Amplitude and phase dynamics in oscillators with distributed-delay coupling

Kyrychko, Yuliya N.; Blyuss, Konstantin B.; Schöll, Eckehard

doi:10.1098/rsta.2012.0466

Cited by 54 publications

(35 citation statements)

References 85 publications

(162 reference statements)

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“…This is common in a variety of systems where the delay times are given by a continuous distribution [48,[52][53][54]. The model can be written as follows:…”

Section: Delayed-feedback Control With Distributed Delaysmentioning

confidence: 99%

“…Nevertheless, most studies have assumed that all the interactions occur with the same time delay and, up to now, little is known about stabilizing UPOs and controlling the synchrony patterns in networks coupled with heterogeneous delays. For instance, the dynamics of an array of chaotic logistic maps coupled with random delay times [47], the effects of heterogeneous delays in the coupling of two excitable neural systems [48,49] or a neural network [50], and amplitude death in the Stuart-Landau system coupled with distributed delays [51][52][53] or periodically modulated delay [54] were investigated.…”

Section: Introductionmentioning

confidence: 99%

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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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“…This is common in a variety of systems where the delay times are given by a continuous distribution [48,[52][53][54]. The model can be written as follows:…”

Section: Delayed-feedback Control With Distributed Delaysmentioning

confidence: 99%

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 analyze the paradigmatic model of Stuart-Landau (SL) oscillators [24,[43][44][45][46][47][48][49],…”

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confidence: 99%

Chimera Death: Symmetry Breaking in Dynamical Networks

2014

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For a network of generic oscillators with nonlocal topology and symmetry-breaking coupling we establish novel partially coherent inhomogeneous spatial patterns, which combine the features of chimera states (coexisting incongruous coherent and incoherent domains) and oscillation death (oscillation suppression), which we call chimera death. We show that due to the interplay of nonlocality and breaking of rotational symmetry by the coupling two distinct scenarios from oscillatory behavior to a stationary state regime are possible: a transition from an amplitude chimera to chimera death via in-phase synchronized oscillations, and a direct abrupt transition for larger coupling strength. Spontaneous symmetry breaking in a complex dynamical system is a fundamental and universal phenomenon which occurs in diverse fields such as physics, chemistry, and biology [1]. It implies that processes occurring in nature favor a less symmetric configuration, although the underlying principles can be symmetric. This intriguing concept has recently gained renewed interest generated by the enormous burst of works on chimera states and on oscillation death, which have emerged independently. In this Letter we draw a relation between these two.Chimera states correspond to the situation when an ensemble of identical elements self-organizes into two coexisting and spatially separated domains with dramatically different behavior, i.e., spatially coherent and incoherent oscillations [2,3]. They have been the subject of intensive theoretical investigations, e.g. [4][5][6][7][8][9][10][11][12][13][14][15]. Experimental evidence of chimeras has only recently been provided for optical [16], chemical [17], mechanical [18] and electronic [19] systems. These peculiar hybrid states may also account for the observation of partial synchrony in neural activity [20], like unihemispheric sleep, i.e., the ability of some birds or dolphins to sleep with one half of their brain while the other half remains aware [21,22]. Chimera states have been initially found for phase oscillators [2], and they typically occur in networks with nonlocal coupling. Recently, for globally coupled oscillators it has been shown that such spatio-temporal patterns can be also connected to the amplitude dynamics (amplitude-mediated chimeras) [23,24]. However, the global coupling topology does not provide a clear notion of space, which is crucial for chimera states. A further open question is whether chimeras can be extended to more general symmetry breaking states.Another fascinating effect which requires the break-up of the system's symmetry as a crucial ingredient is oscillation death which refers to stable inhomogeneous steady states (IHSS) which are created through the coupling of self-sustained oscillators. This regime occurs when a ho- * corresponding author: schoell@physik.tu-berlin.de mogeneous steady state splits into at least two distinct branches -upper and lower -which represent a newly created IHSS [25][26][27][28]. For a network of coupled elements oscillation death ...

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“…In networks with broad distributions, spiking is not possible any longer and initial excitations die out fast, leading to global amplitude death. This behavior is similar to the case of only two coupled oscillators with distributed delay in the coupling, where the regime of amplitude death increases with the width of the delay kernel [Atay, 2003a;Kyrychko et al, 2011Kyrychko et al, , 2013. Large regular networks and to some extent small-world networks are less prone to undergo global amplitude death, since they allow more easily for stable subnetwork spiking or travelling disruptions.…”

Section: Resultsmentioning

confidence: 53%

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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Amplitude and phase dynamics in oscillators with distributed-delay coupling

Cited by 54 publications

References 85 publications

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

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

Chimera Death: Symmetry Breaking in Dynamical Networks

Controlling Synchronization Patterns in Complex Networks

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