Study On Flocking Behavior Of Thermodynamic Cucker-Smale System Under Hierarchical Leadership

There is a rich variety of flocking phenomena in nature.Flocking behavior means that the multi-intelligent system achieves the goal of a consistent overall motion state only through the interactions and simple rules between particles in a group composed of a certain number of agents.In the flocking behavior of thermodynamic Cucker Smale（TCS）systems,particle motions are not only mechanical,but also influenced by temperatures.This article studies the flocking behavior of the TCS-type systems under three cases:（1）The flocking behavior of the TCS model with multiplicative white noise under hierarchical leadership is studied.Flock movements are affected by the uncertain factors in the external environment.When the noise is too large,the flocking motion of the system will be broken.White noise can be used to simulate the uncertainty interference.Through the boundedness of temperature expectation and velocity expectation and the It?οformula,it is proved that the time progressive consistency of temperature and velocity of the model corresponding to a hierarchical population consisting of two particles holds.The above results are extended to the whole system,and it is proved that the system can realize time progressive flock motion when the white noise does not exceed a certain limit.Finally,numerical simulations verify the correctness of the results.The system has a tolerance limit to noise in the environment.The movement of particles has a certain degree of adaptability.The stronger the interactions between particles,the shorter the time required for the system to achieve flocking.（2）The effect of symmetrical and asymmetrical time delays on flocking in TCS systems under hierarchical leadership is studied.In the actual scene,communication delays occur frequently and have an impact on the interactions between particles in flocking motion.First,it is proved that the temperature and velocity of particles in the system are bounded.Then,the consistency of the temperature and velocity of the model corresponding to a hierarchical population consisting of two particles holds.Finally,the above results are extended to the whole system by using mathematical induction.It is proved that the TCS system with symmetric time delays can achieve flock motion when the communication time delay satisfy certain conditions.When the system parametersβ₁,β₂ satisfy certain conditions,the TCS system with asymmetric time delay can achieve flock motion unconditionally These results are replicated in numerical simulations.（3）The flock movement of the TCS system with symmetric communication time delays and free perturbations under hierarchy is studied.The motion of each particle in the group has a certain degree of freedom,and the velocity variation does not only depend on the interactions between particles.The small perturbation function f_i（t）is added to the velocity variation of the model to simulate the variability of particle motion,which makes the system more general in its application.First,the boundedness of the temperature and velocity of the particles in the system is proved.Then,it is shown that the temperature and velocity of a hierarchical group composed of two particles corresponding to the model can reach flocking by using Gronwall inequality.Finally,the conclusion is extended to the whole system through mathematical induction.It is shown that the system can achieve flocking when the time delays and free perturbations satisfy certain conditions.The particle velocity has a certain degree of autonomy will and the system can still achieve cluster motion by adding a small disturbance function to the velocity variation of the model.The damping-type functions are imported into the system,and the validity of the results is verified by numerical simulations.