Center manifolds and contractions on a scale of Banach spaces
The paper contains a new proof of the existence of a \(C^ k\)-center manifold at a nonhyperbolic equilibrium point of a finite-dimensional vectorfield of class \(C^ k\). The main idea is that (after an appropriate globalization of the problem) such center manifold contains all solutions which have a certain maximal exponential growth at infinity. This leads to a fixed point problem on a scale of Banach spaces, indexed by the exponent; the difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on this scale of Banach spaces, such that a direct application of the implicit function theorem is not possible. The proof is then based on an abstract theorem which gives the differentiable dependence on parameters of the fixed point of uniform contractions on embedded Banach spaces and under conditions satisfied in particular by the center manifold problem.
existence of a \(C^ k\)-center manifold at a nonhyperbolic equilibrium point of a finite-dimensional vectorfield of class \(C^ k\)
fixed point problem on a scale of Banach spaces
fixed point of uniform contractions on embedded Banach spaces
scientific article; zbMATH DE number 4007068
scientific article