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2016 | Buch

The Parameterization Method for Invariant Manifolds

From Rigorous Results to Effective Computations

verfasst von: Àlex Haro, Marta Canadell, Jordi-Lluis Figueras, Alejandro Luque, Josep Maria Mondelo

Verlag: Springer International Publishing

Buchreihe : Applied Mathematical Sciences

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. An Overview of the Parameterization Method for Invariant Manifolds
Abstract
This introductory chapter starts by providing an overview of the literature of the parameterization method. After that, it introduces unified formulations of the parameterization method for invariant manifolds of fixed points and for invariant tori in different contexts. These formulations are the basis of the subsequent chapters. This chapter can be considered a reading guide of the rest of the book.
Àlex Haro
Chapter 2. Seminumerical Algorithms for Computing Invariant Manifolds of Vector Fields at Fixed Points
Abstract
This chapter discusses computational aspects of invariant manifolds of vector fields at fixed points. It is focused on algorithms and implementations, since the theory is well established in many classical textbooks and in the foundational papers of the parameterization method. Special emphasis is given to the computation of semi-local expansions of invariant manifolds, for which algorithms are provided, based on the algebraic manipulation of power series and novel Automatic Differentiation techniques. The chapter illustrates the methodology with three detailed examples, which are: the 2D stable manifold of the origin of the Lorenz system, the 4D center manifold of a collinear point of the Restricted Three-Body Problem, and a 6D partial normal form in the same problem that allows the generation of Conley’s transit and non-transit trajectories associated to any object of the center manifold.
Àlex Haro, Josep-Maria Mondelo
Chapter 3. The Parameterization Method for Quasi-Periodic Systems: From Rigorous Results to Validated Numerics
Abstract
This chapter developes the “from theory-to algorithms-to computations-to validations” program for response tori in quasi-periodically forced systems. First, it provides a full proof of a Kantorovich-like theorem for invariant tori in discrete quasi-periodic systems. The proof of this theorem leads to several algorithms for the computation of invariant tori in this context, that are also detailed. Next, it is explained a computer assisted methodology for the validation of numerical results based on the previous a posteriori theorem. The chapter ends with three examples: validation of saddle invariant tori on the verge of breakdown, computation of a rigorous upper bound of the measure of Cantor-like spectra of a discrete Schrödinger operator, and validation of an attracting torus that by direct double precision seems to be a strange nonchaotic attractor.
Jordi-Lluís Figueras, Àlex Haro
Chapter 4. The Parameterization Method in KAM Theory
Abstract
This chapter is devoted to the parameterization method in KAM theory, also referred to as KAM theory without action-angle coordinates. The chapter states and proves a KAM theorem in a posteriori format, with explicit bounds suitable to be applied in an effective and quantitative way. The reader can skip the proof without losing the flavor of the application of the method. We have included full descriptions of the derived algorithms, and applications to the examples that follow, which are: application of the theorem (by hand calculations) to obtain persistence of the golden invariant curve for tiny values of the parameter of the standard map, numerical continuation of this same curve up to values close to the breakdown, and computation of 2D KAM tori in the Froeschlé map.
Àlex Haro, Alejandro Luque
Chapter 5. A Newton-like Method for Computing Normally Hyperbolic Invariant Tori
Abstract
This chapter presents some ideas of normally hyperbolic manifold theory, and focuses on the algorithmic application of the parameterization method in such context. The parameterization method is applied to the computation of several normally hyperbolic invariant manifolds, in the following examples: computation of an attracting invariant curve in a 2D- Fattened Arnold Family, computation of a saddle invariant curve in a 3D- Fattened Arnold Family, and the computation of a 2D normally hyperbolic invariant cylinder in the Froeschlé map.
Marta Canadell, Àlex Haro
Backmatter
Metadaten
Titel
The Parameterization Method for Invariant Manifolds
verfasst von
Àlex Haro
Marta Canadell
Jordi-Lluis Figueras
Alejandro Luque
Josep Maria Mondelo
Copyright-Jahr
2016
Electronic ISBN
978-3-319-29662-3
Print ISBN
978-3-319-29660-9
DOI
https://doi.org/10.1007/978-3-319-29662-3