232 research outputs found
Amplitude and phase effects on the synchronization of delay-coupled oscillators
We consider the behavior of StuartâLandau oscillators as generic limit-cycle oscillators when they
are interacting with delay. We investigate the role of amplitude and phase instabilities in producing
symmetry-breaking/restoring transitions. Using analytical and numerical methods we compare the
dynamics of one oscillator with delayed feedback, two oscillators mutually coupled with delay, and
two delay-coupled elements with self-feedback. Taking only the phase dynamics into account, no
chaotic dynamics is observed, and the stability of the identical synchronization solution is the same
in each of the three studied networks of delay-coupled elements. When allowing for a variable
oscillation amplitude, the delay can induce amplitude instabilities. We provide analytical proof that,
in case of two mutually coupled elements, the onset of an amplitude instability always results in
antiphase oscillations, leading to a leader-laggard behavior in the chaotic regime. Adding selffeedback with the same strength and delay as the coupling stabilizes the system in the transverse
direction and, thus, promotes the onset of identically synchronized behaviorWe would like to thank T. Erneux, E. Schöll,
S. Yanchuk, and P. Perlikowski for helpful discussions.
O.D. acknowledges the Research Foundation Flanders
FWO-Vlaanderen for a fellowship and for project support.
This work was partially supported by the Interuniversity Attraction Poles program of the Belgian Science Policy OfïŹce,
under Grant No. IAP VI-10 âphotonics@be,â by MICINN
Spain under project DeCoDicA Grant No. TEC2009-
14101 ,, and by the project PHOCUS EU FET Open Grant
No. 240763 .Peer reviewe
Heterogeneous Delays in Neural Networks
We investigate heterogeneous coupling delays in complex networks of excitable
elements described by the FitzHugh-Nagumo model. The effects of discrete as
well as of uni- and bimodal continuous distributions are studied with a focus
on different topologies, i.e., regular, small-world, and random networks. In
the case of two discrete delay times resonance effects play a major role:
Depending on the ratio of the delay times, various characteristic spiking
scenarios, such as coherent or asynchronous spiking, arise. For continuous
delay distributions different dynamical patterns emerge depending on the width
of the distribution. For small distribution widths, we find highly synchronized
spiking, while for intermediate widths only spiking with low degree of
synchrony persists, which is associated with traveling disruptions, partial
amplitude death, or subnetwork synchronization, depending sensitively on the
network topology. If the inhomogeneity of the coupling delays becomes too
large, global amplitude death is induced
The Kuramoto model: A simple paradigm for synchronization phenomena
Synchronization phenomena in large populations of interacting elements are the subject of intense research efforts in physical, biological, chemical, and social systems. A successful approach to the problem of synchronization consists of modeling each member of the population as a phase oscillator. In this review, synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. A rigorous mathematical treatment, specific numerical methods, and many variations and extensions of the original model that have appeared in the last few years are presented. Relevant applications of the model in different contexts are also included
Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators
In recent years there has been an increasing interest in studying
time-delayed coupled networks of oscillators since these occur in many real
life applications. In many cases symmetry patterns can emerge in these
networks, as a consequence a part of the system might repeat itself, and
properties of this subsystem are representative of the dynamics on the whole
phase space. In this paper an analysis of the second order N-node time-delay
fully connected network is presented which is based on previous work by Correa
and Piqueira \cite{Correa2013} for a 2-node network. This study is carried out
using symmetry groups. We show the existence of multiple eigenvalues forced by
symmetry, as well as the existence of Hopf bifurcations. Three different models
are used to analyze the network dynamics, namely, the full-phase, the phase,
and the phase-difference model. We determine a finite set of frequencies
, that might correspond to Hopf bifurcations in each case for critical
values of the delay. The map is used to actually find Hopf bifurcations
along with numerical calculations using the Lambert W function. Numerical
simulations are used in order to confirm the analytical results. Although we
restrict attention to second order nodes, the results could be extended to
higher order networks provided the time-delay in the connections between nodes
remains equal.Comment: 41 pages, 18 figure
Restoration of rhythmicity in diffusively coupled dynamical networks
We acknowledge financial support from the National Natural Science Foundation of China (No. 11202082, No. 61203235, No. 11371367 and No. 11271290), the Fundamental Research Funds for the Central Universities of China under Grant No. 2014QT005, IRTG1740(DFG-FAPESP), and SERB-DST Fast Track scheme for young scientist under Grant No. ST/FTP/PS-119/2013, NSF CHE-0955555 and Grant No. 229171/2013-3 (CNPq).Peer reviewedPublisher PD
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