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The asymptotic phase $\theta$ of an initial point $x$ in the stable manifoldof a limit cycle identifies the phase of the point on the limit cycle to whichthe flow $\phi_t(x)$ converges as $t\to\infty$. The infinitesimal phaseresponse curve (iPRC) quantifies the change in timing due to a smallperturbation of a limit cycle trajectory. For a stable limit cycle in a smoothdynamical system the iPRC is the gradient $\nabla_x(\theta)$ of the phasefunction, which can be obtained via the adjoint of the variational equation.For systems with discontinuous dynamics, the standard approach to obtaining theiPRC fails. We derive a formula for the infinitesimal phase response curves(iPRCs) of limit cycles occurring in piecewise smooth (Filipov) dynamicalsystems of arbitrary dimension, subject to a transverse flow condition.Discontinuous jumps in the iPRC can occur at the boundaries separatingsubdomains, and are captured by a linear matching condition. The matchingmatrix, $M$, can be derived from the saltation matrix arising in the associatedvariational problem. For the special case of linear dynamics away fromswitching boundaries, we obtain an explicit expression for the iPRC. We presentexamples from cell biology (Glass networks) and neuroscience (central patterngenerator models). We apply the iPRCs obtained to study synchronization andphase-locking in piecewise smooth limit cycle systems in which synchronizationarises solely due to the crossing of switching manifolds.

The Infinitesimal Phase Response Curves of Oscillators in Piecewise Smooth Dynamical Systems