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 EMS Tracts in Mathematics 2013; 225 pp; hardcover Volume: 21 ISBN-10: 3-03719-124-4 ISBN-13: 978-3-03719-124-8 List Price: US$78 Member Price: US$62.40 Order Code: EMSTM/21 In this book, dynamical systems are investigated from a geometric viewpoint. Admitting an invariant manifold is a strong geometric property of a dynamical system. This text presents rigorous results on invariant manifolds and gives examples of possible applications. In the first part, discrete dynamical systems in Banach spaces are considered. Results on the existence and smoothness of attractive and repulsive invariant manifolds are derived. In addition, perturbations and approximations of the manifolds and the foliation of the adjacent space are treated. In the second part, analogous results for continuous dynamical systems in finite dimensions are established. In the third part, the theory developed is applied to problems in numerical analysis and to singularly perturbed systems of ordinary differential equations. The mathematical approach is based on the so-called graph transform, already used by Hadamard in 1901. The aim is to establish invariant manifold results in a simple setting that provides quantitative estimates. The book is targeted at researchers in the field of dynamical systems interested in precise theorems that are easy to apply. The application part might also serve as an underlying text for a student seminar in mathematics. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Researchers interested in dynamical systems. Table of Contents Discrete Dynamical Systems--Maps Existence Perturbation and approximation Smoothness Foliation Smoothness of the foliation with respect to the base point Continuous Dynamical Systems--ODEs A general result for the time-T map Invariant manifold results Applications Fixed points and equilibria The one-step method associated to a linear multistep method Invariant manifolds for singularly perturbed ODEs Runge-Kutta methods applied to singularly perturbed ODEs Invariant curves of perturbed harmonic oscillators Blow-up in singular perturbations Application of Runge-Kutta methods to differential-algebraic equations Appendices Bibliography Index