Numerical computation of the normal behaviour of invariant curves of n-dimensional maps
Numerical computation of the normal behaviour of invariant curves of n-dimensional maps
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DOI：
10.1088/0951-7715/14/5/303
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发表时间：
2001-09-01
期刊：
影响因子：
1.7
通讯作者：
Jorba, A
中科院分区：
文献类型：
--
作者：
Jorba, A
我们描述了一种数值方法计算线性化正常行为的不变曲线的一个R-n，n大于或等于2的同态。在可约的情况下，该方法不仅计算正常的特征值，无论是椭圆或双曲线，但也相应的特征方向，这是一阶近似不变流形（稳定，不稳定和中心）的曲线。此外，该方法似乎能够检测的不可约性-如果这是案件-的线性化system.The方法的输入是不变的曲线-包括其旋转数-以及计算的映射和其微分的数值过程。因此，这种方法可以很容易地用于常微分方程的庞加莱截面。由于所使用的近似的谱特性，对于足够光滑的情况，该过程的收敛非常快。我们注意到，该方法也是有效的计算正常行为的环面的更高的维度。最后，作为例子，我们研究了一些具体问题中出现的不变曲线的稳定性。特别是，我们计算一个给定的不变曲线的六维辛映射的不稳定流形。
We describe a numerical method for computing the linearized normal behaviour of an invariant curve of a diffeomorphism of R-n, n greater than or equal to 2. In the reducible case, the method computes not only the normal eigenvalues-either elliptic or hyperbolic-but also the corresponding eigendirections, that are the first-order approximation to the invariant manifolds (stable, unstable and central) around the curve. Moreover, the method seems to be able to detect the nonreducibility-if this is the case-of the linearized system.The input of the method is the invariant curve-including its rotation number-as well as a numerical procedure for computing the map and its differential. Hence, this method can be easily used on Poincare sections of ordinary differential equations. Due to the spectral character of the approximations used, the convergence of the process is very fast for sufficiently smooth cases. We note that the method is also valid for computing the normal behaviour of tori of higher dimensions. Finally, as examples, we study the stability of the invariant curves that appear in some concrete problems. In particular, we compute the unstable manifold for a given invariant curve of a six-dimensional symplectic map.