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.
On the co--existence of maximal and whiskered tori for the planetary three--body problem
本文讨论了在三体问题的相空间的一个区域内，稳定和不稳定的准周期{\sc kam}环面{\it共存}的可能性。{论证}沿着{\sc kam}理论，特别是}在相空间的同一区域产生两个非光滑相关的正则坐标系，对于“适当退化”的系统，Nekhorossev和Mi{\displaystyle Mi}{\displaystyle {\frac {i}enko和Fomenko的定理预见了这种可能性。这两个坐标系是雅可比节点的经典简化的替代方案，例如，在~[V.I.~ Arnold，Small Replacators and problems of stability of motion in classical and celestial mechanics，18，85（1963）; p. 141].
Rapid GPU-Assisted Search and Parameterization-Based Refinement and Continuation of Connections between Tori in Periodically Perturbed Planar Circular Restricted 3-Body Problems
Non-autonomous KAM theory for lower dimensional invariant tori (II): Normally parabolic case
Universitat de Barcelona（巴塞罗那大学）；Centre de Mathématiques Laurent Schwartz (CMLS), CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎综合理工学院中央数学实验室）
A new fast multiple-shooting method for computing periodic orbits in symplectic maps leveraging simultaneous Floquet vector computation to avoid large linear systems
