PMC

Δ = 0.5 and γ = 0.5. Rows (a–c) correspond to κ = 2.05, 2.5, and 4.0, respectively, i.e., to the points A, B, and C of . The arrangement of panels are the same as in .

The FD is fixed as Δ = 0.5 and γ = 0.5. (a) The order parameter r vs the coupling strength κ. Letters A, B, C denotes the three conditions that will be analyzed in the next Figure. (b) Given fixed central frequency of FD Δ = 0.5, the critical values of vs the width of FD γ.

(a) r vs Δ (see text for definitions). Letters A, B, C denotes the three conditions that will be analyzed in the next Figure. The inset reports r vs p, i.e., the proportion of conformist oscillators in the ensemble, which (for any given γ) is entirely controlled by Δ. (b) Given γ, Δ_c comes out to be inversely proportional to κ.

Horizontal lines are time, solid lines refer to the numerical solutions of Eq. , and dashed lines are the analytical predictions of Eq. . The three log-linear plots refer to the cases Δ > 0 (a), Δ = 0 (b), and Δ < 0 (c). 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.

Horizontal lines are time, solid lines refer to the direct numerical solutions of Eq. with κ = 1.0, and dashed lines are the fitted straight lines with slope k. Note that (a–c) are semi-log plots, while (d) is double-log. To effectively suppress the fluctuation of the order parameter, the number of oscillator N = 25600000. The formulae and parameters for the three FDs are as follows. Uniform: g(ω) = 1 for |ω − Δ| < 0.5, and 0 otherwise. Δ = 0.1. Gaussian: with Δ = 0.1 and γ = 0.5. Triangle: g(ω) = (γ − |ω − Δ|)/γ² for |ω − Δ| < γ, and 0 otherwise. Δ = 0.1 and γ = 0.5.

. For a Lorentzian frequency distribution, the continuous spectrum is the whole imaginary axis. The solid squares denote, instead, the discrete eigenvalues. The purple ovals in (a,c–e) mark the ghost (fake) eigenvalues predicted by the linear theory, which are actually valid only on the right half complex plane. As discussed in the text, such ghost eigenvalues remarkably control the decaying rate of order parameter r(t) in the neutrally stable regimes, exactly as in the Landau damping context. It should be pointed out that the ghost eigenvalue in (d,e) could also be above the real axis, depending on parameters. For better visualization, we do not plot the other irrelevant (fake) eigenvalues.

κ = 4.5 and γ = 0.5. Rows (a–c) correspond to Δ = 0.226, 0.26, and 0.5, respectively, i.e., to the points A, B, and C of . Column 1 plots the order parameter in the complex plane after a transient stage, while columns 2–3 correspond to the snapshots of the distributions of the instantaneous phases and the frequencies at t = 2,500. In (a), the conformists slightly prevail over the contrarians, and only a small part of conformists form a coherent cluster, rotating at a certain frequency along the unit circle. In (b), the coherent cluster of conformists continuously expands, while the contrarians begin forming a coherent cluster. Both clusters rotate at the same frequency along the unit circle, with a constant phase difference between them. In (c) more conformists are present in the system. The cluster of the conformists further enlarges, leading to the increase of the order parameter. The inset in (c2) shows the phase distribution at that moment. The phase difference between two peaks is 1.2π.