Bifurcation analysis and structural stability of simplicial oscillator populations

Abstract

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 $S{O}_2$ 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.

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Cited by in corpus (17)

• Networks beyond pairwise interactions: structure and dynamics
• Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes
• The effect of heterogeneity on hypergraph contagion models
• Unified treatment of synchronization patterns in generalized networks with higher-order, multilayer, and temporal interactions
• D-dimensional oscillators in simplicial structures: odd and even dimensions display different synchronization scenarios
• Spectrum of extensive multiclusters in the Kuramoto model with higher-order interactions
• Consensus on simplicial complexes, or: The nonlinear simplicial Laplacian
• Collective dynamics of heterogeneously and nonlinearly coupled phase oscillators
• Synchronization in temporal simplicial complexes
• Deeper but smaller: Higher-order interactions increase linear stability but shrink basins
• Memory selection and information switching in oscillator networks with higher-order interactions
• Multistability in coupled oscillator systems with higher-order interactions and community structure
• Collective dynamics on higher-order networks
• Universal scaling and phase transitions of coupled phase oscillator populations
• Simplicial cascades are orchestrated by the multidimensional geometry of neuronal complexes
• Tiered synchronization in coupled oscillator populations with interaction delays and higher-order interactions
• Higher-order synchronization on the sphere