Multistable Attractors in a Network of Phase Oscillators with Three-Body Interactions

Three-body interactions have been found in physics, biology, and sociology. To investigate their effect on dynamical systems, as a first step, we study numerically and theoretically a system of phase oscillators with a three-body interaction. As a result, an infinite number of multistable synchronized states appear above a critical coupling strength, while a stable incoherent state always exists for any coupling strength. Owing to the infinite multistability, the degree of synchrony in an asymptotic state can vary continuously within some range depending on the initial phase pattern.

Figure 1

Two examples of multistability arising from three-body interaction. (a) Raster plot of the spikes in a network of the Hodgkin-Huxley neurons with short-term heterosynaptic plasticity ($N=500$). The different-colored dots represent the firings in the networks starting from different initial conditions. Neurons are sorted in ascending order of the firing rate. The firing rate of each neuron is shown in (b). (c) Time evolution of the order parameter $R$ for three different initial conditions in the phase-oscillator systems with nonuniform random coupling ($N=500$). Phase distributions of oscillators at $t=0$ and $t=1000$ are shown on the left and right, respectively.

Figure 2

(a) Time evolution of the order parameter $R$ from three different initial conditions in the mean field model with $N=10000$ and $K=3$. (b,c,d) ${\omega }_i-{\varphi }_i$ relationship for different initial conditions at $t=10000$.

Figure 3

(a) For some $q(\theta )$, the self-consistent equation $R=S[R,K,q(\theta )]$ has two solutions, $r_1<r_2$ (brown dashed line). Setting $\alpha$ appropriately makes $R=S[R,K,\alpha q(\theta )]$ have only one solution $R=r<r_2$ (red solid line). Note that $S^{\prime }[r,K,\alpha q(\theta )]=1$. (b) If we have $S[r,K,q_1(\theta )]=r$ and $S^{\prime }[r,K,q_1(\theta )]=s_1\le 1$ (blue short-dashed line) and $S[r,K,q_2(\theta )]=r$ and $S^{\prime }[r,K,q_2(\theta )]=s_2\ge 1$ (green long-dashed line), we can obtain $q(\theta )$ with which $S[r,K,q(\theta )]=r$ and $S^{\prime }[r,K,q(\theta )]=1$ hold (red solid line). (c,d,e) $q_2(\theta )$ which maximizes $S^{\prime }[r,K,q(\theta )]$ (e) under the constraint $S[r,K,q(\theta )]=r$ (d) is given by $q_2(\theta )=2\Theta [W(\theta ,r,K)-w_2]-1$ (green long-dashed line) where $w_2$ is set to satisfy $S[r,K,q_2(\theta )]=r$ (c). Under the same constraint, $S^{\prime }[r,K,q(\theta )]$ is minimized by $q_1(\theta )=2\Theta [w_1-W(\theta ,r,K)]-1$ (blue short-dashed line) where $S[r,K,q_1(\theta )]=r$. (f) Attainable region of the order parameter $R$ (shaded orange region). Note that the incoherent state $R=0$ is stable for any $K$ (red solid line). (g) Phase diagram of the system of phase oscillators when the strength of two-body and three-body interactions are changed. The symbol in each region is a schematic representation of the attainable values of the order parameter $R$. Gray lines represent the range of $R$ from 0 to 1. The attainable values and ranges of $R$ are indicated by black circles and boxes, respectively.