Rotating structures in a parabolic functional-differential equation

2010

Cybern Syst Anal

Self Cite

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...

“…The existence of a central manifold for Eq. (7) in the vicinity of ( , ) l R H 1 1 0 Î´is proved as in [16,17] …”

Section: Existence and Stabilitymentioning

confidence: 75%

Section: Introductionmentioning

confidence: 99%

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

2010

Cybern Syst Anal

Self Cite

“…Following [17][18][19][20][21], we indicate conditions under which the existence of approximate periodic solutions yields the existence of a periodic solution. In this case, we obtain more general results than in [8].…”

Section: Introductionmentioning

confidence: 65%

“…Periodic bifurcation solutions for an arbitrary domain and a general nondegenerate smooth transformation were constructed in [5][6][7]. The method of central manifolds was used in [8] for the investigation of the bifurcation of the birth of rotating structures in the case where the transformation of a disk is the product of radial contraction and rotation by a constant angle. In the present paper, for the investigation of this case, we develop a method for the construction of approximate solutions periodic in t in which the single-frequency method [9][10][11] is used together with the formalism of construction of central manifolds for systems invariant with respect to the rotation group of a circle [12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Bifurcations of rotating structures in a parabolic functional differential equation

Lykova

2006

Nonlinear Oscill

Self Cite

We investigate the Andronov-Hopf bifurcation of the birth of a periodic solution from a space-homogeneous stationary solution of the Neumann problem on a disk for a parabolic equation with a transformation of space variables in the case where this transformation is the composition of a rotation by a constant angle and a radial contraction. Under general assumptions, we prove a theorem on the existence of a rotating structure, deduce conditions for its orbital stability, and construct its asymptotic form.

“…Methods of constructing periodic solutions for an arbitrary domain and nondegenerate smooth transformation were developed in [9][10][11]. The central manifold method was used in [12][13][14][15] to study the bifurcations of rotating structures in a ring and a circle for rotation, and in a circle for rotation together with radial contraction. As shown in [16], a method based on approximate periodic solutions [17] can be used to study bifurcation rotating structures.…”

mentioning

confidence: 99%

Optical buffering in stationary structures

2008

Cybern Syst Anal

Self Cite

517.9The local dynamics of a nonlinear parabolic equation on a circle with a shifted spatial argument and a small diffusion is studied. It is proved that the interaction of stationary structures satisfies the 1:3 principle. The number of stable stationary structures increases as the diffusion coefficient tends to zero. INTRODUCTIONRecently, there have been many studies on nonlinear optics, which is due to the wide use of optical systems in information technologies [1]. The key advantages of optical methods for data storage and conversion are parallel signal processing and high performance. Owing to their natural benefits, optical systems are used to create elements of content-addressable memory, pattern recognition systems, and learning analog computers (see [1][2][3], etc.).One of the most popular nonlinear optical systems is a system consisting of a thin layer of nonlinear Kerr-type medium and a two-dimensional feedback loop that can be organized in different ways [1,3]. The fundamental feature of such systems is that the external feedback loop can be used to directly influence the nonlinear dynamics of the system by using prisms, lenses, dynamic holograms, and other devices for controlled transformation of spatial variables. Experiments show that even elementary transformations (rotation, rotation with contraction, reflection) allow the self-organization of a light field to be implemented in a great variety of ways: multipetal rotational waves, optical spirals, stationary structures, switching waves, etc. (see [1,4] and references therein). Note that field stochastization (optical turbulence) was observed in the systems under certain conditions [5, 1].Parabolic functional differential equations with transformed arguments of the unknown function, used to model optical systems with two-dimensional feedback, are a new class of equations for studying the structurization phenomenon. Structurization in these equations has been studied in the mathematical literature since the 1990s. The autowave phenomena were substantiated, for a fixed-angle rotation in a circle or a ring, using the Andronov-Hopf bifurcation theory in [6][7][8], where multipetal rotational waves were described and analyzed for stability. Methods of constructing periodic solutions for an arbitrary domain and nondegenerate smooth transformation were developed in [9][10][11]. The central manifold method was used in [12][13][14][15] to study the bifurcations of rotating structures in a ring and a circle for rotation, and in a circle for rotation together with radial contraction. As shown in [16], a method based on approximate periodic solutions [17] can be used to study bifurcation rotating structures. The bifurcations of stationary structures were studied in [12,18].Parabolic equations with transformed argument and small diffusion are of interest for studying optical turbulence. The method of quasinormal forms for these equations with rotation was applied in [19,20] to describe the dynamics of traveling waves and slowly varying structures. The optical ...