Ott-Antonsen attractiveness for parameter-dependent oscillatory systems

Pietras, Bastian and Daffertshofer, Andreas (2016) Ott-Antonsen attractiveness for parameter-dependent oscillatory systems. Chaos, 26 (10): 103101. ISSN 1054-1500

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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.

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Copyright 2016 American Institute of Physics. The following article appeared in Chaos, 26 (10), 2016 and may be found at 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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09 Dec 2016 09:48
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31 Dec 2023 00:45