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Published in:
An Overview of the Parameterization Method for Invariant Manifolds Read chapter Read first chapter

2016 | OriginalPaper | Chapter

2. Seminumerical Algorithms for Computing Invariant Manifolds of Vector Fields at Fixed Points

Authors : Àlex Haro, Josep-Maria Mondelo

Published in: The Parameterization Method for Invariant Manifolds

Publisher: Springer International Publishing

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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.

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previous chapter An Overview of the Parameterization Method for Invariant Manifolds
next chapter The Parameterization Method for Quasi-Periodic Systems: From Rigorous Results to Validated Numerics
Footnotes
1
We have used Brent’s method in order to save coding. The use of Newton’s method would reduce computing time by requiring less iterations.
 
2
The performance can be improved through the use of Taylor methods, that are built on automatic differentiation techniques. See, e.g., [Sim01, ABBR12, JZ05].
 
3
The value given is in double precision as obtained from the DE406 JPL ephemeris file [Sta98].
 
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Metadata
Title
Seminumerical Algorithms for Computing Invariant Manifolds of Vector Fields at Fixed Points
Authors
Àlex Haro
Josep-Maria Mondelo
Copyright Year
2016
Book
The Parameterization Method for Invariant Manifolds

Print ISBN: 978-3-319-29660-9

Electronic ISBN: 978-3-319-29662-3

Copyright Year: 2016

DOI
https://doi.org/10.1007/978-3-319-29662-3_2

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