Chaos in networks with time-delayed couplings

Kinzel, Wolfgang

doi:10.1098/rsta.2012.0461

Cited by 13 publications

(15 citation statements)

References 22 publications

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“…The condition s in 0 > 0 restricts the value of m to the range 0 m < (2n − 1)/2, thus l 0. The period of the Hopf bifurcation oscillations is determined using (22) and is given by…”

Section: Hopf Bifurcations Of the Steady Statementioning

confidence: 99%

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Tuning the period of square-wave oscillations for delay-coupled optoelectronic systems

2014

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We analyze the response of two delay-coupled optoelectronic oscillators. Each oscillator operates under its own delayed feedback. We show that the system can display square-wave periodic solutions that can be synchronized in phase or out of phase depending on the ratio between self-and cross-delay times. Furthermore, we show that multiple periodic synchronized solutions can coexist for the same values of the fixed parameters. As a consequence, it is possible to generate square-wave oscillations with different periods by just changing the initial conditions.

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“…The condition s in 0 > 0 restricts the value of m to the range 0 m < (2n − 1)/2, thus l 0. The period of the Hopf bifurcation oscillations is determined using (22) and is given by…”

Section: Hopf Bifurcations Of the Steady Statementioning

confidence: 99%

“…The effect of the ratio between different delay times on the synchronization properties of two mutually coupled chaotic lasers was recently investigated in [22]. Here we concentrate on the multiplicity of stable periodic regimes generated by the mutually coupled OEOs rather than their synchronization efficiency as they are chaotic.…”

Section: Introductionmentioning

confidence: 99%

Tuning the period of square-wave oscillations for delay-coupled optoelectronic systems

2014

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“…This process is very similar to a Bernoulli map [16,17], X n+1 =(aX n ) mod 1, which is non-chaotic for a<1. The variable X n stands not for the neuronal response latency L, but for L'(L n ).…”

Section: (B)mentioning

confidence: 99%

Chaotic and non-chaotic phases in experimental responses of a single neuron

Marmari¹,

Vardi²,

Kanter³

2014

EPL

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Consistency and predictability of brain functionalities depend on reproducible activity of a single neuron. We identify a reproducible non-chaotic neuronal phase where deviations between concave response latency profiles of a single neuron do not increase with the number of stimulations. A chaotic neuronal phase emerges at a transition to convex latency profiles which diverge exponentially, indicating irreproducible response timings. Our findings are supported by a quantitative mathematical framework and found robust to periodic and random stimulation patterns. In addition, these results put a bound on the neuronal temporal resolution which can be enhanced below a millisecond using neuronal chains.

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“…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%

“…In the following, the MSF is derived for a delay-coupled network [Dhamala et al, 2004;Choe et al, 2010;Kinzel et al, 2009;Heiligenthal et al, 2011;Kinzel, 2013;D'Huys et al, 2013]. If the delay time is set to zero, the results of [Pecora and Carroll, 1998] for a network without delayed coupling are recovered.…”

Section: Derivation Of the Master Stability Functionmentioning

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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Chaos in networks with time-delayed couplings

Cited by 13 publications

References 22 publications

Tuning the period of square-wave oscillations for delay-coupled optoelectronic systems

Tuning the period of square-wave oscillations for delay-coupled optoelectronic systems

Chaotic and non-chaotic phases in experimental responses of a single neuron

Controlling Synchronization Patterns in Complex Networks

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