Synchronization of networks of chaotic units with time-delayed couplings

Kinzel, Wolfgang; Englert, A.; Reents, G.; Zigzag, M.; Kanter, Ido

doi:10.1103/physreve.79.056207

Cited by 96 publications

(94 citation statements)

References 41 publications

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“…Time delays are always present in coupled systems due to the finite signal propagation time. These time lags give rise to complex dynamics and have been shown to play a key role in the synchronization behavior of systems [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], see also the review [43]. In Ref.…”

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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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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“…In the last two decades dynamics on network received a growing amount of interest [Dhamala et al, 2004;Zigzag et al, 2009;Choe et al, 2010;Chavez et al, 2005;Sorrentino and Ott, 2007;Albert et al, 2000;Kinzel et al, 2009;Lehnert et al, 2011a;Keane et al, 2012]. One of the central topics of dynamics on networks is synchronization [Strogatz and Stewart, 1993;Rosenblum et al, 1996;Pikovsky et al, 2001;Pecora and Carroll, 1998;Mosekilde et al, 2002;Wang and Chen, 2002;Arenas et al, 2006aArenas et al, , 2008Balanov et al, 2009;Omelchenko et al, 2010;Schöll, 2013].…”

Section: Dynamics On Networkmentioning

confidence: 99%

“…Time delay in neural networks emanates from the finite speed of the transmission of an action potential between two neurons where the propagation velocity of an action potential varies between 1 to 100 mm/ms depending on the diameter of the axon and whether the fibers are myelinated or not [Koch, 1999]. The influence of delay on the dynamics on networks has been investigated by several authors Kestler et al, 2008;Kinzel et al, 2009;Englert et al, 2010;Zigzag et al, 2010;Flunkert et al, 2010;Rosin et al, 2010;Englert et al, 2011;Kanter et al, 2011b;Heiligenthal et al, 2011;Flunkert et al, 2013b;Popovych et al, 2011;Lücken et al, 2013;Kinzel, 2013;D'Huys et al, 2013;Kantner and Yanchuk, 2013;D'Huys et al, 2014] Depending on the context, delay can play a constructive or a destructive role. For example, time-delayed feedback control (TDFC) is a well established control method to control unstable periodic orbits embedded in chaotic attractors as well as unstable fixed points [Pyragas, 1992;Ahlborn and Parlitz, 2004;Rosenblum and Pikovsky, 2004;Hövel and Schöll, 2005;Schöll and Schuster, 2008;Grebogi, 2010].…”

Section: Adaptive Control Of Uncoupled Systems and Networkmentioning

confidence: 99%

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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 concept of small-world networks by Watts and Strogatz was also included [1,6]. The recent innovation is the introduction of time delays to network models [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] which allows to study the influence of the limited speed of information processing on the network dynamics. This speed limitation is indeed present in neuronal communication as the action potential propagates with the speed of tens of meters per second which is a significant aspect if the physical size of nerve tissue taken into consideration.…”

Section: Introductionmentioning

confidence: 99%

Spatial evolution of Hindmarsh–Rose neural network with time delays

Łepek¹,

Fronczak²

2018

Nonlinear Dyn

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Spatial relations between neurons in the network with time delays play a crucial role in determining dynamics of the system. During the development of the nervous system, different types of neurons group together to enable specific functions of the network. Right spatial distances, thus right time delays between cells are crucial for an appropriate functioning of the system. To model the process of neural migration, we proposed simple but effective model of network spatial evolution based on Hindmarsh-Rose neurons and Metropolis-Hastings Monte Carlo algorithm. Under the specific assumptions and using appropriate parameters of the neural evolution, the network can converge to the desirable state giving the opportunity of achieving large variety of spectra. We show that there is a specific range of network size in space which allows it to generate assumed output. A network or generally speaking a system with time delays (corresponding to the arrangement in the physical space) of specific output properties has a specific spatial dimension that allows it to function properly.

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Synchronization of networks of chaotic units with time-delayed couplings

Cited by 96 publications

References 41 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

Controlling Synchronization Patterns in Complex Networks

Spatial evolution of Hindmarsh–Rose neural network with time delays

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