  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 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. 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.
