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April 2012, 32(4): 1309-1353. doi: 10.3934/dcds.2012.32.1309
Computation of whiskered invariant tori and their associated manifolds: New fast algorithms
Gemma Huguet 1, , Rafael de la Llave 2, and Yannick Sire 3,
Center for Neural Science, New York University, New York, NY 10003, United States
Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712-1082, United States
Université Paul Cézanne, Laboratoire LATP UMR 6632, Marseille, France
Received September 2010 Revised August 2011 Published October 2011
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.
Keywords: Quasi-periodic solutions, whiskered KAM tori, numerical computation., invariant manifolds.
Mathematics Subject Classification: Primary: 70K43; Secondary: 37J4.
Citation: Gemma Huguet, Rafael de la Llave, Yannick Sire. Computation of whiskered invariant tori and their associated manifolds: New fast algorithms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1309-1353. doi: 10.3934/dcds.2012.32.1309
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Gemma Huguet Rafael de la Llave Yannick Sire