Uniform-in-time transition from discrete to continuous dynamics in the Kuramoto synchronization

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.

On the generic complete synchronization of the discrete Kuramoto model

“…which is analogous to the result in [16]. Then, Ha et al [17] verified the uniformin-time convergence of (2) to (1) during h → 0 and exponential synchronization of (discrete-time) identical oscillators, whose corresponding result was also provided in [8,16,19]. After [20] studied a synchronization and phase-locked state of the Kuramoto model for more generic setting, Zhang and Zhu established a corresponding stability theory in [38] for discrete gradient flow to show that (2) exhibits an asymptotic phase-locking for sufficiently small ν κ and h (see Definition 2.3 for the definition of phase-locked state).…”

Emergent behaviors of discrete Lohe aggregation flows

“…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].…”

Emergent behaviors of discrete Lohe aggregation flows

“…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-intime 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, [3,12,13,18,20,25,27], 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].…”