Open AccessHyperbolic almost periodic solutions and toroidal limit sets
Author(s) -
George R. Sell
Publication year - 1977
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.74.8.3124
Subject(s) - diffeomorphism , torus , jacobian matrix and determinant , mathematics , manifold (fluid mechanics) , dimension (graph theory) , order (exchange) , mathematical analysis , limit (mathematics) , pure mathematics , differential equation , combinatorics , mathematical physics , geometry , mechanical engineering , finance , engineering , economics
We consider an almost periodic solution φ(t ) of an autonomous differential equationx ′ =f (x ). Letk denote the topological dimension of the hullH (φ). By considering the linearized equationx ′ =A (t )x alone [whereA (t ) is the Jacobian matrix off evaluated along φ(t )], one can derive a sufficient condition in order that φ(t ) be quasi-periodic and that the hullH (φ) be diffeomorphic to ak -dimensional torusTk . The proof is based upon an extension of the center manifold theorem to nonautonomous nonperiodic differential systems.