Emergent behaviors of thermodynamic Cucker–Smale ensemble with a unit-speed constraint

We propose the thermodynamic Cucker–Smale system with unit-speed constraints (TCSUS) and study its emergent dynamics with rigorous proofs. For this, three dissipative differential inequalities are derived in terms of $ L^\infty $-diameters of the position–velocity–temperature to verify mono-cluster flocking in the TCSUS model under suitable frameworks using two approaches. Motivated by the first approach, we provide a sufficient framework for bi-cluster flocking to the TCSUS model. Unlike to the mono-cluster flocking behavior of TCSUS, the decay rate of velocity–temperature for each cluster group is not exponential and is dependent on the decay rate of the corresponding communication weight, respectively.