Delay-Induced Multistability Near a Global Bifurcation

Hizanidis, Johanne; Aust, Roland; Schöll, Eckehard

doi:10.1142/s0218127408021348

Cited by 40 publications

(22 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…INTRODUCTION bifurcation: Type-I excitability occurs in systems close to a saddle-node infinite period (SNIPER) bifurcation, while type-II arises around a Hopf bifurcation. As a model for type-I, I use the normal form of the SNIPER bifurcation [Hu et al, 1993;Hizanidis et al, 2008]; as an example for type-II, I investigate the FitzHugh-Nagumo system [FitzHugh, 1961;Nagumo et al, 1962]. …”

Section: Model Systemsmentioning

confidence: 99%

“…The aim of this Chapter is to study the type-I and type-II excitability on two generic models, the saddle-node infinite period (SNIPER) bifurcation model, also known as the SNIC (saddle-node bifurcation on invariant cycle) model [Hu et al, 1993;Hizanidis et al, 2008], and the FitzHugh-Nagumo model [FitzHugh, 1961;Nagumo et al, 1962] In Sec. 4.1, I explain the notion of excitability and the differences between the two types.…”

Section: Couplingmentioning

confidence: 99%

“…Hence, this normal form of the SNIPER bifurcation will be investigated [Hu et al, 1993;Hizanidis et al, 2008], mathematically represented by two first-order differential equations:…”

Section: Type-i Excitabilitymentioning

confidence: 99%

See 2 more Smart Citations

Controlling Synchronization Patterns in Complex Networks

Lehnert¹

2016

Springer Theses

View full text Add to dashboard Cite

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...

show abstract

Section: Model Systemsmentioning

confidence: 99%

Section: Couplingmentioning

confidence: 99%

See 1 more Smart Citation

Controlling Synchronization Patterns in Complex Networks

Lehnert¹

2016

Springer Theses

View full text Add to dashboard Cite

show abstract

“…In these cases the deterministic system was slightly below a Hopf bifurcation, i.e., a local bifurcation. Less is known on how control affects systems near a global bifurcation, like the SNIPER [17].…”

Section: Ns) T Marks the Pulse Duration In (A) Dashed (Red) Curves mentioning

confidence: 99%

Control of noise‐induced spatiotemporal patterns in superlattices

Hizanidis

Schöll

2008

Phys. Status Solidi (c)

View full text Add to dashboard Cite

We investigate a semiconductor superlattice exhibiting complex dynamics of electron accumulation and depletion fronts (travelling field domains) in the presence of noise and control. In the uncontrolled system noise is able to induce electron charge front motion through the device when the system is prepared near a global bifurcation which yields it highly sensitive to fluctuations. The associated current density oscillations exhibit a maximum of coherence at an optimal noise intensity, a phenomenon known as coherence resonance. We show how control in the form of a time‐delayed feedback signal affects the noise‐induced dynamics. We observe that control induces a limit cycle in a wide range of the control parameter space and therefore destroys the effect of coherence resonance. For low noise intensity, time‐delayed feedback control enhances the regularity (coherence) of the noisy oscillations considerably. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

show abstract

“…[42,43,44,45,46,47,48,49,50,51,52,53,54] as well as by numerical bifurcation analysis, e.g. [55,56]. Time-delayed feedback can also stabilize fixed points using single [57,48,49] or multiple delay times [58,59,50].…”

Section: Introductionmentioning

confidence: 99%