Computation of whiskered invariant tori and their associated manifolds: New fast algorithms

Gemma Huguet; Rafael de la Llave; Yannick Sire

doi:10.3934/dcds.2012.32.1309

Article Contents

2012, Volume 32, Issue 4: 1309-1353. Doi: 10.3934/dcds.2012.32.1309

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Computation of whiskered invariant tori and their associated manifolds: New fast algorithms

1.
Center for Neural Science, New York University, New York, NY 10003
2.
Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712-1082
3.
Université Paul Cézanne, Laboratoire LATP UMR 6632, Marseille

Received: August 31, 2010

Revised: July 31, 2011

Published: March 2012

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Abstract

We present efficient (low storage requirement and low operation count) algorithms for the computation of several invariant objects for Hamiltonian dynamics, namely KAM tori (i.e diffeomorphic copies of tori such that the motion on them is conjugated to a rigid rotation) both Lagrangian tori(of maximal dimension) and whiskered tori (i.e. tori with hyperbolic directions which, together with the tangents to the torus and the symplectic conjugates span the whole tangent space). We also present algorithms to compute the invariant splitting and the invariant manifolds of whiskered tori. We present the algorithms for both discrete-time dynamical systems and differential equations.
The algorithms do not require that the system is presented in action-angle variables nor that it is close to integrable and are backed up by rigorous a-posteriori bounds. We will report on the implementation results elsewhere.

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Mathematics Subject Classification: Primary: 70K43; Secondary: 37J40.

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