Systems consisting of a large number of phase oscillators, each with a different natural oscillation frequency, are of interest in diverse natural and physical settings. These include groups of interacting insects, humans, neuronal cells, Josephson junctions, etc. Starting with the paradigmatic Kuramoto model introduced in 1975, appropriate phase-oscillator-based models have been proposed for these different situations. In this talk we first show that these models typically admitt an exact reduction to low dimensional dynamics [1], i.e., the full many-oscillator models can be reduced to a small number of ordinary differential equations. This reduction is achieved by postulating a special form for the distribution function of the oscillator phases. In spite of this assumed special form, it was numerically found (e.g., [2,3]) that all long term behavior of the full systems (e.g., their attractors and bifurcations) was apparently captured by the low dimensional descriptions. This puzzle is now rigorously understood [4]: the special solutions form a manifold in the space of distribution functions, and all initial conditions are attracted to this manifold. This talk will discuss these recent developments, including applications, development of the special solutions for a general class of models, and discussion of the attractive nature of the manifold of special solutions.
[1] E.Ott and T.M.Antonsen, Chaos, vol. 18, 037113 (2008).
[2] T.M.Antonsen, et al., Chaos, Vol. 18, 037112 (2008).
[3] E.Martens, et al., Phys.Rev.E, vol. 79, 026204 (2009).
{4} E.Ott and T.M.Antonsen, Chaos, vol. 19, 023117 (2007).
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