Shape versus Timing: Linear Responses of a Limit Cycle with Hard Boundaries under Instantaneous and Static Perturbation
Shape versus Timing: Linear Responses of a Limit Cycle with Hard Boundaries under Instantaneous and Static Perturbation
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DOI：
10.1137/20m1344974
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发表时间：
2019-06
影响因子：
2.1
通讯作者：
Yangyang Wang;Jeffrey P. Gill;H. Chiel;P. Thomas
中科院分区：
文献类型：
--
作者：
Yangyang Wang;Jeffrey P. Gill;H. Chiel;P. Thomas
当产生节律行为的动力系统在硬极限内运行时，它们可能会表现出具有滑动分量的极限环，即与约束表面接触和断开接触的封闭孤立周期轨道。例子包括运动中的脚跟-地面相互作用，神经网络中的发射率整流和粘滑振荡器。在许多节律系统中，对外部扰动的鲁棒性包括对极限环轨迹的形状和时间的响应。现有的无穷小相位响应曲线（iPRC）和变分分析的方法，以及建立量化的时间和形状的变化，分别为光滑的系统。这些工具最近被扩展到非光滑动力学与横向交叉边界。在这项工作中，我们进一步扩展iPRC方法的非光滑系统的滑动组件，这使我们能够预测弱耦合粘滑振子的同步特性。我们观察到一个新的功能的等时线在一个平面的极限环与硬滑动边界：非光滑扭结的渐近相函数，起源于点的极限环顺利离开约束表面，并传播远离硬边界到内部的域。此外，经典的变分分析忽略了时间信息，并限于瞬时扰动。通过定义“无限小形状响应曲线”（iSRC），我们将振荡器的定时灵敏度来描述这个振荡器的形状响应参数扰动。为了提取时序信息，我们还开发了一个“本地时序响应曲线”（lTRC），在任何给定的区域内的极限环的时序灵敏度的措施。我们证明在一个具体的例子中，考虑到在非光滑系统的局部时序灵敏度大大提高了精度的iSRC全球时序分析的iPRC。
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 global timing analysis given by the iPRC.