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We present an analytical description for the collective dynamics ofoscillator ensembles with higher-order coupling encoded by simplicialstructure, which serves as an illustrative and insightful paradigm for brainfunction and information storage. The novel dynamics of the system, includingabrupt desynchronization and multistability, are rigorously characterized andthe critical points that correspond to a continuum of first-order phasetransitions are found to satisfy universal scaling properties. Moreimportantly, the underlying bifurcation mechanism giving rise to multipleclusters with arbitrary ensemble size is characterized using a rigorousspectral analysis of the stable cluster states. As a consequence of $SO_2$group symmetry, we show that the continuum of abrupt desynchronizationtransitions result from the instability of a collective mode under thenontrivial antisymmetric manifold in the high dimensional phase space.

Bifurcation analysis and structural stability of simplicial oscillator populations