Wysham Derin B, Meiss James D
University of Colorado at Boulder, Applied Mathematics, Boulder, Colorado 80309-0526, USA.
Chaos. 2006 Jun;16(2):023129. doi: 10.1063/1.2200159.
We develop an iterative technique for computing the unstable and stable eigenfunctions of the invariant tori of diffeomorphisms. Using the approach of Jorba [Nonlinearity 14, 943 (2001)], the linearized equations are rewritten as a generalized eigenvalue problem. Casting the system in this light allows us to take advantage of the speed of eigenvalue solvers and create an efficient method for finding the first-order approximations to the invariant manifolds of the torus. We present a numerical scheme based on the power method that can be used to determine the behavior normal to such tori, and give some examples of the application of the method. We confirm the qualitative conclusions of the Melnikov calculations of Lomeli and Meiss [Nonlinearity 16, 1573 (2003)] for a volume-preserving mapping.
我们开发了一种迭代技术，用于计算微分同胚不变环面的不稳定和稳定本征函数。采用约尔巴[《非线性》14, 943 (2001)]的方法，将线性化方程重写为广义特征值问题。从这个角度看待系统，使我们能够利用特征值求解器的速度，并创建一种有效的方法来找到环面不变流形的一阶近似。我们提出了一种基于幂法的数值方案，可用于确定垂直于此类环面的行为，并给出了该方法的一些应用示例。对于一个保体积映射，我们证实了洛梅利和迈斯[《非线性》16, 1573 (2003)]的梅尔尼科夫计算的定性结论。
