Computation of maximal local (un)stable manifold patches by the parameterization method
Computation of maximal local (un)stable manifold patches by the parameterization method
复制标题
通过参数化方法计算最大局部（不稳定）流形补丁
DOI：
10.1016/j.indag.2015.11.001
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
J. D. M. James
中科院分区：
文献类型：
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作者：
M. Breden;J. Lessard;J. D. M. James
在这项工作中，我们开发了一些自动程序，用于计算微分方程平衡的局部（不稳定）稳定流形的高阶多项式展开式。我们的方法结合了经过验证的截断误差界限，并相对于某些指定的约束最大化多项式近似的图像大小。更准确地说，我们使用流形计算在很大程度上取决于特征向量的缩放：实际上，我们研究了这些缩放对确定验证误差范围的估计的精确影响。特征向量缩放和误差估计之间的这种关系在我们的自动过程中起着核心作用。为了说明这些方法的实用性，我们提出了几种应用，包括 Lorenz 和 FitzHugh-Nagumo 系统中不变流形的可视化以及悬索桥问题中（不稳定）流形的自动连续方案。在目前的工作中，我们明确处理特征值满足特定非共振条件的情况。
In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size of the image of the polynomial approximation relative to some specified constraints. More precisely we use that the manifold computations depend heavily on the scalings of the eigenvectors: indeed we study the precise effects of these scalings on the estimates which determine the validated error bounds. This relationship between the eigenvector scalings and the error estimates plays a central role in our automatic procedures. In order to illustrate the utility of these methods we present several applications, including visualization of invariant manifolds in the Lorenz and FitzHugh–Nagumo systems and an automatic continuation scheme for (un)stable manifolds in a suspension bridge problem. In the present work we treat explicitly the case where the eigenvalues satisfy a certain non-resonance condition.