Armbruster, Dieter; Guckenheimer, John; Holmes, Philip Kuramoto-Sivashinsky dynamics on the center-unstable manifold. (English) Zbl 0687.34036 SIAM J. Appl. Math. 49, No. 3, 676-691 (1989). The dynamical behavior of solutions of the Kuramoto-Sivashinsky equation \[ u_ t+\alpha u_{xxxx}+u_{xx}+(u_ x)^ 2=0 \] with periodic boundary conditions on a spatial interval [0,h] is studied where the length h of the interval acts as bifurcation parameter. After a Galerkin projection to an infinite system of complex first order ordinary differential equations a bifurcation analysis is carried out for the solutions branching off the trivial solution. The main part of the paper is concerned with the reduction of this system to a two-(complex- )dimensional local center-unstable manifold near the second bifurcation point and the detailed analysis of the resulting O(2)-equivariant system. It turns out that the bifurcations and dynamical behavior of the fourth- order approximation to the reduced system agree well, qualitatively and quantitatively, with numerical simulations of the full partial differential equation. Reviewer: B.Aulbach Cited in 55 Documents MSC: 34C30 Manifolds of solutions of ODE (MSC2000) 34C40 Ordinary differential equations and systems on manifolds 35B32 Bifurcations in context of PDEs 37G99 Local and nonlocal bifurcation theory for dynamical systems Keywords:Kuramoto-Sivashinsky equation; bifurcation parameter; Galerkin projection; local center-unstable manifold PDF BibTeX XML Cite \textit{D. Armbruster} et al., SIAM J. Appl. Math. 49, No. 3, 676--691 (1989; Zbl 0687.34036) Full Text: DOI OpenURL