Computation of whiskered invariant tori and their associated manifolds: new fast algorithms
By G. Huguet, R. Llave, Yannick Sire. 2010
66 citations recorded by the source corpus.
In this paper we present efficient algorithms for the computation of several invariant objects for Hamiltonian dynamics. More precisely, we consider KAM tori (i.e diffeomorphic copies of the torus 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). In the case of whiskered tori, we also present algorithms to compute the invariant splitting and the invariant manifolds associated to the splitting. We present them both for the case of discrete time and for differential equations. The algorithms are based on a Newton method to solve an appropriately chosen functional equation that expresses invariance. The algorithms are efficient: if we discretize the objects by $N$ elements, one step of the Newton method requires only O(N) storage and $O(N \ln(N))$ operations. Furthermore, if the object we consider is of dimension $\ell$, we only need to compute functions of $\ell$ variables, independently of what is the dimension of the phase space. The algorithms do not require that the system is presented in action-angle variables nor that it is close to integrable. The algorithms are backed up by rigorous \emph{a-posteriori} bounds which state that if the equations are solved with a small residual and some explicitly computable condition numbers are not too big, then, there is a true solution which is close to the computed one. The algorithms apply both to primary (i.e non-contractible) and secondary tori (i.e. contractible to a torus of lower dimension, such as islands). They have already been implemented. We will report on the technicalities of the implementation and the results of running them elsewhere.
Subfields: Condensed Matter Physics, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/6a7ada0568b0c5cb7c1bd06ef2a4e5da44ccf1be
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In Condensed Matter Physics
In Mathematical Physics
In Statistical and Nonlinear Physics
Citations in the corpus
- Computation of maximal local (un)stable manifold patches by the parameterization method (2015, methodology): “The parameterization method can also be used to compute stable/unstable manifolds associated with periodic orbits of differential equations [10, 37, 38], as well as stable/unstable manifolds associated with invariant circles/tori [39, 40].”