| The focking studied in this article describes the phenomenon of autonomous particle swarm changing from disordered motion to ordered motion through limited environmental information and simple rules.For example,focks of birds form a group and fy in the air at almost the same speed.This phenomenon has aroused great interest of biologists and ecologists.At the same time,this phenomenon has also been widely studied in the felds of natural science and engineering technology,such as automated agricultural machinery systems for agricultural production and intelligent transportation systems for urban planning.Therefore,in order to describe the focking better in real life,this article extends the popular Cucker Smale(C-S)system in recent years.We establish two important systems by introducing harmonic potential feld,multiplicative noise,and time delay.And we provide suffcient conditions for the system to exhibit focking.First,this article studies the non-delayed system in theory.And then we introduce a macro-micro decomposition to decompose the system into two parts: one system describes the macroscopic dynamics and the other system describes the microscopic fuctuations.Next,we construct a suitable Lyapunov functional for the system,and provide suffcient conditions for the collective dynamics.On this basis,we further study the delayed system.By comparing the delayed system with the corresponding non-delayed system,we construct the Lyapunov functional.Using the stability theory,we prove that if the delay is uniformly bounded in a suffciently small time,and the corresponding non-delayed system is exponentially stable almost surely,the speed and position of the delayed system will eventually converge.Finally,the two systems studied in this paper are numerically simulated by using the Euler methods through Matlab.Thus,the accuracy of the theoretical analysis results in this paper is verifed numerically. |
