Study On Flocking Under Hierarchical Leadership

The phenomena of flocking can be seen everywhere in life,such as the birds flying in the sky,the fishes parading in the water,the ant colonies on the ground,and so on.,they may start at random,but eventually they form a regular whole,the speed of all individuals is the same,and the relative position remains unchanged.In addition to the careful study of biologists,scholars in other fields are also very interested in this phenomenon.Scholars in the field of mathematics tried to use mathematical theory to study the inherent laws of the flocking and made many contributions to this.In 2007,Cucker and Smale proposed two discrete and continuous classic models for studying flocking.Shen extended Cucker and Smale's model to the existence of a hierarchy.Since the model of the first hierarchical system was established in an idealized environment,this imposes higher requirements on the external environment.In order to make the model more in line with the actual situation.The thesis extends the model under the hierarchy to the case of time delay and noise.The main contents are as follows:In real life,due to problems such as limited transmission bandwidth and speed limitation,the communication time lag between agents is unavoidable.Therefore,it is necessary and meaningful to analyze the influence of communication delay on the system.Chapters 3of the thesis mainly analyzes the situation of asymmetry time-delay for discrete model under hierarchical system..Using the properties of norm and mathematical induction,proves that even if the discrete model with asymmetric time delay,the system can still achieve flocking.The analysis of the asymptotic delay with the continuous model under hierarchical system is similar to that of the discrete model,therefore,the fourth chapter of the thesis mainly analyzes the case of symmetrical delays in the continuous model under the hierarchy system,using the properties of mathematical induction and quadratic functions,it is proved that when the symmetric delays of the continuous model satisfy certain conditions,system can achieve flocking.And in the simulation,the thesis illustrated that when the delay was too large,the system eventually failed to reach the flocking.In reality,In addition to the flocking may be affected by the time delay,it may also be influenced by external uncertainties(the effect of noise).For example,a flock of birds will be affected by the flow of wind in the course of a flight.When a fish group marches in the water,it will be affected by the flow of water,etc.When the wind or water flows through Large,it will destroy the consistency of the cluster,so it is very meaningful to study the impact of noise.The fifth chapter of the thesis considers the existence of white noise.Using Ito formula and mathematical induction,it is proved that the system can achieve flocking when the noise intensity meets certain conditions.And in the simulation,it is illustrated that when the noise intensity is too large,the system eventually fails to achieve flocking.The theorems of the article are all numerical simulations using Matlab,which proves the accuracy of the theoretical results.