Simulating and analyzing the dynamics of collective movement with elastic interactions and leadership influence.

This paper investigates the simulation and analysis of collective self-propelled motion in the presence of a leader, a phenomenon commonly observed in natural and artificial systems, such as bird flocks or robotic swarms. Using a dynamic approach based on the Langevin equation, the study models interactions between particles and a leader through elastic restoring forces and incorporates stochastic noise to simulate random fluctuations. These forces ensure group cohesion, avoid collisions, and mimic real-world conditions where unpredictability plays a significant role. The simulation starts with particles randomly distributed in a two-dimensional space. As the leader moves at a constant velocity, particles adjust their positions and velocities to follow it. The results demonstrate a two-phase evolution: an initial rapid clustering around the leader due to strong elastic forces, followed by a gradual alignment of particle velocities to match the leader. The velocity profile, represented by a linear increase followed by an exponential decay, captures the transition from disorder to order. Key parameters such as the spring constant k_{1}, which governs the leader-particle interaction, and the diffusion coefficient D, which regulates stochastic noise, are shown to significantly influence the system dynamics. High interaction strength accelerates synchronization and stabilizes the group, while weaker interactions lead to slower clustering and potential dispersion. Phase diagrams and susceptibility analyses confirm that the leader's role is critical in initiating and maintaining collective behavior. This study highlights the emergence of coordinated motion in a leader-follower context, offering insights into biological systems and inspiring applications in swarm robotics, where efficient and cohesive movement is essential.