The Kuramoto–Sivashinsky Equation. A Local Attractor Filled with Unstable Periodic Solutions

Resumo

A periodic boundary value problem is considered for one of the first versions of the Kuramoto–Sivashinsky equation, which is widely known in mathematical physics. Local bifurcations in a neighborhood of spatially homogeneous equilibrium points are studied in the case when they change stability. It is shown that the loss of stability of homogeneous equilibrium points leads to the occurrence of a two-dimensional local attractor on which all solutions are periodic functions of time, except for one spatially inhomogeneous state. The spectrum of frequencies of this family of periodic solutions fills the entire number line, and all of them are unstable in the sense of Lyapunov’s definition in the metric of the phase space (the space of initial conditions) of the corresponding initial boundary value problem. As the phase space, a Sobolev functional space natural for this boundary value problem is chosen. Asymptotic formulas are given for periodic solutions filling the two-dimensional attractor. To analyze the bifurcation problem, analysis methods for an infinite-dimensional dynamical system are used: the integral (invariant) manifold method combined with the methods of the Poincaré normal form theory and asymptotic methods. Analyzing the bifurcations for the periodic boundary value problem is reduced to analyzing the structure of the neighborhood of the zero solution to the homogeneous Dirichlet boundary value problem for the equation under consideration.