Rotating waves in parabolic functional differential equations with rotation of spatial argument and time delay

Razgulin, A. V.; Romanenko, T. E.

doi:10.1134/s0965542513110109

Cited by 18 publications

(5 citation statements)

References 33 publications

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“…Based on these theories, there have been many subsequent studies on symmetry. Firstly, some researchers were concerned about nonlinear optical systems, which can effectively characterize optical problems such as circular diffraction [20][21][22]. Besides, a Hopfield-Cohen-Grossberg network consisting of n identical elements also has a certain symmetry, which has been studied in [23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation Analysis in a Class of Scalar Functional Differential Equations

Stech¹

2020

Physical Mathematics and Nonlinear Partial Differential Equations

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Circular domains frequently appear in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations around equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction-diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.

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Section: Introductionmentioning

confidence: 99%

Hopf Bifurcation Analysis in a Class of Scalar Functional Differential Equations

Stech¹

2020

Physical Mathematics and Nonlinear Partial Differential Equations

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“…However, it is necessary to remember that feedback circuit implementation by devices with the frame transfer architecture can cause a significant (compared to the nonlinearity relaxation time τ) transport delay. This effect is not considered in the investigated model, although the presence of the transport delay (latency) in such systems may cause the appearance of space-time instabilities [20][21][22][23].…”

Section: Discussionmentioning

confidence: 99%

Phase distortion suppression in a nonlinear optical system with integral feedback

Larichev¹,

Nikolaev²,

Pavlov³

et al. 2015

Laser Phys.

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We consider a mathematical model of an optical system where the phase modulation introduced by a nonlinear optical element involves a phase shift in an additional time-integral feedback loop, which is determined by the gain coefficient G. In the case of = G 0, this model describes the dynamics of a nonlinear interferometer, where the effect of phase distortion compensation has been numerically and experimentally investigated before. In this paper, the results of 2D numerical simulations are presented for the case > G 0. It is shown that different types of small-amplitude stationary phase inhomogeneities can be fully compensated with time. The effect of the partial suppression of dynamic distortions given in the form of traveling waves and Kolmogorov-spectrum atmospheric irregularities is demonstrated; the quality of the distortion compensation is analyzed, depending on the spatiotemporal characteristics of the waves. The obtained results confirm the potential of using the optical feedback configuration with an additional integral circuit for suppression of both static and dynamic phase distortions.

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“…Among the natural physical phenomena that can be taken into account in the mathematical model are interference of the input and feedback light fields [6], and free propagation diffraction in the feedback loop [5]. In the most general case, ð1.1Þ is a delayed nonlinear partial functional differential equation [6]. The magnitude of nonlocalities together with the input light field intensity form an effective toolkit for controlling the dynamics of the system, which is crucial for applications (see [7,8]).…”

Section: Introductionmentioning

confidence: 99%

“…Depending on the configuration of the feedback loop, expression for the complex amplitude A FB can bring nonlocal interactions into ð1.1Þ: time delay and/or spatial nonlocality (see [3,4]). Among the natural physical phenomena that can be taken into account in the mathematical model are interference of the input and feedback light fields [6], and free propagation diffraction in the feedback loop [5]. In the most general case, ð1.1Þ is a delayed nonlinear partial functional differential equation [6].…”

Section: Introductionmentioning

confidence: 99%

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Rotating Waves in a Model of Delayed Feedback Optical System with Diffraction

Budzinskiy

2016

IIS

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We study a delayed parabolic functional differential equation on a circle that is coupled with an initial value problem for the Schrodinger equation. Such equations arise as models of nonlinear optical systems with a timedelayed feedback loop, when diffusion of molecular excitation and diffraction are taken into account. The goal of this paper is to prove the existence of spatially inhomogeneous rotating-wave solutions bifurcating from homogeneous equilibria. We pass to a rotating coordinate system and seek an inhomogeneous solution to an ordinary functional differential equation. We find the solution in the form of a small parameter expansion and explicitly compute the first-order coefficients. We also provide examples of parameters that satisfy the constraints imposed throughout the analysis.

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Rotating waves in parabolic functional differential equations with rotation of spatial argument and time delay

Cited by 18 publications

References 33 publications

Hopf Bifurcation Analysis in a Class of Scalar Functional Differential Equations

Hopf Bifurcation Analysis in a Class of Scalar Functional Differential Equations

Phase distortion suppression in a nonlinear optical system with integral feedback

Rotating Waves in a Model of Delayed Feedback Optical System with Diffraction

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