Particle swarm motion model on Riemannian manifolds
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 mo-tion 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 homoge-neous 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.
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