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Kuramoto-Sivashinsky dynamics on the center-unstable manifold. (English) Zbl 0687.34036

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

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
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