Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form
Parameterization of Invariant Manifolds for Periodic Orbits I: Efficient Numerics via the Floquet Normal Form
复制标题
周期轨道不变流形的参数化 I：通过 Floquet 范式的高效数值
DOI：
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发表时间：
2015
影响因子：
2.1
通讯作者：
J. Mireles
中科院分区：
文献类型：
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作者：
R. Castelli;J. Lessard;J. Mireles
本文提出了一种计算双曲周期轨道（不）稳定流形的Fourier-Taylor展开式的有效数值方法。该方法的三个特点是：（1）我们获得了不变流形以及流形上的动力学的精确表示，（2）它允许自然的后验误差分析，（3）它不需要数值积分的向量场。我们的方法是基于不变流形的参数化方法，并研究了一个特定的偏微分方程，它表征了一个图表映射的流形。该方法只需要一些温和的非共振条件成立。目前的工作的新奇在于，我们利用Floquet规范形式，以有效地计算傅立叶-泰勒展开。给出了一些例子计算，包括相空间维数高达10和流形是二维和三维的流形。我们还讨论了计算周期到周期连接轨道.
We present an efficient numerical method for computing Fourier--Taylor expansions of (un)stable manifolds associated with hyperbolic periodic orbits. Three features of the method are that (1) we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) it admits natural a posteriori error analysis, and (3) it does not require numerically integrating the vector field. Our approach is based on the parameterization method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the manifold. The method requires only that some mild nonresonance conditions hold. The novelty of the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier--Taylor expansion. A number of example computations are given including manifolds in phase space dimension as high as ten and manifolds which are two and three dimensional. We also discuss computations of cycle-to-cycle connecting orbits ...