In this paper, we study the singular Cucker-Smale (C-S) model on the real line. For long range case, i.e. β < 1, we prove the uniqueness of the solution in the sense of Definition 2.1 and the unconditional flocking emergence. Moreover, the sufficient and necessary condition for collision and sticking phenomenon will be provided. For short range case, i.e. β > 1, we construct the uniform-in-time lower bound of the relative distance between particles and provide the sufficient and necessary condition for the emergence of multi-cluster formation. For critical case, i.e. β = 1, we show the uniform lower bound of the relative distance and unconditional flocking emergence. These results provide a complete classification of the collective behavior for C-S model on the real line.(In fact, recently, the C-S model with singular interaction (1.2) attracts a lot of attention from various area. This is mainly due to that the Coulomb type interaction will automatically generates the repulsion and leads to the avoidance of collision [7], which is more physical and very important for application in engineering such as formation of unmanned aerial vehicles. However, this singular communication weight causes a lot of difficulty in the mathematical analysis. For instance, the uniqueness of the solution to (1.1) cannot be guaranteed by the fundamental theory of ODE, because the vector field on the right hand side of (1.1) is no more Lipschitz. Therefore, comparing to the extensive study on regular communication weight, there are very few works concerning on the C-S model with singular interaction: flocking dynamics and mean-field limit [25], avoidance of collision [1,7], global existence of weak solutions in particle and kinetic level [6,35,39,40]. More precisely,in [39], the author constructed the global existence of weak solution without uniqueness. Meanwhile, the author in [39] found the finite time flocking phenomenon but only in two particles system. In [6,7], the authors proved the collision avoidance for C-S model in any finite time, but the uniform-in-time lower bound is still unknown. Moreover, in [26,22], the authors studied the C-S model with regular short range interaction and constructed a sufficient and necessary condition for the emergence of mono-cluster and multi-cluster formation, respectively. While, it is not clear wether the similar results can be obtained in the singular case. Therefore, may we address three natural questions as follows,• (Q1) Can we derive the uniqueness of the solution to the C-S model (1.1) with singular communication (1.2)? Moreover, can we find the sufficient and necessary condition for emergence of finite time flocking in N particle system?• (Q2) Can we construct the uniform-in-time lower bound between two particles of the C-S model with singular interaction, so that the collision avoidance occurs when time tends to infinity and asymptotical equilibrium state can be constructed?• (Q3) Can we obtain the sufficient and necessary condition for the emergence of mono-cluster and multi-cluster...

No abstract

Numerical simulations for continuous dynamical systems are based on their discretized models in time and space, and we take it for granted that the simulation results for the discrete dynamical systems as what is happening in the corresponding continuous dynamical systems. In general, this is valid in any finite time window under suitable conditions on system parameters. In this paper, we present a smooth transition from discrete dynamics to continuous dynamics for the ensemble of Kuramoto oscillators which is valid in the whole time interval by utilizing the exponential synchronization estimates and confining the initial phase in a compact set of the state space. We perform this uniform-in-time transition for the discrete Kuramoto model and its augmented second-order extension.