The Kuramoto--Sivashinsky equation. A local attractor filled with unstable periodic solutions

A. N. Kulikov ; D. A. Kulikov

Modelirovanie i analiz informacionnyh sistem, Tome 25 (2018) no. 1, pp. 92-101.

Résumé

A periodic boundary value problem is considered for one version of the Kuramoto–Sivashinsky equation, which is widely known in mathematical physics. Local bifurcations in a neighborhood of the spatially homogeneous equilibrium points in the case when they change stability are studied. It is shown that the loss of stability of homogeneous equilibrium points leads to the appearance of a two-dimensional attractor on which all solutions are periodic functions of time, except one spatially inhomogeneous state. A spectrum of frequencies of the given family of periodic solutions fills the entire number line, and they are all unstable in a sense of Lyapunov definition in the metric of the phase space (space of initial conditions) of the corresponding initial boundary value problem. It is chosen the Sobolev space as the phase space. For the periodic solutions which fill the two-dimensional attractor, the asymptotic formulas are given. In order to analyze the bifurcation problem it was used analysis methods for infinite-dimensional dynamical systems: the integral (invariant) manifold method, the Poincare normal form theory, and asymptotic methods. The analysis of bifurcations for periodic boundary value problem was reduced to analysing the structure of the neighborhood of the zero solution of the homogeneous Dirichlet boundary value problem for the considered equation.

Détail

Keywords: the Kuramoto-Sivashinsky equation, periodic boundary value problem, stability, attractor, asymptotic formulas.
Mots-clés : local bifurcations

Bibliographie

[1] Kuramoto Y., Chemical oscillations, waves and turbulence, Springer, Berlin, 1984 | MR

[2] Sivashinsky G. I., “Weak turbulence in periodic flows”, Physica D: Nonlinear Phenomena, 17:2 (1985), 243–255 | DOI | MR

[3] Akhmediev N., A. Ankevich, Dissipativnye solitony, Fizmatlit, M., 2008

[4] Armbruster D., Guckenheimer J., Holmes P., “Kuramoto–Sivashinsky Dynamics on the Center-Unstable Manifold”, SIAM J. Appl. Math., 49:3 (1989), 676–691 | DOI | MR

[5] Kevrekidis I.G., Nicolaenko B., Scovel J.C., “Back in the saddle again: A computer assisted study of the Kuramoto–Sivashinsky equation”, SIAM J. Appl. Math., 50:3 (1990), 760–790 | DOI | MR

[6] Nicolaenko B., Scheurer B., Temam R., “Some global dynamical properties of the Kuramoto–Sivashinsky equations: Nonlinear stability and attractors”, Physica D: Nonlinear Phenomena, 16:2 (1985), 155–183 | DOI | MR

[7] Changpin Li, Zhonghua Y., “Bifurcation of two-dimensional Kuramoto–Sivashinsky equation”, Appl. Math.- JCU, 13:3 (1998), 263–270 | DOI | MR

[8] Kulikov A. N, Kulikov D. A, “Formation of wavy nanostructures on the surface of flat substrates by ion bombardment”, Computational Mathematics and Mathematical Physics, 52:5 (2012), 800–814 | DOI | MR

[9] Kulikov A.N., Kulikov D.A., “Bifurcations of spatially inhomogeneous solutions in two boundary value problems for the generalized Kuramoto–Sivashinsky equation”, Vestnik MIFI, 3:4 (2014), 408–415 (in Russian) | DOI | MR

[10] Kulikov A.N., Kulikov D.A., “Inhomogeneous solutions for a modified Kuramoto–Sivashinsky equation”, Journal of Mathematical Sciences, 219:2 (2016), 173–183 | DOI | MR

[11] Kulikov A.N., “The attractors of two boundary value problems for a modifieded nonlinear telegraph equation”, Rus. J. Nonlin. Dyn., 4:1 (2008), 57–68 (in Russian) | MR

[12] Glyzin S.D., Kolesov A.Yu, “Primer attraktora, sostoyashchego iz neustoychivykh po Lyapunovu periodicheskikh traektoriy”, Modeling and Analysis of Information Systems, 15:2 (2008), 94–95 (in Russian)

[13] Kulikov A.N., Kulikov D.A., “The existence of attractor formed by the unstable solutions”, Abstracts of international conference reports “New Trends in Nonlinear Dynamics”, JarGU, Jaroslavl, 2017, 50–51 (in Russian)