We recently derived a method, local orthogonal rectification (LOR), that provides a natural and useful geometric frame for analyzing dynamics relative to a base curve in the phase plane for two-dimensional systems of ODEs (Letson and Rubin, SIAM J. Appl. Dyn. Syst., 2018). This work extends LOR to apply to any embedded base manifold in a system of ODEs of arbitrary dimension and establishes a corresponding system of LOR equations for analyzing dynamics within the LOR frame, which maps naturally back to the original phase space. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. In addition to developing a general theory for LOR that makes use of a given normal frame, we show how to construct a normal frame that conveniently simplifies the computations involved in LOR. Finally, we illustrate the utility of LOR by showing that a blow-up transformation on the LOR equations provides a useful decomposition for studying trajectories' behavior relative to the embedded base manifold and by using LOR to identify canard behavior near a fold of a critical manifold in a two-timescale system.

中文翻译：

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局部正交校正：推导自然坐标以研究相对于歧管的流量

我们最近推导了一种方法，即局部正交整流（LOR），该方法为分析ODE的二维系统（Letson和Rubin，SIAM J.Appl。 Dyn。Syst。，2018）。这项工作扩展了LOR，以应用于任意尺寸的ODE系统中的任何嵌入式基础流形，并建立了一个相应的LOR方程系统，用于分析LOR框架内的动力学，该动力学自然地映射回原始相空间。LOR方程对基础流的几何属性进行编码，并且通常在嵌入式歧管的局部邻域之外仍然有效。除了开发利用给定正常框架的LOR的一般理论外，我们还将展示如何构建可方便地简化LOR中涉及的计算的正常框架。最后，我们通过显示LOR方程的爆炸变换为研究轨迹的运动提供了有用的分解，说明了LOR的效用。