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Higher-dimensional generalizations of the Watanabe-Strogatz transform for vector models of synchronization. (English) Zbl 1398.34072

The Watanabe-Strogatz transform applies to networks of all-to-all coupled identical phase oscillators coupled through a sinusoidal function of oscillator phase. It reduces the number of differential equations needed to describe a network of \(N\) oscillators from \(N\) to 3, along with the values of \(N-3\) conserved quantities. The state of an oscillator in the Kuramoto model can be described by a \(d\)-dimensional unit vector lying on the unit sphere \(S^{d-1}\) for \(d=2\). This paper generalises the Watanabe-Strogatz transform to “oscillators” whose state is described by \(d\)-dimensional unit vectors for arbitrary \(d>2\). A network of \(N\) oscillators is then described by \(d(d+1)/2\) equations rather than \(N(d-1)\) and there exist a large number of conserved quantities constructed from inner products of vector variables. Complete synchronisation is discussed and the reduced equations are used to determine a condition for the stability of this state. Several generalisations of the Möbius map which preserve the unit ball are also presented.

MSC:

34D06 Synchronization of solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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