Zhuchun Li

Emergent Dynamics of the Cucker–Smale Flocking Model and Its Variants

In this chapter, we present the Cucker–Smale-type flocking models and discuss their mathematical structures and flocking theorems in terms of coupling strength, interaction topologies, and initial data. In 2007, two mathematicians Felipe Cucker and Steve Smale introduced a second-order particle model which resembles Newton’s equations in N-body system and present how their simple model can exhibit emergent flocking behavior under sufficient conditions expressed only in terms of parameters and initial data. After Cucker–Smale’s seminal works in [31, 32], their model has received lots of attention from applied math and control engineering communities. We discuss the state of the art for the flocking theorems to Cucker–Smale-type flocking models.

Emergent phenomena in an ensemble of Cucker–Smale particles under joint rooted leadership

We present an emergent flocking estimate in a group of interacting Cucker–Smale particles under the joint rooted leadership via the discrete-time Cucker–Smale model. It is well known that the network topology regulates the emergence of flocking, and the rooted leadership topology is the most general topology with a leader–follower structure. When the network topology satisfies a weaker rooted leadership, in which the union of neighbor graphs on infinite time-blocks is under rooted leadership, we show that the asymptotic flocking can be achieved from some class of initial configurations by generalizing the earlier result of Li and Xue using the (sp) matrix theory in Ref. 36.

Large-Time Dynamics of Kuramoto Oscillators under the Effects of Inertia and Frustration

We study the intricate interplay between inertial effect and interaction frustration in an ensemble of Kuramoto oscillators. In particular, we discuss how asymptotic synchronization can arise from the competition between synchronization factors such as strong coupling strength and desynchronization factors such as inertia and frustration. We provide several frameworks in terms of system parameters and initial configurations that guarantee the emergence of complete synchronization. For a restricted class of initial configurations, we show that asymptotic complete synchronization occurs exponentially fast and its exponential decay rate depends on the strength of system parameters such as coupling, inertia, and frustration. We also provide several numerical simulations and compare these with analytical results.

Flocking behavior of the Cucker-Smale model under rooted leadership in a large coupling limit

We present an asymptotic flocking estimate for the Cucker-Smale flocking model under the rooted leadership in a large coupling limit. For this, we reformulate the Cucker-Smale model into a fast-slow dynamical system involving a small parameter which corresponds to the inverse of a coupling strength. When the coupling strength tends to infinity, the spatial configuration will be frozen instantaneously, whereas the velocity configuration shrinks to the global leader’s velocity immediately. For the rigorous explanation of this phenomenon, we use Tikhonov’s singular perturbation theory. We also present several numerical simulations to confirm our analytical theory.

Asymptotic synchronous behavior of Kuramoto type models with frustrations

We present a quantitative asymptotic behavior of coupled Kuramoto oscillators with frustrations and give some sufficient conditions for the parameters and initial condition leading to phase or frequency synchronization. We consider three Kuramoto-type models with frustrations. First, we study a general case with nonidentical oscillators; i.e., the natural frequencies are distributed. Second, as a special case, we study an ensemble of two groups of identical oscillators. For these mixture of two identical Kuramoto oscillator groups, we study the relaxation dynamics from the mixed stage to the phase-locked states via the segregation stage. Finally, we consider a Kuramoto-type model that was recently derived from the Van der Pol equations for two coupled oscillator systems in the work of Luck and Pikovsky [27]. In this case, we provide a framework in which the phase synchronization of each group is attained. Moreover, the constant frustration causes the two groups to segregate from each other, although they have the same natural frequency. We also provide several numerical simulations to confirm our analytical results.

Uniqueness and well-ordering of emergent phase-locked states for the Kuramoto model with frustration and inertia

We discuss the uniqueness and well-ordering property of phase-locked states emerged from some admissible class of initial configurations for the Kuramoto model under the effect of frustration and inertia. Our results rely on the nonlinear stability and structure of phase-locked states for the Kuramoto model. When the coupling strength is sufficiently large and the diameter of initial phase configuration is sufficiently small, we show that the emergent phase configurations are stable in l∞-norm with respect to initial configurations and they tend to the unique collision-free phase-locked state up to rotation. Moreover, we verify that the geometric shape of the emergent phase-locked state is invariant under the effect of inertia. We provide several numerical examples and compare them with our analytical results.

Emergent behavior of Cucker–Smale flocking particles with heterogeneous time delays

Abstract We analyze the flocking behavior of a Cucker–Smale-type system with a delayed coupling, where delays are information processing and reactions of individuals. By constructing a system of dissipative differential inequalities together with a continuity argument, we provide a sufficient condition for the flocking behavior showing the velocity alignment between particles as time goes to infinity when the maximum value of time delays is sufficiently small.

Emergent Dynamics of Kuramoto Oscillators with Adaptive Couplings: Conservation Law and Fast Learning

We study an emergent dynamics of the Kuramoto oscillators with adaptive couplings. In the Kuramoto model, pairwise coupling strengths are assumed to be constant and uniform over all interaction pai...