November  2006, 6(6): 1261-1300. doi: 10.3934/dcdsb.2006.6.1261

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms

1. 

Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain

2. 

School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA 30332-0160, United States

Received  July 2005 Revised  May 2006 Published  August 2006

In this paper we develop several numerical algorithms for the computation of invariant manifolds in quasi-periodically forced systems. The invariant manifolds we consider are invariant tori and the asymptotic invariant manifolds (whiskers) to these tori.
    The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34].
    The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties.
    The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.
Citation: Àlex Haro, Rafael de la Llave. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1261-1300. doi: 10.3934/dcdsb.2006.6.1261
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