This paper demonstrates several sufficient frameworks for the mono-cluster flocking, the non-emergence of mono-cluster flocking and the multi-cluster flocking of the thermodynamic Cucker–Smale model with a unit-speed constraint (say TCSUS). First, in a different way than ^[2], we present the admissible data for the mono-cluster flocking of TCSUS to occur. Second, we prove that when the coupling strength is less than some positive value, mono-cluster flocking does not occur in the TCSUS system with an integrable communication weight. Third, motivated from the study on coupling strengths where the mono-cluster flocking does not occur, we investigate appropriate sufficient frameworks to derive the multi-cluster flocking of the TCSUS system.

This paper demonstrates several sufficient frameworks for the multi-cluster flocking behavior of the fractional Cucker–Smale (CS) model. For this, we first employ the Caputo fractional derivative instead of the usual derivative to propose the fractional CS model with the memory effect. Then, using mathematical tools based on fractional calculus, we present suitable sufficient conditions in terms of properly separated initial data close to the multi-cluster, and well-prepared system parameters for the multi-cluster flocking of the fractional system to emerge. Finally, we offer several numerical simulations and compare them with the analytical results.

We illustrate finite-in-time flocking in the thermodynamic Cucker–Smale (TCS) model. First, we extend the original TCS model to allow for a continuous vector field with a locally Lipschitz continuity. Then, within this system, we derive appropriate dissipative inequalities concerning the position-velocity-temperature using several preparatory estimates. Subsequently, based on initial data and system parameters, we formulate sufficient conditions to guarantee the desired finite-time flocking in each case where the communication weight conditions are divided into two scenarios: one with a positive lower bound and another with nonnegativity and monotonicity. Finally, we provide several numerical simulations and compare them with the analytical results.