Presentation Information
[C08-03]Flocking of the revised Cucker-Smale model with visual field
*Yujie Gao1 (1. The university of Tokyo (Japan))
Keywords:
flocking behavior,visual field,complex network
The physical description and theoretical analysis of flocking behavior have garnered significant attention over the past two decades. Researchers have focused on understanding formation mechanisms of such phenomena. In 1995, Vicsek introduced a physical model to describe collective behavior which is called Vicsek model. Later, mathematicians Felipe Cucker and Stephen Smale developed the more comprehensive C-S model. Based on the C-S framework, this research explores flocking behavior with limited perceptual abilities, particularly restricted visual field. We propose a natural and effective mathematical model to characterize these visual constraints: η=(H/(σˆ2+y)ˆβ)max{c+cosθ, 0}, where c are constants and θ means the angle between direction of motion and position vector. When calculating the cosθ, we use the definition cosθ = <vi,xj-xi>/|vi||xj-xi|.
Based on our mathematical model, we use some technical methods in the field of graph theory and dynamic system. Finally, we conducted a quantitative analysis of the sufficient conditions for the formation of flocking behavior in the C-S model with visual field. we show the one of sufficient conditions which is related to the visual field in flocking behavior: Λ(0) ≦ Kε02(0), where Λ(0) = maxi,j |vi − vj | at initial time point, and ε0(0) = minj { c + cosθij } at t=0. From this sufficient condition, we are able to understand that not only does the initial energy function (or so called initial difference in individual velocities) influence the formation of flocking, but the relative positions of individuals at the initial time also play a significant role in flocking formation. This conclusion aligns with our understanding. Finally, we take some computer simulation to verify the correction of our conclusions and take some physical analysis about pahse transition.
In this study, the construction of the visual field function generalizes the C-S model beyond only considering distance as the factor. The construction of the visual field function highlights that interaction intensity is influenced not only by Euclidean or topological distance but also by the relative position of neighbors with respect to the visual field center. Mathematically, this can be expressed as the inner product of relative positions and velocity direction. This study can inspired researchers consider more mathematical functions on perception limitation such as hearing limitation and chemical signal which depend on different species. This construction will enable biologists to better understand and apply these theoretical analyses.
Based on our mathematical model, we use some technical methods in the field of graph theory and dynamic system. Finally, we conducted a quantitative analysis of the sufficient conditions for the formation of flocking behavior in the C-S model with visual field. we show the one of sufficient conditions which is related to the visual field in flocking behavior: Λ(0) ≦ Kε02(0), where Λ(0) = maxi,j |vi − vj | at initial time point, and ε0(0) = minj { c + cosθij } at t=0. From this sufficient condition, we are able to understand that not only does the initial energy function (or so called initial difference in individual velocities) influence the formation of flocking, but the relative positions of individuals at the initial time also play a significant role in flocking formation. This conclusion aligns with our understanding. Finally, we take some computer simulation to verify the correction of our conclusions and take some physical analysis about pahse transition.
In this study, the construction of the visual field function generalizes the C-S model beyond only considering distance as the factor. The construction of the visual field function highlights that interaction intensity is influenced not only by Euclidean or topological distance but also by the relative position of neighbors with respect to the visual field center. Mathematically, this can be expressed as the inner product of relative positions and velocity direction. This study can inspired researchers consider more mathematical functions on perception limitation such as hearing limitation and chemical signal which depend on different species. This construction will enable biologists to better understand and apply these theoretical analyses.