(untitled) (Analysis in Theory and Applications, 2024)
Noting that the particles communicate with each other within a finite distance, in the paper, we investigate the hierarchical model with free will and cut-off function (linear or nonlinear). As our observation, the cut-off size r is a sensitive coefficient to achieve the flocking behavior. Besides we will use energy function to achieve some sufficient conditions to achieve a flock. The free-will represents the tendency of the particle itself to move. It will either facilitate or prevent cluster generation. Some numerical simulations will validate our conclusions.