Generalized Ginzburg-Landau equations, slaving principle and center manifold theorem

Wunderlin, A.; Haken, Hermann

doi:10.1007/bf01292661

Cited by 55 publications

(20 citation statements)

References 2 publications

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“…To reduce of the degree of freedom of the rate equations, we extend Haken's adiabatic elimination method by using the center manifold reduction theory [41,48,49].…”

Section: Discussionmentioning

confidence: 99%

Probing the Ginzburg-Landau Potential for Lasers Using Higher-order Photon Correlations

Takemura,

Takiguchi,

Notomi

2019

Preprint

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The lasing transition of "class-A" lasers, where the photon lifetime is much longer than the carrier lifetime, is analogous to the second-order phase transition and, in addition, there is the Ginzburg-Landau (GL) type theory. We theoretically examine the validity of the GL theory for lasers in terms of various parameters, particularly, the ratio between photon and carrier lifetimes. For this purpose, we propose a higher-order photon correlation (g (q) ) measurement to check whether or not the photon statistics are described by the GL theory. The measurement of g (q) has an important experimental advantage because of its quantum efficiency independence. We found that the applicability of the GL theory of lasers is broader than conventionally imagined. Namely, for low-β lasers, GL theory is applicable even when the photon lifetime is comparable to the carrier lifetime. We show that our finding can be understand in the framework of center manifold reduction, which is an example of the "slaving principle" proposed by Haken.

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“…To reduce of the degree of freedom of the rate equations, we extend Haken's adiabatic elimination method by using the center manifold reduction theory [41,48,49].…”

Section: Discussionmentioning

confidence: 99%

Probing the Ginzburg-Landau Potential for Lasers Using Higher-order Photon Correlations

Takemura,

Takiguchi,

Notomi

2019

Preprint

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“…The centre manifold theorem [28,29,111,247] has long been used as a major tool in the spirit of model-order reduction. Using the terminology introduced in [159,316], the long-term dynamics is driven by the central modes and reduction to the centre manifold allows an adiabatic elimination of the slave coordinates.…”

Section: Invariant Manifolds For Dynamical Systemsmentioning

confidence: 99%

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

2021

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This paper aims at reviewing nonlinear methods for model order reduction in structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear-based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappings and reduced dynamics. These developments are first recalled following a historical perspective, where the main applications were first oriented toward structural models that can be expressed thanks to partial differential equations. They are then replaced in the more general context of the parametrisation of invariant manifold that allows unifying the approaches. Then, the specific case of structures dis-C. Touzé (B)

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“…Equation (4.16) in the case of L =Li leads to ma(" lm;11m1, 12m2 ll m1 12m2 =(mi+m2)aim'. i, , i, , xi, , xi m (4.18) (2) and the coefficients aj. j,j have to vanish for (mi+mz)+m.…”

Section: Equationsmentioning

confidence: 99%

Static, wavelike, and chaotic thermal convection in spherical geometries

Friedrich

Haken

1986

Phys. Rev. A

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The onset of convection in a Auid contained between a hot inner and a cold outer spherical boundary in a radially symmetric gravity field is examined. Due to the spherical symmetry there are usually (21+1) unstable modes belonging to the Ith irreducible representation of the O(3) group.The actual value of l is determined by a properly defined aspect ratio. There are special values of this aspect ratio where two groups of modes simultaneously become unstable. The instability involving the two groups of modes with I =1 and 1=2 is considered by investigating the corresponding generalized Ginzburg-Landau equations. By varying the Rayleigh number and the aspect ratio a variety of interesting spatio-temporal behavior is observed. The occurring static, time periodic, and quasiperiodic fluid motions as well as the various types of chaotic Auid Aow are examined in detail.

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Generalized Ginzburg-Landau equations, slaving principle and center manifold theorem

Cited by 55 publications

References 2 publications

Probing the Ginzburg-Landau Potential for Lasers Using Higher-order Photon Correlations

Probing the Ginzburg-Landau Potential for Lasers Using Higher-order Photon Correlations

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

Static, wavelike, and chaotic thermal convection in spherical geometries

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