标题： 抛物线和双曲线标度之间的交叉，群体附近振荡模式和共振 显示英文标题

标题： Crossover between parabolic and hyperbolic scaling, oscillatory modes and resonances near flocking

摘要： 对动力学方程的稳定性和分岔分析表明，二维Vicsek模型的群体运动分岔表现出抛物线行为和双曲行为之间的相互作用。 对于小于某个较大值的盒子尺寸，群体运动在噪声的临界值处从均匀无序状态连续出现。 由于质量守恒，描述群体运动状态的振幅方程包括一个标量方程，用于描述相对于均匀粒子密度（粒子数除以盒子面积）的密度扰动，以及一个矢量方程，用于描述电流密度。 这两个方程包含两个时间尺度。 在较短的时间尺度上，它们是一个双曲系统，其中时间和空间以相同的方式缩放。 在较长的、扩散的时间尺度上，方程是抛物线的。 分岔解依赖于角度，并且在空间上是均匀的，如同通常的叉型分岔的正规形式。 我们证明，在后者解附近进行线性化时，可以用双曲时间尺度下的克莱因-戈登方程来描述。 然后，在分岔解附近存在持续的振荡，具有许多不相容的频率，它们会导致临界噪声的偏移，并与对齐规则的周期性强迫产生共振。 这些预测通过Vicsek模型的直接数值模拟得到了验证。 显示英文摘要

摘要： A stability and bifurcation analysis of a kinetic equation indicates that the flocking bifurcation of the two-dimensional Vicsek model exhibits an interplay between parabolic and hyperbolic behavior. For box sizes smaller than a certain large value, flocking appears continuously from a uniform disordered state at a critical value of the noise. Because of mass conservation, the amplitude equations describing the flocking state consist of a scalar equation for the density disturbance from the homogeneous particle density (particle number divided by box area) and a vector equation for a current density. These two equations contain two time scales. At the shorter scale, they are a hyperbolic system in which time and space scale in the same way. At the longer, diffusive, time scale, the equations are parabolic. The bifurcating solution depends on the angle and is uniform in space as in the normal form of the usual pitchfork bifurcation. We show that linearization about the latter solution is described by a Klein-Gordon equation in the hyperbolic time scale. Then there are persistent oscillations with many incommensurate frequencies about the bifurcating solution, they produce a shift in the critical noise and resonate with a periodic forcing of the alignment rule. These predictions are confirmed by direct numerical simulations of the Vicsek model.