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This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online.
The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.
List of contents
An Overview of the Parameterization Method for Invariant Manifolds.- Seminumerical Algorithms for Computing Invariant Manifolds of Vector Fields at Fixed Points.- The Parameterization Method for Quasi-Periodic Systems: From Rigorous Results to Validated Numerics.- The Parameterization Method in KAM Theory.- A Newton-like Method for Computing Normally Hyperbolic Invariant Tori.
Summary
This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online.
The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.
Additional text
“This book provides a unified formulation of the parametrization method (PM), which can be used to study different problems in the theory of dynamical systems. … The monograph can be recommended to mathematicians, scientists and engineers who are interested in the theory and application of dynamical systems as a reference book. It can also be used as a textbook for graduate students in courses on the mathematical theory of dynamical systems.” (Martin Hermann, Mathematical Reviews, March, 2017)
“In this monograph, the authors present a unified formulation of the parametrization method applied in some specific contexts … . The monograph contains an extensive bibliography and is recommended to researchers specialized in the theoretical and computational methods for the study of dynamical systems.” (Rodica Luca, zbMATH, 1372.37002, 2017)
Report
"This book provides a unified formulation of the parametrization method (PM), which can be used to study different problems in the theory of dynamical systems. ... The monograph can be recommended to mathematicians, scientists and engineers who are interested in the theory and application of dynamical systems as a reference book. It can also be used as a textbook for graduate students in courses on the mathematical theory of dynamical systems." (Martin Hermann, Mathematical Reviews, March, 2017)
"In this monograph, the authors present a unified formulation of the parametrization method applied in some specific contexts ... . The monograph contains an extensive bibliography and is recommended to researchers specialized in the theoretical and computational methods for the study of dynamical systems." (Rodica Luca, zbMATH, 1372.37002, 2017)
