Dynamics of phase oscillators with generalized frequency-weighted coupling

Heterogeneous coupling patterns among interacting elements are ubiquitous in real systems ranging from physics, chemistry to biology communities, which have attracted much attention during recent years. In this paper, we extend the Kuramoto model by considering a particular heterogeneous coupling scheme in an ensemble of phase oscillators, where each oscillator pair interacts with different coupling strength that is weighted by a general function of the natural frequency. The Kuramoto theory for the transition to synchronization can be explicitly generalized, such as the expression for the critical coupling strength. Also, a self-consistency approach is developed to predict the stationary states in the thermodynamic limit. Moreover, Landau damping effects are further revealed by means of linear stability analysis and resonance poles theory below the critical threshold, which turns to be far more generic. Our theoretical analysis and numerical results are consistent with each other, which can help us understand the synchronization transition in general networks with heterogenous couplings.

Figure 1

Different scenarios of the decay of $\delta r\left(t\right)$ with respect to different frequency distributions $g\left(\omega \right)$ and frequency-weighted functions $f\left(\omega \right)$ below the critical threshold ($K<K_c$). The horizontal axes are time, solid lines refer to the direct numerical solutions of Eq. (5), and dashed lines are the fitted curve lines with exponent $s_d$. (a) $K=0$, the red is $g\left(\omega \right)=\frac{1}{\pi ({\omega }^2+1)}$, the green is $g\left(\omega \right)=\frac{1}{2},\omega \in (-1,1)$, and the blue is $g\left(\omega \right)=1-\left|\omega \right|,\omega \in (-1,1)$, the inset is the expression of $\delta r\left(t\right)$. (b) $K=1$, with $f\left(\omega \right)=\omega ,g\left(\omega \right)=\frac{1}{\pi ({(\omega -\mathrm{\Delta })}^2+1)}$, the red $\mathrm{\Delta }=1.0$, the blue $\mathrm{\Delta }=0$, and the pink $\mathrm{\Delta }=-1.0$, respectively, the inset is the decay exponent $s_d$. (c) $K=1.2$, with $f\left(\omega \right)=\left|\omega \right|,g\left(\omega \right)=\frac{1}{\pi ({\omega }^2+1)}$, and $s_d=-0.1298$ which is calculated numerically. (d) $K=0.8$, with $f\left(\omega \right)=\omega$. $g\left(\omega \right)=\frac{1}{2},\omega \in (-1,1)$, and $s_d=-0.07433$ calculated numerically. All curves belong to the neutrally stable regime of the incoherent state predicted by linear theory. In the numerical simulations, the initial states of the system are set in the fully coherent states, and the total number of oscillators is $N=100000$.