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首页> 外文期刊>SIAM journal on applied dynamical systems >A New Frame for an Old (Phase) Portrait: Finding Rivers and Other Flow Features in the Plane
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A New Frame for an Old (Phase) Portrait: Finding Rivers and Other Flow Features in the Plane

机译：旧（阶段）肖像的新帧：在平面中查找河流和其他流量功能

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摘要

In this work, we introduce a new change of coordinates, which we term local orthogonal rectification, or LOR, that can be applied at any selected curve in the phase space of a dynamical system. LOR, based on the Frenet frame, yields a coordinate system, the LOR frame, which allows us to rigorously study curves that are nearly invariant in the flow. We use the LOR approach to derive a novel definition for rivers, long-recognized but poorly understood trajectories that locally attract other orbits yet need not be related to invariant manifolds or other familiar phase space structures, and to identify rivers within several example systems. We show that our new definition of rivers connects nicely with the curvature of trajectories and that our approach provides a way to handle the "ghost" phenomenon of falsely identified candidate structures described in past analysis of the zero-curvature locus of a flow. We also apply the LOR approach to locate the periodic orbit in a FitzHugh-Nagumo system and to study its parameter dependence. Finally, we extend the approach to an asymptotic setting that, as we show, can be used to track invariant manifolds in multiple-timescale systems far from the limit of strong timescale separation.
机译：在这项工作中，我们引入了坐标的新变化，我们术语局部正交整流或者可以应用于动态系统的相位空间中的任何所选曲线。基于FRENET框架，LOR产生坐标系，LOR框架，这使我们能够严格地研究流量几乎不变的曲线。我们利用LOR方法导出河流的新型定义，长期识别但众所周知的轨迹，局部地吸引其他轨道，但不需要与不变的歧管或其他熟悉的相空间结构相关，并在几个示例系统中识别河流。我们表明，我们的河流的新定义与轨迹的曲率很好地连接，我们的方法提供了一种处理过去分析流动零曲率轨迹的错误识别候选结构的“Ghost”现象的方法。我们还应用LOR方法来定位Fitzhugh-Nagumo系统中的周期性轨道，并研究其参数依赖。最后，我们将方法扩展到渐近设置，正如我们所示，可以用来跟踪远期间系统中的不变歧管，远离强大的时间尺度分离的极限。

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