Populations of globally coupled phase oscillators are described in the thermodynamic limit by kinetic equations for the distribution densities or, equivalently, by infinite hierarchies of equations for the order parameters. Ott and Antonsen [Chaos 18, 037113 (2008)] have found an invariant finite-dimensional subspace on which the dynamics is described by one complex variable per population. For oscillators with Cauchy distributed frequencies or for those driven by Cauchy white noise, this subspace is weakly stable and, thus, describes the asymptotic dynamics. Here, we report on an exact finite-dimensional reduction of the dynamics outside of the Ott–Antonsen subspace. We show that the evolution from generic initial states can be reduced to that of three complex variables, plus a constant function. For identical noise-free oscillators, this reduction corresponds to the Watanabe–Strogatz system of equations [Watanabe and Strogatz, Phys. Rev. Lett. 70, 2391 (1993)]. We discuss how the reduced system can be used to explore the transient dynamics of perturbed ensembles.
Skip Nav Destination
Article navigation
November 2022
Research Article|
November 08 2022
Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott–Antonsen and Watanabe–Strogatz theories
Rok Cestnik
;
Rok Cestnik
a)
(Conceptualization, Formal analysis, Investigation, Methodology, Validation, Visualization, Writing – original draft)
Department of Physics and Astronomy, University of Potsdam
, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
a)Author to whom correspondence should be addressed: [email protected]
Search for other works by this author on:
Arkady Pikovsky
Arkady Pikovsky
(Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Supervision, Validation, Visualization, Writing – original draft)
Department of Physics and Astronomy, University of Potsdam
, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam-Golm, Germany
Search for other works by this author on:
a)Author to whom correspondence should be addressed: [email protected]
Chaos 32, 113126 (2022)
Article history
Received:
June 27 2022
Accepted:
October 10 2022
Citation
Rok Cestnik, Arkady Pikovsky; Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott–Antonsen and Watanabe–Strogatz theories. Chaos 1 November 2022; 32 (11): 113126. https://doi.org/10.1063/5.0106171
Download citation file:
Pay-Per-View Access
$40.00
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Citing articles via
Response to music on the nonlinear dynamics of human fetal heart rate fluctuations: A recurrence plot analysis
José Javier Reyes-Lagos, Hugo Mendieta-Zerón, et al.
Reliable detection of directional couplings using cross-vector measures
Martin Brešar, Ralph G. Andrzejak, et al.
Synchronization in spiking neural networks with short and long connections and time delays
Lionel Kusch, Martin Breyton, et al.
Related Content
A two-frequency-two-coupling model of coupled oscillators
Chaos (August 2021)
Repulsively coupled Kuramoto-Sakaguchi phase oscillators ensemble subject to common noise
Chaos (March 2019)
Low-dimensional dynamics in non-Abelian Kuramoto model on the 3-sphere
Chaos (August 2018)