Existence of a central manifold for the initial boundary value problem with transformation of variables

A. A. Kornuta

V. I. Vernadsky Crimean Federal University, Simferopol

Abstract: In bifurcation theory, central manifolds are fundamental for studying dynamical systems in the vicinity of equilibrium points located near critical situations. The central manifold theorem reduces the study of small solutions to an infinite-dimensional problem of the form containing linear and nonlinear parts that are close enough to a trivial solution to the study of small solutions of a reduced system with finite dimension. Solutions on the central manifold are described by a finite-dimensional system of ordinary differential equations called a reduced system.
The paper considers an initial boundary value problem on a circle for a parabolic functional differential equation with a rotation transformation of a spatial variable and a periodicity condition. Earlier, using a central manifold in the vicinity of a spatially homogeneous stationary solution, the author obtained asymptotic representations of spatially inhomogeneous solutions and traveling wave-type solutions in the initial boundary value problem with the Neumann boundary condition on a ring with rotation transformation and radial compression, in the problem on a circle with rotation transformation. In the problem on a circle with a rotation transformation and a periodicity condition on the boundary, a representation for a spatially inhomogeneous stationary solution is obtained.
This paper provides a proof of the existence of a central manifold. The scheme proposed in the work of Haragus M. is used and Iooss G. «Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamic Systems», which is based on proving some auxiliary statements and obtaining estimates of the resolvent of the operator $L$.
The proofs of the conditions necessary to fulfill the central manifold theorem corresponding to the case when the unstable part of the spectrum of the operator $L$ corresponding to the eigenvalues with a positive real part is empty are considered. Estimates of the resolvent of the operator $L$ are also obtained. The key is the statement about the finiteness of the part of the spectrum of the operator $L$ corresponding to an eigenvalue with a zero real part (the central spectrum).

Keywords: dynamical system, initial boundary value problem, center manifold, central spectrum, unstable and stable spectrum

UDC: 517.95

MSC: 35K10, 35K55, 35R10