相互作用粒子系统的噪声、分叉和建模
Noise, Bifurcations, and Modeling of Interacting Particle Systems
摘要：
我们考虑随机模式的通信系统，或耦合，自推进粒子在存在噪声和通信时间延迟。对于足够大的环境噪声，存在平移状态和具有静止质心的旋转状态之间的过渡。时间延迟通信产生取决于粒子之间的耦合幅度的分叉模式。利用一个大数极限下的平均场模型，我们证明了在存在通信延迟和耦合振幅的情况下，完整的分岔是如何展开的。相对于质心，模式可以被描述为平移之间的过渡，围绕一个固定点的旋转，或旋转的群，其中的质心经历了一个从稳定状态到极限环的霍普夫分岔。对于大量的粒子，将给出一些随机模式的例子。
英文摘要：
We consider the stochastic patterns of a system of communicating, or coupled, self-propelled particles in the presence of noise and communication time delay. For sufficiently large environmental noise, there exists a transition between a translating state and a rotating state with stationary center of mass. Time delayed communication creates a bifurcation pattern dependent on the coupling amplitude between particles. Using a mean field model in the large number limit, we show how the complete bifurcation unfolds in the presence of communication delay and coupling amplitude. Relative to the center of mass, the patterns can then be described as transitions between translation, rotation about a stationary point, or a rotating swarm, where the center of mass undergoes a Hopf bifurcation from steady state to a limit cycle. Examples of some of the stochastic patterns will be given for large numbers of particles.
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