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 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.

Higher-order synchronization on the sphere

“…The largest value of d that we consider is d = 5 for the combined system (12), in order to confirm that it has properties similar to the d = 3 case. The main difference compared with d = 3 is that as the two-body forces increase in relative strength, the transition to a completely synchronized configuration is discontinuous.…”

“…For a general description of synchronization with respect to higher-order interactions we refer to [4] (section 6) also [5] (section 4) and [9], and note that extensions of the Kuramoto model to higher-order networks have been extensively investigated, but generally with trajectories restricted to the unit circle S 1 [7,[10][11][12][13][14]]. An exception is [15] where an extended D-dimensional Kuramoto model on the sphere is regarded as a three-body system with symmetric connectivity coefficients, in contrast to the antisymmetric couplings in our determinantal systems.…”

“…As will be shown, the system synchronizes for any coupling coefficients, whether positive or negative, and the three-body forces prevail unless the two-body coupling is positive and sufficiently strong by comparison to the three-body coupling. The specific system we consider, the d = 3 case of (12), is given by:…”

Tiered synchronization in coupled oscillator populations with interaction delays and higher-order interactions

“…Another property that has attracted more attention in recent years is the presence of higherorder interactions [13][14][15][16][17] , most notably motivated by applications neuroscience [18][19][20][21] and physics 22,23 . In fact, the effect of higher-order interactions have already been investigated in both synchronization [24][25][26][27][28][29][30][31][32][33][34][35] and other kinds of collective behavior [36][37][38][39] .…”

“…Appendix A: Alternative derivation of Eqs. (28), ( 31)- (35) We begin our alternative derivation with Eq. (15).…”