Travelling wave dynamics in a nonlinear interferometer with spatial field transformer in feedback

Grigorieva, Elena V.; Haken, Hermann; Kashchenko, S. A.; Pelster, Axel

doi:10.1016/s0167-2789(98)00196-1

Cited by 33 publications

(27 citation statements)

References 12 publications

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“…We find coefficients of expansions (11) and (12). To this end, we substitute expansions (11) and (12) in Eq.…”

Section: Existence and Stabilitymentioning

confidence: 90%

“…To this end, we substitute expansions (11) and (12) in Eq. (7) and equate coefficients multiplying z with identical powers.…”

Section: Existence and Stabilitymentioning

confidence: 99%

“…Methods of construction of periodic solutions for an arbitrary domain and a nondegenerate smooth transformation are developed in [8,9]. The method of quasinormal forms for a parabolic functional-differential equation with rotation for the case of small diffusion is used in [10,11] for the description of the dynamics of running waves and slowly varying structures. The optical bufferness of running waves is established in [12][13][14][15].…”

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confidence: 99%

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Dynamics of stationary structures in a parabolic problem with reflected spatial argument

Belan

2010

Cybern Syst Anal

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Properties of stationary structures in a nonlinear optical resonator with an inversion transformer in its two-dimensional feedback are investigated. This system is mathematically described by a scalar parabolic equation with an inversion transformation of its spatial argument and with Neumann conditions on a segment. The evolution of forms of stationary structures and their stability are investigated. It is proved that the number of stable stationary structures increases with lengthening the segment. The central manifold method and Galerkin method are used. INTRODUCTIONAt the present time, an extension of investigations in the field of nonlinear optics is caused by the intensive use of optical systems in information technologies (see [1][2][3][4] and bibliographies of these works).A system consisting of a thin layer of a Kerr-type nonlinear medium and a variously organized two-dimensional feedback loop is acknowledged to be among the most popular nonlinear optical systems. A basic distinctive feature of such systems is that an external feedback loop can be used for the direct influence on the nonlinear dynamics of a system by means of controlled transformation of spatial variables that is performed by prisms, lenses, dynamic holograms, and other devices.Parabolic functional-differential equations with transforming the arguments of the sought-for function that are used for modelling optical systems with a two-dimensional feedback form a new class of equations for investigating the structure formation phenomenon. The observable autowave phenomena for the transformation of rotation by a fixed angle in a circle or a ring is mathematically substantiated in [5-7] on the basis of the Andronov-Hopf bifurcation theory. Methods of construction of periodic solutions for an arbitrary domain and a nondegenerate smooth transformation are developed in [8,9]. The method of quasinormal forms for a parabolic functional-differential equation with rotation for the case of small diffusion is used in [10,11] for the description of the dynamics of running waves and slowly varying structures. The optical bufferness of running waves is established in [12][13][14][15]. Following [12,13], recall that, in the phase space of some infinitely dimensional dynamic system, the bufferness phenomenon is realized if, by a suitable choice of parameters, it is possible to guarantee the existence of any fixed number of attractors of the same type in it. The method of central manifolds is used in [16,17] for the investigation of bifurcations of rotating structures in a ring and a circle for the case of rotation and also in a circle for rotation transformation together with radial compression. Based on [18], rotating structures are analyzed in [19,20] by the method of construction of approximate periodic solutions. A local bifurcation theory was used in [21,3] to construct stationary structures and analyze their stability for a parabolic equation on a segment with reflection transformation.A nonlinear interferometer with the mirror reflection of the fiel...

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“…We find coefficients of expansions (11) and (12). To this end, we substitute expansions (11) and (12) in Eq.…”

Section: Existence and Stabilitymentioning

confidence: 90%

“…To this end, we substitute expansions (11) and (12) in Eq. (7) and equate coefficients multiplying z with identical powers.…”

Section: Existence and Stabilitymentioning

confidence: 99%

mentioning

confidence: 99%

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Dynamics of stationary structures in a parabolic problem with reflected spatial argument

Belan

2010

Cybern Syst Anal

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“…It represents a technical procedure to deduce the normal form, once the bifurcation type is known, by using the knowledge of how the respective quantities depend on the smallness parameter ε = (R − R c )/R c . Although the multiple scale method was originally developed for ordinary differential equations [24,25,26], it can be also applied to delay differential equations (see, for instance, the treatment in [27]). …”

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confidence: 99%

Synergetic System Analysis for the Delay-Induced Hopf Bifurcation in the Wright Equation

Schanz¹,

Pelster²

2003

SIAM J. Appl. Dyn. Syst.

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“…Исследованию динамических свойств уравнений с пространственно-распреде-ленными параметрами посвящено значительное число работ (см., например, [1,2,3,4,5]). В работе [4] изучались динамические свойства пространственно-распределен-ного комплексного уравнения, которое будем называть уравнением Курамото с про-странственно-распределенным управлением …”

Section: Introductionunclassified

Dynamics of the Kuramoto equation with spatially distributed control

Kashchenko

2016

Communications in Nonlinear Science and Numerical Simulation

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Travelling wave dynamics in a nonlinear interferometer with spatial field transformer in feedback

Cited by 33 publications

References 12 publications

Dynamics of stationary structures in a parabolic problem with reflected spatial argument

Dynamics of stationary structures in a parabolic problem with reflected spatial argument

Synergetic System Analysis for the Delay-Induced Hopf Bifurcation in the Wright Equation

Dynamics of the Kuramoto equation with spatially distributed control

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