On the generic complete synchronization of the discrete Kuramoto model

Abstract

We study the emergent behavior of discrete-time approximation of the finite-dimensional Kuramoto model. Compared to Zhang and Zhu's recent work in [38], we do not rely on the consistency of one-step foward Euler scheme but analyze the discrete model directly to obtain sharper and more explicit result. More precisely, we present the optimal condition for the convergence and order preserving for identical oscilators with generic initial data. Then, we give the exact convergence rate of the identical oscillators to their limit under the reasonable assuption on time step. Finally, we provide an alternative proof of the asymptotic phase-locking of nonidentical oscillators which can be applied whenever the given Lyapunov functional is continuous and all zeros are isolated.

Emergent behaviors of discrete Lohe aggregation flows

“…(3.9) Note that for |x| 1, one has arctan(x) ≈ x. So (3.9) 2 becomes the discrete Kuramoto model [11,45,55]:…”

“…Emergent dynamics of (3.10) and uniform-in-time transition to the continuous dynamics have been discussed in recent literature, e.g., exponential synchronization [11] for some restricted initial configuration, complete synchronization [45,55] for a generic initial configuration and uniform-in-time transition from discrete dynamics to continuous dynamics [21].…”

“…The emergent dynamics of the Kuramoto model (1.1) on the unit circle has been extensively studied in literature, to name a few, [3,12,13,18,20,23,27], and its first-order discretized model for (1.1) based on the forward first-order Euler method was also addressed in [11,21,45,55] from the viewpoint of emergent dynamics. As high-dimensional generalizations of the Kuramoto model, several first-order models have been proposed on specific manifolds, to name a few, the Lohe sphere model on d-sphere S d [9,10,22,38,33,39,40,42,48,56], the Lohe matrix model [6,14,15,16,25,30,37] on the unitary group and the Lohe tensor model on the space of tensors with the same rank and size [28,29].…”

Collective behaviors of second-order nonlinear consensus models with a bonding force

“…The emergent dynamics of the second-order model (1.1) has been studied in [27] for a class of restricted initial data, whereas the emergent dynamics of the first-order model (1.2) has been extensively studied from diverse perspectives, e.g., complete synchronization [6,11,13,18,25], critical coupling strength [19], uniform mean field limit [23,33], gradient flow formulation [43], discretized model [30,39], kinetic Kuramoto model [4,5,8], etc. Next, we consider a finite ensemble of Cucker-Smale flocking particles whose mechanical states are given by position and velocity.…”

Emergent behaviors of discrete Lohe aggregation flows

“…The emergent dynamics of the Kuramoto model (1.1) on the unit circle has been extensively studied in literature, to name a few, [3,12,13,18,20,25,27], and its first-order discretized model for (1.1) based on the forward first-order Euler method was also addressed in [11,21,45,55] from the viewpoint of emergent dynamics. As high-dimensional generalizations of the Kuramoto model, several first-order models have been proposed on some manifolds, to name a few, the swarm sphere model on d-sphere S d [9,10,22,38,33,39,41,40,48,56], the Lohe matrix model [6,14,15,16,25,30,37] on the unitary group and the Lohe tensor model on the space of tensors with the same rank and size [28,29].…”