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


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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