黎曼流形上的粒子群运动模型

Particle swarm motion model on Riemannian manifolds

自然界中生物群的协同运动模式多种多样,运动空间的复杂性对该多样性起着关键作用.为描述生物群的协调运动,人们基于欧氏空间提出了多种粒子群运动模型.然而,这些模型往往只能描述简单的运动模式.本文基于Morse势和黎曼流形提出了一个新模型,该模型可以描述任意运动空间中的群体运动,且易于工程实现.本文分别以球面、环面、Möbius带及类丘陵曲面为代表仿真分析了黎曼曲面的紧致性、亏格、可定向性及高斯曲率等性质对模型的影响.结果显示,紧致性导致粒子群聚合得更加紧密对称,非零亏格导致群体中粒子的运动速度趋于一致并阻止粒子群形成类涡流形态,不可定向性导致粒子群在任何条件下都趋于分散,而高斯曲率对粒子群运动行为的影响较小.此外,本文还研究了当曲面包含一个、两个及三个障碍物时模型的输出,结果显示,当势强度足够大时,粒子群能够趋向目标,也能包围目标或绕行障碍物.综上,相比已有模型,该模型能够描述更为丰富的运动模式.

The motion patterns of biological swarms in nature are diverse and even amazing.The complexity of spaces in which biological swarms move plays a key role to this diversity.To model the synergistic motion of biological swarms,many particle swarm motion models are introduced and exploited.However,by now these models can still describe some simple motion patterns of biological swarms.In this paper we introduce a new motion model based on the Morse potentials and Riemannian manifolds.This model can describe the motion of biological swarms in any kind of space and can be easily implemented in engineering.To numerically explore the effect of the geometry of Reimann surface on the model,we take sphere surface,torus,Möbius band and hilly terrain-like surface as a representation of compactness,genus,non-orientability and Gaussian curvature,respectively.It is shown that compactness can help the particle swarm aggregate into homogeneous pattern,non-zero genus can make agents’velocities become identical and thus prevent the emergence of vortex,non-orientability can diffuse the particle swarm,and Gaussian curvature has little influence on the model.Meanwhile,we also check the performance of the model with Reimann surfaces including one,two and three obstacles.It is shown that the particle swarm can move directly to the target or enclose the target and avoid the obstacles for suitable potential strengths.To summarize,our model can describe more motion patterns of biological swarms than the known models.