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

TL;DR

This paper introduces efficient, rigorous algorithms based on Newton methods for computing invariant tori and their manifolds in Hamiltonian systems, applicable to both discrete and continuous dynamics without requiring near-integrability.

Contribution

The paper presents novel fast algorithms for computing whiskered invariant tori and their manifolds, with efficiency and applicability to general Hamiltonian systems.

Findings

Algorithms require O(N) storage and O(N log N) operations per step.

Applicable to both primary and secondary tori, in discrete and continuous systems.

Backed by rigorous a-posteriori validation ensuring proximity to true solutions.

Abstract

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.