SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 701-744, January 2021.
When dynamical systems that produce rhythmic behaviors operate within hard limits, they may exhibit limit cycles with sliding components, that is, closed isolated periodic orbits that make and break contact with a constraint surface. Examples include heel-ground interaction in locomotion, firing rate rectification in neural networks, and stick-slip oscillators. In many rhythmic systems, robustness against external perturbations involves response of both the shape and the timing of the limit cycle trajectory. The existing methods of infinitesimal phase response curve (iPRC) and variational analysis are well established for quantifying changes in timing and shape, respectively, for smooth systems. These tools have recently been extended to nonsmooth dynamics with transversal crossing boundaries. In this work, we further extend the iPRC method to nonsmooth systems with sliding components, which enables us to make predictions about the synchronization properties of weakly coupled stick-slip oscillators. We observe a new feature of the isochrons in a planar limit cycle with hard sliding boundaries: a nonsmooth kink in the asymptotic phase function, originating from the point at which the limit cycle smoothly departs the constraint surface and propagating away from the hard boundary into the interior of the domain. Moreover, the classical variational analysis neglects timing information and is restricted to instantaneous perturbations. By defining the “infinitesimal shape response curve" (iSRC), we incorporate timing sensitivity of an oscillator to describe the shape response of this oscillator to parametric perturbations. In order to extract timing information, we also develop a “local timing response curve" (lTRC) that measures the timing sensitivity of a limit cycle within any given region. We demonstrate in a specific example that taking into account local timing sensitivity in a nonsmooth system greatly improves the accuracy of the iSRC over the global timing analysis given by the iPRC.

中文翻译：

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形状与时序：瞬时和静态扰动下具有硬边界的极限环的线性响应

SIAM Journal on Applied Dynamical Systems，第 20 卷，第 2 期，第 701-744 页，2021 年 1 月。
当产生节奏行为的动力系统在硬限制内运行时，它们可能会表现出带有滑动分量的极限循环，即与约束表面建立和断开接触的封闭孤立周期轨道。示例包括运动中的脚跟-地面相互作用、神经网络中的发射率校正和粘滑振荡器。在许多节奏系统中，对外部扰动的鲁棒性涉及极限循环轨迹的形状和时间的响应。现有的无限小相位响应曲线 (iPRC) 和变分分析方法已被很好地建立起来，分别用于量化平滑系统的时间和形状变化。这些工具最近已扩展到具有横向交叉边界的非光滑动力学。在这项工作中，我们进一步将 iPRC 方法扩展到具有滑动组件的非光滑系统，这使我们能够预测弱耦合粘滑振荡器的同步特性。我们在具有硬滑动边界的平面极限环中观察到等时线的一个新特征：渐近相函数中的非光滑扭结，起源于极限环平滑地离开约束面并从硬边界传播到域的内部。此外，经典变分分析忽略了时间信息并且仅限于瞬时扰动。通过定义“无穷小形状响应曲线”(iSRC)，我们结合了振荡器的时间灵敏度来描述该振荡器对参数扰动的形状响应。