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A key challenge of nonlinear dynamics and network science is to understandhow higher-order interactions influence collective dynamics. Although manystudies have approached this question through linear stability analysis, lessis known about how higher-order interactions shape the global organization ofdifferent states. Here, we shed light on this issue by analyzing the richpatterns supported by identical Kuramoto oscillators on hypergraphs. We showthat higher-order interactions can have opposite effects on linear stabilityand basin stability: they stabilize twisted states (including full synchrony)by improving their linear stability, but also make them hard to find bydramatically reducing their basin size. Our results highlight the importance ofunderstanding higher-order interactions from both local and globalperspectives.

Deeper but smaller: Higher-order interactions increase linear stability but shrink basins