Bifurcation analysis and structural stability of simplicial oscillator populations

Understand

We present an analytical description for the collective dynamics of oscillator ensembles with higher-order coupling encoded by simplicial structure, which serves as an illustrative and insightful paradigm for brain function and information storage.

• The novel dynamics of the system, including abrupt desynchronization and multistability, are rigorously characterized and the critical points that correspond to a continuum of first-order phase transitions are found to satisfy universal scaling properties.
• More importantly, the underlying bifurcation mechanism giving rise to multiple clusters with arbitrary ensemble size is characterized using a rigorous spectral analysis of the stable cluster states.
• As a consequence of $SO_2$ group symmetry, we show that the continuum of abrupt desynchronization transitions result from the instability of a collective mode under the nontrivial antisymmetric manifold in the high dimensional phase space.