Geodesic
Stability and local bifurcations of single-mode equilibrium states of the Ginzburg--Landau variational equation
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 33 (2023) no. 2, pp. 240-258

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One of the versions of the generalized variational Ginzburg-Landau equation is considered, supplemented by periodic boundary conditions. For such a boundary value problem, the question of existence, stability, and local bifurcations of single-mode equilibrium states is studied. It is shown that in the case of a nearly critical threefold zero eigenvalue, in the problem of stability of single-mode spatially inhomogeneous equilibrium states, subcritical bifurcations of two-dimensional invariant tori filled with spatially inhomogeneous equilibrium states are realized. The analysis of the stated problem is based on such methods of the theory of infinite-dimensional dynamical systems as the theory of invariant manifolds and the apparatus of normal forms. Asymptotic formulas are obtained for the solutions that form invariant tori.
Keywords: Ginzburg–Landau variational equation, boundary value problem, stability, asymptotic formulas.
Mots-clés : bifurcations 
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     author = {D. A. Kulikov},
     title = {Stability and local bifurcations of single-mode equilibrium states of the {Ginzburg--Landau} variational equation},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {240--258},
     publisher = {mathdoc},
     volume = {33},
     number = {2},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2023_33_2_a3/}
}
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D. A. Kulikov. Stability and local bifurcations of single-mode equilibrium states of the Ginzburg--Landau variational equation. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 33 (2023) no. 2, pp. 240-258. http://geodesic.mathdoc.fr/item/VUU_2023_33_2_a3/