Persistence of Saddle Behavior in the Nonsmooth Limit of Smooth Dynamical Systems

Models such as those involving abrupt changes in the Earth's reflectivity due to ice melt and formation often use nonlinear terms (e.g., hyperbolic tangent) to model the transition between two states. For various reasons, these models are often approximated by "simplified" discontinuous piecewise-linear systems that are obtained by letting the length of the transition region limit to zero. The smooth versions of these models may exhibit equilibrium solutions that are destroyed as parameter changes transition the models from smooth to nonsmooth. Using one such model as motivation, we explore the persistence of local behavior around saddle equilibria under these transitions. We find sufficient conditions under which smooth models with saddle equilibria can become nonsmooth models with "zombie saddle" behavior, and analogues of stable and unstable manifolds persist.