Flocking Behavior For A Class Of Cucker-Smale Model With Perturbation

In recent years,the study of flocking behavior has become more and more popular with some scientists and researchers.Surprisingly,the C-S model can well reveal the operating mechanism and mathematical principles of birds in limited environmental information,so the C-S model has developed significantly in the fields of mathematics,dynamics,and control since its birth.In 2017,Ru Lining and Xue Xiaoping provided an orderly relationship for the initial state of the C-S model with hierarchical leadership.The results show that the occurrence of multi-cluster flocking behavior depends on the given ordered relationship.In addition,the model uses a constant parameter β to measure the strength of the interaction between particles,where the particles represent birds.Inspired by this conclusion,this paper adds a perturbation term to it,which studies and analyzes the influence of perturbation on the flocking behavior of the above model.For the C-S model with perturbation constructed in this paper,mono-cluster flocking behavior is demonstrated by the constructed energy function and the Gronwall inequality,and the proof of multi-cluster flocking behaviors is performed by using analytical methods such as contradiction and induction.The results of this paper show that when β=0.5,the above-mentioned C-S system has conditional flocking behavior.The condition is mainly maintained by perturbation and initial conditions,and the symmetrical and asymmetric influence functions are selected to prove respectively.When β>0.5,the C-S system under the level of perturbation leads to multicluster flocking behavior.The above research results may be applied in the fields of signal transmission and flight formation.In addition,this paper also introduces several numerical experiments to verify the validity of the above conclusions.