SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 58-84, January 2020.
A recently introduced technique, local orthogonal rectification (LOR), provides a way to derive coordinate systems that are tailored to locating objects of interest within flows in arbitrary dimensions. In this work, we apply LOR to identify periodic orbits and study the transient dynamics nearby. In the LOR method, the standard approach of finding periodic orbits by solving for fixed points of a Poincaré return map is replaced by the solution of a boundary value problem with fixed endpoints, and the computation provides information about the stability of the identified orbit. We detail the general method and derive theory to show that once a periodic orbit has been identified with LOR, the LOR coordinate system allows us to characterize the stability of the periodic orbit, to continue the orbit with respect to system parameters, to identify invariant manifolds attendant to the periodic orbit, and to compute the asymptotic phase associated with points in a neighborhood of the periodic orbit in a novel way. All of this analysis can be done in a computationally synergistic manner within the “one-stop shop” of the LOR framework. We illustrate these ideas, along with the importance of the invariant manifolds for organizing the flow in the approach to a stable periodic orbit in $\mathbb{R}^3$, using the Goodwin oscillator and a polynomial system featuring a period-doubling bifurcation.

中文翻译：

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用于周期性动力学分析的LOR：一站式服务

SIAM应用动力系统杂志，第19卷，第1期，第58-84页，2020年1月。
最近引入的一种技术，即局部正交校正（LOR），提供了一种导出坐标系的方法，该坐标系专门用于在任意尺寸的流中定位感兴趣的对象。在这项工作中，我们将LOR应用于识别周期性轨道并研究附近的瞬态动力学。在LOR方法中，通过求解庞加莱返回图的固定点来查找周期轨道的标准方法被具有固定端点的边值问题的求解所替代，并且该计算提供了有关已识别轨道的稳定性的信息。我们详细介绍了一般方法并推导了理论，以表明一旦用LOR识别了周期性轨道，LOR坐标系就使我们能够表征周期性轨道的稳定性，并根据系统参数继续轨道，以确定与周期轨道相关的不变流形，并以新颖的方式计算与周期轨道附近的点相关的渐近相位。所有这些分析都可以在LOR框架的“一站式服务”内以计算协同的方式完成。我们使用古德温振荡器和具有倍增周期分叉的多项式系统，阐明了这些思想，以及不变流形对于组织到\ mathbb {R} ^ 3 $的稳定周期轨道的流动组织流动的重要性。 。