- 제목: On the emergent dynamics of generalized inertial spin models

- 초록: The inertial spin model describes flocking dynamics with collective turning of an ensemble of self-propelled particles moving with constant speed. In this thesis, we study three generalizations of the inertial spin model. First, we study quantitative flocking estimates for two thermo-mechanical inertial spin models. More precisely, we present a sufficient framework leading to exponential flocking of each thermo-mechanically consistent inertial spin model. For this, we derive a system of differential inequalities for relative kinetic and spin energies, when thermal and linear velocities are confined in an invariant cone, and then we explicitly obtain exponential flocking under the proposed frameworks. Second, we study exponential flocking of the inertial spin model with delayed communication. Compared with the non-delayed model, delayed interactions due to the finite speed of communication generate an additional memory effect. To handle this effect, we introduce a generalized energy functional that incorporates the delay effect. Using the time evolution of this energy functional, we show that relative velocities and spins tend to zero exponentially fast under suitable conditions on the system parameters and initial data, together with a smallness condition on the maximal delay. The main novelty is that, even in the presence of time delays, the pairwise alignment condition of the previous result for the non-delayed model is replaced by a weaker mean-velocity a priori condition. We also remove the large damping condition required in that result. Third, we introduce a finite-agent high-dimensional inertial spin (HIS) model which generalizes the classical three-dimensional inertial spin model and study its emergent behaviors. For the generalization, we replace the cross product with multiplication by agent-wise skew-symmetric spin matrices, and identify a new set of constants of motion. We provide two frameworks leading to the collective behaviors of the HIS model, yielding qualitative and quantitative emergent dynamics in terms of system parameters and initial data. We also perform a set of numerical simulations and compare them with analytical results.