Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes

Collective behavior in large ensembles of dynamical units with nonpairwise interactions may play an important role in several systems ranging from brain function to social networks. Despite recent work pointing to simplicial structure, i.e., higher-order interactions between three or more units at a time, their dynamical characteristics remain poorly understood. Here we present an analysis of the collective dynamics of such a simplicial system, namely coupled phase oscillators with three-way interactions. The simplicial structure gives rise to a number of novel phenomena, most notably a continuum of abrupt desynchronization transitions with no abrupt synchronization transition counterpart, as well as extensive multistability whereby infinitely many stable partially synchronized states exist. Our analysis sheds light on the complexity that can arise in physical systems with simplicial interactions like the human brain and the role that simplicial interactions play in storing information.

Figure 1

Synchronization profiles. The order parameters (a) $r_1$ and (b) $r_2$ as a function of the coupling strength $K$ for various asymmetry values $\eta$. Blue, red, green, orange, and purple circles represent results obtained from direct simulations of Eq. (2) with $N={10}^5$ oscillators for $\eta =1$, 0.95, 0.9, 0.85, and 0.8, respectively.

Figure 2

Synchronization branches. For order parameters (a) $r_1$ and (b) $r_2$, the synchronization branches predicted by Eqs. (10) and (11) are plotted as solid curves for asymmetries $\eta =1$, 0.95, 0.9, 0.85, and 0.8 as well as the minimum branches [predictions for which are given in Eq. (15)]. Results obtained from direct simulations with $N={10}^5$ oscillators are plotted in blue, red, green, orange, and purple circles for each value of $\eta$ and black circles represent minimum branches.

Figure 3

Abrupt desynchronization transition. (a) The critical coupling strength $K_c$ at which the abrupt desynchronization transition occurs as a function of the asymmetry $\eta$. The solid curve represents the theoretical prediction given by Eq. (15) and black circles represent observations from direct simulation with $N={10}^5$ oscillators. (b) Illustration of the distribution $f(\theta )$ of phases as the system passes though the abrupt desynchronization transition. Here $\eta =0.9$ with $K_c\approx 6.47$. Distributions for $K=6.7$, 6.6, 6.5, and 6.4 are plotted in solid blue, dot-dashed red, dotted green, and dashed orange curves.

Figure 4

1-simplex layer dynamics. The order parameter ${\rho }_1$ describing synchronization of $\varphi$ oscillators in the 1-simplex layer as a function of the driving strength $d$. Connected circles represent numerical simulations with $10^5$ oscillators. 2-simplex and 1-simplex coupling are $k=12$ and $\kappa =3$, and asymmetry values $\eta =1$, 0.95, 0.9, 0.85, and 0.8 are colored blue, red, green, orange, and purple, respectively. Dashed horizontal lines denote the values of $r_1$ for each $\eta$ and the dotted vertical line indicated $d=\kappa$.