Geodesic
Bifurcations of invariant tori in second-order quasilinear evolution equations in Hilbert spaces and scenarios of transition to turbulence
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 1, Tome 168 (2019), pp. 45-52.

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In this paper, we consider second-order quasilinear differential equations in a separable Hilbert space for which the well-known Landau–Hopf scenario of transition to turbulence can be realized. We prove increasing of the control parameter leads to the consequtive appearance of invariant tori of increasing dimensions. In this case, the invariant torus of the largest possible dimension appears to be attractive. The results are obtained by using methods of the qualitative theory of dynamical systems with an infinite-dimensional space of initial conditions: the method of integral manifolds, the theory of normal forms, and also asymptotic methods of analysis of dynamical systems.
Keywords: quasilinear differential equation, Hilbert space, stability, normal form, Landau–Hopf scenario.
Mots-clés : bifurcation 
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     author = {A. N. Kulikov},
     title = {Bifurcations of invariant tori in second-order quasilinear evolution equations in {Hilbert} spaces and scenarios of transition to turbulence},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
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A. N. Kulikov. Bifurcations of invariant tori in second-order quasilinear evolution equations in Hilbert spaces and scenarios of transition to turbulence. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference "Geometric Methods in Control Theory and Mathematical Physics" dedicated to the 70th anniversary of S.L. Atanasyan, the 70th anniversary of I.S. Krasilshchik, the 70th anniversary of A.V. Samokhin, and the 80th anniversary of V.T. Fomenko. S.A. Esenin Ryazan State University, Ryazan, September 25–28, 2018. Part 1, Tome 168 (2019), pp. 45-52. http://geodesic.mathdoc.fr/item/INTO_2019_168_a5/

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