The Stuart-Landau oscillator generalized to $D>2$ dimensions has $\mathrm{SO}(D$) rotational symmetry. We study the collective dynamics of a system of $K$ such oscillators of dimensions $D=3$ and 4, with coupling chosen to either preserve or break rotational symmetry. This leads to emergent dynamical phenomena that do not have analogs in the well-studied case of $D=2$. Further, the larger number of internal parameters allows for the exploration of different forms of heterogeneity among the individual oscillators. When rotational symmetry is preserved there can be various forms of synchronization as well as multistability and partial amplitude death, namely, the quenching of oscillations within a subset of variables that asymptote to the same constant value. The oscillatory dynamics in these cases are characterized by phase locking and phase drift. When the coupling breaks rotational symmetry we observe partial synchronization (when a subset of the variables coincide and oscillate) and partial oscillation death (when a subset of variables asymptote to different stationary values), as well as the coexistence of these different partial quenching phenomena.

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• Collective dynamics
• Coupled oscillators
• Synchronization
• Synchronization transition