scholarly journals On the stability of the Kuramoto model of coupled nonlinear oscillators

2004 ◽  
Author(s):  
A. Jadbabaie ◽  
N. Motee ◽  
M. Barahona
Keyword(s):  
Nonlinear Oscillators ◽  
Kuramoto Model ◽  
Coupled Nonlinear Oscillators ◽  
The Stability ◽  
The Kuramoto Model

NONLINEAR DYNAMICS OF DIGITAL PHASE-LOCKED LOOPS WITH DELAY

1994 ◽  
Vol 04 (03) ◽  
pp. 715-726 ◽  
Author(s):  
MARIA DE SOUSA VIEIRA ◽  
ALLAN J. LICHTENBERG ◽  
MICHAEL A. LIEBERMAN
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic Behavior ◽  
Nonlinear Oscillators ◽  
Coupled Systems ◽  
Phase Locked Loops ◽  
Bifurcation Diagrams ◽  
Digital Phase ◽  
Coupled Nonlinear Oscillators ◽  
Digital Phase Locked Loops ◽  
The Stability

We investigate numerically and analytically the nonlinear dynamics of a system consisting of two self-synchronizing pulse-coupled nonlinear oscillators with delay. The particular system considered consists of connected digital phase-locked loops. We find mapping equations that govern the system and determine the synchronization properties. We study the bifurcation diagrams, which show regions of periodic, quasiperiodic and chaotic behavior, with unusual bifurcation diagrams, depending on the delay. We show that depending on the parameter that is varied, the delay will have a synchronizing or desynchronizing effect on the locked state. The stability of the system is studied by determining the Liapunov exponents, indicating marked differences compared to coupled systems without delay.

scholarly journals Distribution of Order Parameter for Kuramoto Model

2015 ◽  
Vol 3 (9) ◽  
pp. 52-68
Author(s):  
Hui Wu ◽  
Dongwook Kim
Keyword(s):  
Order Parameter ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Kuramoto Model ◽  
Chemical System ◽  
Phase Oscillators ◽  
Numerical Result ◽  
Coupled Nonlinear Oscillators ◽  
Large Populations ◽  
The Kuramoto Model

The synchronization in large populations of interacting oscillators has been observed abundantly in nature, emergining in fields such as physical, biological and chemical system. For this reason, many scientists are seeking to understand the underlying mechansim of the generation of synchronous patterns in oscillatory system. The synchronization is analyzed in one of the most representative models of coupled phase oscillators, the Kuramoto model. The Kuramoto model can be used to understand the emergence of synchronization in nextworks of coupled, nonlinear oscillators. In particular, this model presents a phase transition from incoherence to synchronization. In this paper, we investigated the distribution of order parameter γ which describes the strength of synchrony of these oscillators. The larger the order parameter γ is, the more extent the oscillators are synchronized together. This order parameter γ is a critical parameter in the Kuramoto model. Kuramoto gave a initial estimate equation for the value of the order parameter by giving the value of the coupling constant. But our numerical results show that the distribution of the order parameter is slightly different from Kuramoto’s estimation. We gave an estimation for the distribution of order parameter for different values of initial conditions. We discussed how the numerical result will be distributed around Kuramoto’s analytical equation.

From centralized to decentralized power system: A space-analysis for France

2020 ◽  
Vol 64 (1-4) ◽  
pp. 73-78
Author(s):  
Raphaël Cluet ◽  
Nadia Maïzi ◽  
Vincent Mazauric
Keyword(s):  
Kinetic Energy ◽  
Power System ◽  
Kuramoto Model ◽  
Energy Planning ◽  
System A ◽  
System Energy ◽  
The Stability ◽  
The Kuramoto Model ◽  
Grid Based

This paper reviews the conditions for ensuring the space consolidation and time reconciliation of all scales involved in power system energy planning. At the upper space scale, a simplified description of the grid based on the Kuramoto model is adopted to assess the stability of the synchronism state of the power system. The consequences of a massive dissemination of renewables for the future of the power system is addressed in the case of France in a long-term planning exercise up to 2050. Attention is paid to how the new capacities are dispatched among the administrative regions. While a favorable impact of sparse generation on the synchronism is obtained, the constraint on the inertia provided by the kinetic energy has to be endogenized to enforce an operable power mix under admissible disturbances.

