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On Higher Dimensional Generalized Kuramoto Oscillator Systems | | Max Lipton
; Renato Mirollo
; Steven H. Strogatz
; | | Date: |
16 Jul 2019 | | Abstract: | The aim of this set of notes is to explain and unify some work by Tanaka [1],
Lohe [2] and Chandra et~al.~[3, 4] on a generalization of Kuramoto oscillator
networks to the case of higher dimensional ’’oscillators.’’ Instead of
oscillators represented by points on the unit circle $S^1$ in ${Bbb R}^2$, the
individual units in the network are represented by points on a higher
dimensional unit sphere $S^{d-1}$ in ${Bbb R}^d$. Tanaka demonstrates in his
2014 paper that the dynamics of such a system can be reduced using M"obius
transformations, similar to the classic case when $d = 2$ [5]. Tanaka also
presents a generalization of the famous Ott-Antonsen reduction for the complex
version of the system [9]. Lohe derives a similar reduction using M"obius
transformations for the finite-$N$ model, whereas Chandra et~al.~concentrate on
the infinite-$N$ or continuum limit system, and derive a dynamical reduction
for a special class of probability densities on $S^{d-1}$, generalizing the
Poisson densities used in the Ott-Antonsen reduction.
The oscillator systems studied in [1]--[4] are intimately related to the
natural hyperbolic geometry on the unit ball $B^d$ in ${Bbb R}^d$; as we shall
show, once this connection is realized, the reduced dynamics, evolution by
M"obius transformations and the form of the special densities in [3] and [4]
all follow naturally. This framework also allows one to see the seamless
connection between the finite and infinite-$N$ cases. In addition, we shall
show that special cases of these networks have gradient dynamics with respect
to the hyperbolic metric, and so their dynamics are especially easy to
describe. | | Source: | arXiv, 1907.7150 | | Services: | Forum | Review | PDF | Favorites | |
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