# Invariant Manifolds in Discrete and Continuous Dynamical Systems

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*Kaspar Nipp; Daniel Stoffer*

A publication of the European Mathematical Society

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.