The discrete time Vicsek model confined by a harmonic potential explains many aspects of swarm formation in insects. We have found exact solutions of this model without alignment noise in two or three dimensions. They are periodic or quasiperiodic (invariant circle) solutions with positions on a circular orbit or on several concentric orbits and exist for quantized values of the confinement. There are period 2 and period 4 solutions on a line for a range of confinement strengths and period 4 solutions on a rhombus. These solutions may have polarization one, although there are partially ordered period 4 solutions and totally disordered (zero polarization) period 2 solutions. We have explored the linear stability of the exact solutions in two dimensions using the Floquet theorem and verified the stability assignments by direct numerical simulations.

中文翻译：

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谐波受限 Vicsek 模型的精确解


受谐波势限制的离散时间 Vicsek 模型解释了昆虫群体形成的许多方面。我们找到了该模型的精确解，在二维或三维中没有对准噪声。它们是周期性或准周期性（不变圆）解，位于圆形轨道或多个同心轨道上，并且存在于约束的量子化值上。一条线上有针对一系列约束强度的第 2 周期和第 4 周期解决方案，菱形上有第 4 周期解决方案。这些解可能具有极化 1，尽管有部分有序的第 4 周期解和完全无序（零极化）的第 2 周期解。我们使用 Floquet 定理探索了精确解在二维中的线性稳定性，并通过直接数值模拟验证了稳定性分配。