Computation of whiskered invariant tori and their associated manifolds: new fast algorithms
Computation of whiskered invariant tori and their associated manifolds: new fast algorithms
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胡须不变环面及其相关流形的计算：新的快速算法
DOI：
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发表时间：
2010
期刊：
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通讯作者：
Y. Sire
中科院分区：
文献类型：
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作者：
G. Huguet;R. Llave;Y. Sire
在本文中，我们提出了有效的算法计算的几个不变的对象的哈密顿动力学。更确切地说，我们考虑KAM环面（即环面的纯拷贝，使得其上的运动与刚性旋转共轭）拉格朗日环面（具有最大维数）和须状环面（即具有双曲方向的环面，连同环面的切线和辛共轭一起跨越整个切空间）。在须环的情况下，我们还提出了算法来计算不变的分裂和不变的流形分裂。我们提出他们的情况下，离散时间和微分方程。该算法是基于牛顿法来解决一个适当选择的功能方程，表示不变性。算法是有效的：如果我们将对象离散为$N$个元素，则牛顿方法的一个步骤仅需要O（N）存储和O（N ln（N））$操作。此外，如果我们考虑的对象是维数$ell$，我们只需要计算$ell$变量的函数，而与相空间的维数无关。该算法不要求系统的作用角变量，也不要求它是接近可积的。该算法是由严格的后验界的状态，如果方程的解决与一个小的残留和一些明确的可计算的条件数是不是太大，然后，有一个真正的解决方案，这是接近计算的。该算法适用于主要（即不可收缩）和次要环面（即可收缩到较低维度的环面，如岛屿）。这些建议已经得到执行。我们将报告实施的技术细节和在其他地方实施的结果。
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.