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Single trajectories were calculated in MATLAB using the standard adaptive Runge–Kutta scheme with relative and absolute error tolerances chosen conservatively at 10−9 and 10−11, respectively. For the parameter scans, the maximal Lyapunov exponents were calculated by integrating the system for T = 105 time units using a standard fourth order Runge–Kutta scheme with fixed time step of Δt = 10−2.
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The Kuramoto–Sakaguchi system of coupled phase oscillators, where interaction between oscillators is determined by a single harmonic of phase differences of pairs of oscillators, has very simple emergent dynamics in the case of identical oscillators that are globally coupled: there is a variational structure that means the only attractors are full synchrony (in-phase) or splay phase (rotating wave/full asynchrony) oscillations and the bifurcation between these states is highly degenerate. Here we show that nonpairwise coupling—including three and four-way interactions of the oscillator phases—that appears generically at the next order in normal-form based calculations can give rise to complex emergent dynamics in symmetric phase oscillator networks. In particular, we show that chaos can appear in the smallest possible dimension of four coupled phase oscillators for a range of parameter values.


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