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Bifurcations of Self-Oscillatory Solutions to a Nonlinear Parabolic Equation with a Rotating Spatial Argument and Time Delay

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Abstract

For a problem arising in nonlinear optics, namely, for an initial-boundary value problem in a disk for a nonlinear parabolic equation with time delay and rotation of spatial argument by a given angle, bifurcations of self-oscillatory solutions from homogeneous equilibrium states are studied. In the plane of basic parameters of the equation, domains of stability (instability) of homogeneous equilibrium states are constructed, and the dynamics of the stability domains is analyzed depending on the delay value. The mechanisms of stability loss of homogeneous equilibrium states are investigated, possible bifurcations of spatially inhomogeneous self-oscillatory solutions and their stability are analyzed, and the dynamics of such solutions near the boundary of a stability domain in the plane of basic parameters of the equation is studied.

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Funding

This work was supported by the Russian Foundation for Basic Research, project no. 19-31-90133.

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  1. Faculty of Mathematics, Yaroslavl State University, 150000, Yaroslavl, Russia

    E. P. Kubyshkin & V. A. Kulikov

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  1. E. P. Kubyshkin
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  2. V. A. Kulikov
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Correspondence to E. P. Kubyshkin or V. A. Kulikov.

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Translated by I. Ruzanova

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Kubyshkin, E.P., Kulikov, V.A. Bifurcations of Self-Oscillatory Solutions to a Nonlinear Parabolic Equation with a Rotating Spatial Argument and Time Delay. Comput. Math. and Math. Phys. 61, 403–423 (2021). https://doi.org/10.1134/S0965542521030118

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