Figure 1

Mean field strength $R$ and frame frequency $\Omega$ for different distributions $g(\omega ,K)$ in dependence on $⟨K⟩$, varied by changing the proportion $p$ of oscillators with different $K_i$. Except (d), couplings have bimodal-$\delta$ distribution: $\Gamma (K)=W_{1-p,p}(K;K_1,K_2)$ with $K_1=-0.5$, $K_2=1$ [in notations of (6)]. In (d), we use trimodal-$\delta$ $\Gamma (K)=W_{\frac{1-p}{2},\frac{1-p}{2},p}(K;K_1,K_2,K_3)$ with $K_1=-1.5$, $K_2=-0.75$, $K_3=2$. Conditional frequency distribution $g(\omega |K)$ differs for each case: (a) $g(\omega |K)\sim e^{-{\omega }^2/2{\sigma }^2}$ with $\sigma =0.05$; (b) $g(\omega |K)=L(\omega ,\gamma /K^2)$ with $\gamma =0.05$; (c) $g(\omega |K)\sim [(\omega +\mu {)}^2+2{\gamma }^2/(1+e^{(\omega +\mu )/{\gamma }^2}){]}^{-1}$, where $\mu$ establishes $⟨\omega ⟩=0$, and $\gamma =0.1$; (d) $g(\omega |K)=L(\omega ,\gamma )$ with $\gamma =0.025$. Theoretical predictions (lines) are compared with numerical simulations (circles). $I=\mathrm{\text{incoherence}}$; $NS=\mathrm{\text{natural state}}$; $TW=\mathrm{\text{traveling wave}}$. Full (dotted) lines imply stable (unstable) states, according to the ESCs. Black lines indicate TW states with $\mathrm{tr}(\widehat{S})<0$, $\det (\widehat{S})\approx 0$ [see Eq. (16)]. The simulations used $N=25600$ oscillators and a 6th order Runge-Kutta algorithm with time step 0.01 s for 500 s. $R$ and $\Omega$ are averages over the last 100 s.Reuse & Permissions