Dongnam Ko

Emergence of bi-cluster flocking for the Cucker–Smale model

We present the mathematical analysis of bi-cluster flocking phenomenon for the short-ranged Cucker–Smale model with some well-prepared initial data. For this, we derive a system of differential inequalities for the functionals measuring the local spatial and velocity fluctuations and differences of local velocity averages, and then estimate the upper bound of spatial fluctuations and the lower bound of the difference between local velocity averages. We explicitly present an admissible class of initial configurations leading to the asymptotic emergence of bi-cluster flocking phenomenon. Unlike global flocking (a mono-cluster flocking configuration in velocity), where the convergence rate is always exponential, the asymptotic convergence to bi-cluster flocking configurations is affected by the far-field decay rate of communication weights so that it can be algebraic.

Emergence of bi-cluster flocking for agent-based models with unit speed constraint

We study the emergence of bi-cluster flocking for a generalized Justh–Krishnaprasad (J–K) model and the Cucker–Smale (C–S) model with the unit speed constraint. We provide a sufficient framework for the emergence of bi-cluster flocking in terms of initial configurations and communication weights between agents. Our analytical results on bi-cluster flocking illustrate that the spatial-decay rate of communication weights is transferred to the temporal decay rate toward the bi-cluster flocking configuration. We also provide several numerical simulations and compare them to our analytical results.

Critical coupling strength of the Cucker–Smale model for flocking

We present a non-trivial positive lower bound for the critical coupling strength of the Cucker–Smale model with a short-range communication weight from the viewpoint of mono-cluster (global) flocking. For a long-range communication weight, it is well known [F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Control 52 (2007) 852–862; S.-Y. Ha and J.-G. Liu, A proof of Cucker–Smale flocking dynamics and mean field limit, Commun. Math. Sci. 7 (2009) 297–325] that as long as the coupling strength is positive, mono-cluster flocking occurs asymptotically for any initial configuration. Hence, the critical coupling strength is simply zero. However, for a short-range communication weight, numerical simulations indicate that for a given initial configuration, mono-cluster flocking is possible only in a large-coupling-strength regime depending on the initial configuration. This suggests the positivity of the critical coupling strength in the sense that if the coupling strength is above the cr...

Emergence of Phase-Locking in the Kuramoto Model for Identical Oscillators with Frustration

We present an exponential synchronization for the finite-dimensional Kuramoto model for identical oscillators under the effect of frustration. In the presence of frustration, the total phase balance law and gradient-flow structure are destroyed. Hence the gradient flow approach cannot be applied for the emergence of phase-locking, and the Lyapunov functional approach based on

Time-asymptotic interactions of two ensembles of Cucker-Smale flocking particles

\ell^2

Collective synchronization of classical and quantum oscillators

-energy and phase diameter only show some results in a restricted set of initial configurations smaller than a half circle. In this paper, we show that an initial phase configuration whose order parameter is bounded below by some positive value, which is only depending on the size of frustration, leads to the complete phase synchronization or the bipolar state exponentially fast. This extends earlier results on the asymptotic phase-locking for the identical oscillators in which initial phase configurations are confined in a half circle.

Emergent Dynamics of a Generalized Lohe Model on Some Class of Lie Groups

We study the time-asymptotic interactions of two ensembles of Cucker-Smale flocking particles. For this, we use a coupled hydrodynamic Cucker-Smale system and discuss two frameworks, leading to mono-cluster and bi-cluster flockings asymptotically depending on initial configurations, coupling strengths, and the far-field decay property of communication weights. Under the proposed two frameworks, we show that mono-cluster and bi-cluster flockings emerge asymptotically exponentially fast and algebraically slow, respectively. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system.We study the time-asymptotic interactions of two ensembles of Cucker-Smale flocking particles. For this, we use a coupled hydrodynamic Cucker-Smale system and discuss two frameworks, leading to mono-cluster and bi-cluster flockings asymptotically depending on initial configurations, coupling strengths, and the far-field decay property of communication weights. Under the proposed two frameworks, we show that mono-cluster and bi-cluster flockings emerge asymptotically exponentially fast and algebraically slow, respectively. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system.