Bifurcations of Invariant Manifolds for a Periodic Boundary Value Problem for a Generalized Version of the Cahn–Hilliard Equation

A periodic boundary value problem for the generalized Cahn–Hilliard equation is considered. It is shown that the boundary value problem under study can have any number of smooth two-dimensional invariant manifolds. This depends, in particular, on the number of roots of the auxiliary algebraic equation. These two-dimensional invariant manifolds are filled with solutions that are periodic in the evolution variable and depend on the spatial variable. These two-dimensional manifolds are unstable as invariant manifolds of the boundary value problem under study. In the extreme case, only one of these manifolds can be a local attractor. Asymptotic formulas in powers of the bifurcation parameter are constructed for the solutions that form them.