Rights statement: Copyright 2016 American Institute of Physics. The following article appeared in Chaos, 26 (10), 2016 and may be found at http://dx.doi.org/10.1063/1.4963371 This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - Ott-Antonsen attractiveness for parameter-dependent oscillatory systems
AU - Pietras, Bastian
AU - Daffertshofer, Andreas
N1 - Copyright 2016 American Institute of Physics. The following article appeared in Chaos, 26 (10), 2016 and may be found at http://dx.doi.org/10.1063/1.4963371 This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics.
PY - 2016/10
Y1 - 2016/10
N2 - The Ott-Antonsen (OA) ansatz [Ott and Antonsen, Chaos 18, 037113 (2008); Chaos 19, 023117 (2009)] has been widely used to describe large systems of coupled phase oscillators. If the coupling is sinusoidal and if the phase dynamics does not depend on the specific oscillator, then the macroscopic behavior of the systems can be fully described by a low-dimensional dynamics. Does the corresponding manifold remain attractive when introducing an intrinsic dependence between an oscillator's phase and its dynamics by additional, oscillator specific parameters? To answer this, we extended the OA ansatz and proved that parameter-dependent oscillatory systems converge to the OA manifold given certain conditions. Our proof confirms recent numerical findings that already hinted at this convergence. Furthermore, we offer a thorough mathematical underpinning for networks of so-called theta neurons, where the OA ansatz has just been applied. In a final step, we extend our proof by allowing for time-dependent and multi-dimensional parameters as well as for network topologies other than global coupling. This renders the OA ansatz an excellent starting point for the analysis of a broad class of realistic settings.
AB - The Ott-Antonsen (OA) ansatz [Ott and Antonsen, Chaos 18, 037113 (2008); Chaos 19, 023117 (2009)] has been widely used to describe large systems of coupled phase oscillators. If the coupling is sinusoidal and if the phase dynamics does not depend on the specific oscillator, then the macroscopic behavior of the systems can be fully described by a low-dimensional dynamics. Does the corresponding manifold remain attractive when introducing an intrinsic dependence between an oscillator's phase and its dynamics by additional, oscillator specific parameters? To answer this, we extended the OA ansatz and proved that parameter-dependent oscillatory systems converge to the OA manifold given certain conditions. Our proof confirms recent numerical findings that already hinted at this convergence. Furthermore, we offer a thorough mathematical underpinning for networks of so-called theta neurons, where the OA ansatz has just been applied. In a final step, we extend our proof by allowing for time-dependent and multi-dimensional parameters as well as for network topologies other than global coupling. This renders the OA ansatz an excellent starting point for the analysis of a broad class of realistic settings.
U2 - 10.1063/1.4963371
DO - 10.1063/1.4963371
M3 - Journal article
VL - 26
JO - Chaos
JF - Chaos
SN - 1054-1500
IS - 10
M1 - 103101
ER -