effect of a time-delayed feedback within a generic model for a saddle-node bifurcation on a limit cycle. Without delay the only attractor below this global bifurcation is a stable node. Delay renders the phase space infinite-dimensional and creates multistability of periodic orbits and the fixed point. Homoclinic bifurcations, period-doubling and saddle-node bifurcations of limit cycles are found in accordance with Shilnikov's theorems.

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“…Unstable periodic orbits with periods T 1 ≈ 5.91679 ("period-1 orbit") and T 2 ≈ 11.82814 ("period-2 orbit") are embedded in the chaotic attractor. As shown in [Balanov et al, 2005] by a bifurcation analysis, application of TDFC with τ = T 1 and 0.24 < K < 2.3 stabilizes the period-1 orbit, and it becomes the only attractor of the system. In [Just et al, 1997] it was predicted analytically by a linear expansion that control using the standard TDFC can only be realized in a finite range of the values of K : At the lower control boundary the limit cycle undergoes a period-doubling bifurcation, and at the upper boundary a Hopf bifurcation occurs generating a stable or an unstable torus from a limit cycle (Neimark-Sacker bifurcation).…”

Section: Stabilization Of An Unstable Periodic Orbit In the Rössler Smentioning

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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Delayed feedback control of chaos: Bifurcation analysis

Cited by 112 publications

References 34 publications

Control of unstable steady states by extended time-delayed feedback

Control of unstable steady states by extended time-delayed feedback

Delay-Induced Multistability Near a Global Bifurcation

Controlling Synchronization Patterns in Complex Networks

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