Low dimensional behavior of large systems of globally coupled oscillators.

By Edward Ott, T. Antonsen. Chaos, 2008

1,055 citations recorded by the source corpus.

It is shown that, in the infinite size limit, certain systems of globally coupled phase oscillators display low dimensional dynamics. In particular, we derive an explicit finite set of nonlinear ordinary differential equations for the macroscopic evolution of the systems considered. For example, an exact, closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function is obtained. Low dimensional behavior is also demonstrated for several prototypical extensions of the Kuramoto model, and time-delayed coupling is also considered.

Subfields: Computational Mechanics, Computer Networks and Communications, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/4bf87101bea0b5f5ef8f5d1bda4703fb22fec460

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In Computer Networks and Communications

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Citations in the corpus

Synchronization in complex networks of phase oscillators: A survey (2014, methodology): “…of K partial , see (Kuramoto, 1984a; Crawford, 1994; Strogatz, 2000; Acebrón et al., 2005; Mirollo and Strogatz, 2007; Balmforth and Sassi, 2000; Ott and Antonsen, 2008; Chiba, 2014) and references therein, the articles (Ermentrout, 1985; Hemmen and Wreszinski, 1993; Mirollo and Strogatz, 2005,…”
The mathematics behind chimera states (2018, methodology): “More precisely, for globally coupled phase oscillators it is known [132, 133] that the Nonlinearity 31 (2018) R121”
The Kuramoto model in complex networks (2015, methodology): “[198] provided a framework for the study of delay heterogeneity on the stability of incoherent states and the dynamics transitions from stable incoherent states to stable coherent states, using the OA ansatz [32].”
A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model (2010, methodology): “Their analysis depends on extensive symmetries of the Kuramoto model found by Ott and Antonsen [36, 37] (see also [32]) and on the special form of g(ω), however, such a diagram seems to be common for any bimodal distributions.”
The dynamics of chimera states in heterogeneous Kuramoto networks (2009, methodology): “With g(ω) from (9) and ω0 = 0 we can use contour integration to perform the integral over ω in (8), obtaining [12,16]”
Explosive transitions in complex networks’ structure and dynamics: Percolation and synchronization (2016, methodology): “Based on the Ott-Antonsen ansatz [241], the dynamics of Eq.”
Coupling functions:universal insights into dynamical interaction mechanisms (2017, background): “…amplitude and oscillation death (Aronson et al., 1990; Koseska et al., 2013a; Schneider et al., 2015; Zakharova et al., 2014), the low-dimensional dynamics of ensembles (Ott and Antonsen, 2008; Watanabe and Strogatz, 1993), and clustering in networks (Ashwin and Timme, 2005; Kori et al., 2014).”
Networks beyond pairwise interactions: structure and dynamics (2020, methodology): “The Watanabe-Strogatz theory provides a lower-dimensional description of all-to-all coupled oscillators with identical frequencies [366, 367], similarly to the Ott-Antonsen theory for distributed frequencies [368].”
Phase reduction approach to synchronisation of nonlinear oscillators (2016, background): “In a recent study, Ott and Antonsen showed that, for the Kuramoto model with Lorentzian frequency distribution, the whole infinite-dimensional system can be drastically reduced to a single ODE for the complex order parameter by putting an ansatz (called the Ott-Antonsen ansatz) for the probability d”
Macroscopic description for networks of spiking neurons (2015, methodology): “According to the Ott-Antonsen (OA) theory, in the thermodynamic limit the dynamics of the class of systems ∂tθ(η, t) = Ω(η, t) + Im [ H(η, t)e−iθ ] , (16) generally converges to the OA manifold (14).”
Coherence–incoherence patterns in a ring of non-locally coupled phase oscillators (2013, methodology): “To this end, we use the OttAntonsen method suggested in [40, 41].”
Low-dimensional dynamics of populations of pulse-coupled oscillators (2013, methodology): “A O ] 20 M ay 2 01 3 Low-dimensional dynamics of populations of pulse-coupled oscillators Diego Pazó1 and Ernest Montbrió2 1Instituto de Fı́sica de Cantabria (IFCA), CSIC-Universidad de Cantabria, 39005 Santander, Spain 2Department of Information and Communication Technologies, Universitat Pompeu ”
Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching (2019, methodology): “θi −H∗eiθi ˙ . (7) Expanding finto its Fourier series f(θ,ω,t) = g(ω) 2π h 1+ P∞ n=1 fˆ n(ω,t)einθ+c.c. i (where c.c. denoted the complex conjugate of the previous term), we follow Ott and Antonsen [29] and ansatz that Fourier coeﬃcients decay genometrically, i.e., fˆ n(ω,t) = αn(ω,t) for some funct”
Spontaneous synchrony in power-grid networks (2013, background): “d ! i . Assuming that i = i + i 0, 5 P ei= P ei + P0 ei , and P mi= P mi + P mi 0 , the linearized equation reads 2H i ! R d20 i dt2 = @P mi @! i !0 i 0 @P ei @! i !0 i Xn j=1 @P ei @ j j ; (5) where we neglect the dependence of the mechanical power on changes in the phase i, and denote ! 0 i =”