The Origin of Stability Indeterminacy in a Symmetric Hamiltonian

1984 ◽  
Vol 51 (2) ◽  
pp. 399-405 ◽  
Author(s):  
M. R. Hyams ◽  
L. A. Month
Keyword(s):  
Perturbation Theory ◽  
Stability Analysis ◽  
Hamiltonian System ◽  
Phase Plane ◽  
Nonlinear Oscillators ◽  
Two Dimensional ◽  
Periodic Motions ◽  
Coupled Nonlinear Oscillators ◽  
The Stability ◽  
Two Degree Of Freedom

The stability and bifurcation of periodic motions in a symmetric two-degree-of-freedom Hamiltonian system is studied by a reduction to a two-dimensional action-angle phase plane, via canonical perturbation theory. The results are used to explain why linear stability analysis will always be indeterminate for the in-phase mode in a class of coupled nonlinear oscillators.

On the Stability Behavor of Bifurcated Normal Modes in Coupled Nonlinear Systems

1989 ◽  
Vol 56 (1) ◽  
pp. 155-161 ◽  
Author(s):  
C. H. Pak
Keyword(s):  
Nonlinear Systems ◽  
Calculus Of Variations ◽  
Normal Modes ◽  
Nonlinear Oscillators ◽  
Coupled Nonlinear Oscillators ◽  
Floquet’S Theory ◽  
Generic Bifurcation ◽  
The Stability ◽  
Floquet's Theory

The stability of bifurcated normal modes in coupled nonlinear oscillators is investigated, based on Synge’s stability in the kinematico-statical sense, utilizing the calculus of variations and Floquet’s theory. It is found, in general, that in a generic bifurcation, the stabilities of two bifurcated modes are opposite, and in a nongeneric bifurcation, the stability of continuing modes is opposite to that of the existing mode, and the stabilities of the two bifurcated modes are equal but opposite to that of the continuing mode. Some examples are illustrated.

scholarly journals Emulating the local Kuramoto model with an injection-locked photonic crystal laser array

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Naotomo Takemura ◽  
Kenta Takata ◽  
Masato Takiguchi ◽  
Masaya Notomi
Keyword(s):  
Photonic Crystal ◽  
Nearest Neighbor ◽  
Multiple Quantum Well ◽  
Kuramoto Model ◽  
Multiple Quantum ◽  
Laser Array ◽  
Novel Strategy ◽  
A Chain ◽  
The One ◽  
The Kuramoto Model

AbstractThe Kuramoto model is a mathematical model for describing the collective synchronization phenomena of coupled oscillators. We theoretically demonstrate that an array of coupled photonic crystal lasers emulates the Kuramoto model with non-delayed nearest-neighbor coupling (the local Kuramoto model). Our novel strategy employs indirect coupling between lasers via additional cold cavities. By installing cold cavities between laser cavities, we avoid the strong coupling of lasers and realize ideal mutual injection-locking with effective non-delayed dissipative coupling. First, after discussing the limit cycle interpretation of laser oscillation, we demonstrate the synchronization of two indirectly coupled lasers by numerically simulating coupled-mode equations. Second, by performing a phase reduction analysis, we show that laser dynamics in the proposed device can be mapped to the local Kuramoto model. Finally, we briefly demonstrate that a chain of indirectly coupled photonic crystal lasers actually emulates the one-dimensional local Kuramoto chain. We also argue that our proposed structure, which consists of periodically aligned cold cavities and laser cavities, will best be realized by using state-of-the-art buried multiple quantum well photonic crystals.

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