A classical approach to the analyticity problem of center manifolds
The author is concerned with ordinary differential equations of the form (1) \(\dot x=-y+X(x,y,z)\), \(\dot y=x+Y(x,y,z)\), \(\dot z=Az+Z(x,y,z)\), (x,y\(\in {\mathbb{R}}\) and \(z\in {\mathbb{R}}^ n)\), where X, Y and Z are analytic and real-valued functions which together with their first derivatives vanish at the origin 0. The matrix A is assumed to have no purely imaginary eigenvalues. Any center manifold equation corresponding to (1) is a two-dimensional system of the form (2) \(\dot x=-y+X(x,y,c(x,y)),\) \(\dot y=x+Y(x,y,c(x,y)),\) where \(z=c(x,y)\) is a function representing locally a center manifold of (1) passing through 0. It can be proved by means of Poincaré's focal values that if \(z=c(x,y)\) is analytic, then the following alternative holds: ''The zero solution of (2) is either a focus or a center.'' However, the center manifold \(z=c(x,y)\) may be non- analytic in general. The main purpose of this paper is to prove the validity of the above alternative in spite of the lacking analyticity for the center manifold.
scientific article; zbMATH DE number 3899333
scientific article