scholarly journals A proof of the Kuramoto conjecture for a bifurcation structure of the infinite-dimensional Kuramoto model

2013 ◽  
Vol 35 (3) ◽  
pp. 762-834 ◽  
Author(s):  
HAYATO CHIBA
Keyword(s):  
Hilbert Space ◽  
Steady State ◽  
Linear Operator ◽  
Continuous Spectrum ◽  
Imaginary Axis ◽  
Center Manifold ◽  
Generalized Functions ◽  
Kuramoto Model ◽  
Space Of Generalized Functions ◽  
The Kuramoto Model

AbstractThe Kuramoto model is a system of ordinary differential equations for describing synchronization phenomena defined as coupled phase oscillators. In this paper, a bifurcation structure of the infinite-dimensional Kuramoto model is investigated. A purpose here is to prove the bifurcation diagram of the model conjectured by Kuramoto in 1984; if the coupling strength $K$ between oscillators, which is a parameter of the system, is smaller than some threshold ${K}_{c} $, the de-synchronous state (trivial steady state) is asymptotically stable, while if $K$ exceeds ${K}_{c} $, a non-trivial stable solution, which corresponds to the synchronization, bifurcates from the de-synchronous state. One of the difficulties in proving the conjecture is that a certain non-selfadjoint linear operator, which defines a linear part of the Kuramoto model, has the continuous spectrum on the imaginary axis. Hence, the standard spectral theory is not applicable to prove a bifurcation as well as the asymptotic stability of the steady state. In this paper, the spectral theory on a space of generalized functions is developed with the aid of a rigged Hilbert space to avoid the continuous spectrum on the imaginary axis. Although the linear operator has an unbounded continuous spectrum on a Hilbert space, it is shown that it admits a spectral decomposition consisting of a countable number of eigenfunctions on a space of generalized functions. The semigroup generated by the linear operator will be estimated with the aid of the spectral theory on a rigged Hilbert space to prove the linear stability of the steady state of the system. The center manifold theory is also developed on a space of generalized functions. It is proved that there exists a finite-dimensional center manifold on a space of generalized functions, while a center manifold on a Hilbert space is of infinite dimension because of the continuous spectrum on the imaginary axis. These results are applied to the stability and bifurcation theory of the Kuramoto model to obtain a bifurcation diagram conjectured by Kuramoto.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

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.

scholarly journals Emergent synchronous beating behavior in spontaneous beating cardiomyocyte clusters

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Kazufumi Sakamoto ◽  
Yoshitsune Hondo ◽  
Naoki Takahashi ◽  
Yuhei Tanaka ◽  
Rikuto Sekine ◽  
...  
Keyword(s):  
Stem Cell ◽  
Embryonic Stem Cell ◽  
Human Embryonic Stem Cell ◽  
Single Cells ◽  
Embryonic Stem ◽  
Kuramoto Model ◽  
Distribution Analysis ◽  
Cell Clusters ◽  
Overdrive Suppression ◽  
The Kuramoto Model

AbstractWe investigated the dominant rule determining synchronization of beating intervals of cardiomyocytes after the clustering of mouse primary and human embryonic-stem-cell (hES)-derived cardiomyocytes. Cardiomyocyte clusters were formed in concave agarose cultivation chambers and their beating intervals were compared with those of dispersed isolated single cells. Distribution analysis revealed that the clusters’ synchronized interbeat intervals (IBIs) were longer than the majority of those of isolated single cells, which is against the conventional faster firing regulation or “overdrive suppression.” IBI distribution of the isolated individual cardiomyocytes acquired from the beating clusters also confirmed that the clusters’ IBI was longer than those of the majority of constituent cardiomyocytes. In the complementary experiment in which cell clusters were connected together and then separated again, two cardiomyocyte clusters having different IBIs were attached and synchronized to the longer IBIs than those of the two clusters’ original IBIs, and recovered to shorter IBIs after their separation. This is not only against overdrive suppression but also mathematical synchronization models, such as the Kuramoto model, in which synchronized beating becomes intermediate between the two clusters’ IBIs. These results suggest that emergent slower synchronous beating occurred in homogeneous cardiomyocyte clusters as a community effect of spontaneously beating cells.

Formally Normal Operators Having no Normal Extensions

1965 ◽  
Vol 17 ◽  
pp. 1030-1040 ◽  
Author(s):  
Earl A. Coddington
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Null Space ◽  
Normal Operator ◽  
Closed Linear Operator ◽  
Normal Operators ◽  
The Given ◽  
Densely Defined ◽  
Made In

The domain and null space of an operator A in a Hilbert space will be denoted by and , respectively. A formally normal operatorN in is a densely defined closed (linear) operator such that , and for all A normal operator in is a formally normal operator N satisfying 35 . A study of the possibility of extending a formally normal operator N to a normal operator in the given , or in a larger Hilbert space, was made in (1).

scholarly journals Investigation of Preferred Orientations in Planar Polycrystals

2008 ◽  
Vol 75 (5) ◽  
Author(s):  
M. R. Tonks ◽  
A. J. Beaudoin ◽  
F. Schilder ◽  
D. A. Tortorelli
Keyword(s):  
Texture Evolution ◽  
Mean Field ◽  
Homogenization Method ◽  
Process Models ◽  
Kuramoto Model ◽  
Taylor Model ◽  
Crystal Plasticity Finite Element ◽  
Polycrystal Plasticity ◽  
Taylor Hypothesis ◽  
The Kuramoto Model

More accurate manufacturing process models come from better understanding of texture evolution and preferred orientations. We investigate the texture evolution in the simplified physical framework of a planar polycrystal with two slip systems used by Prantil et al. (1993, “An Analysis of Texture and Plastic Spin for Planar Polycrystal,” J. Mech. Phys. Solids, 41(8), pp. 1357–1382). In the planar polycrystal, the crystal orientations behave in a manner similar to that of a system of coupled oscillators represented by the Kuramoto model. The crystal plasticity finite element method and the stochastic Taylor model (STM), a stochastic method for mean-field polycrystal plasticity, predict the development of a steady-state texture not shown when employing the Taylor hypothesis. From this analysis, the STM appears to be a useful homogenization method when using representative standard deviations.

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