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

TL;DR

This paper investigates how saddle equilibria in smooth dynamical systems persist or transform into nonsmooth models with 'zombie saddle' behavior as the transition region shrinks, providing conditions for their persistence.

Contribution

It introduces conditions under which saddle equilibria in smooth models persist as nonsmooth 'zombie saddles' during the limit transition, extending understanding of nonsmooth dynamical systems.

Findings

Sufficient conditions for saddle persistence in nonsmooth limits

Existence of 'zombie saddle' behavior in nonsmooth models

Persistence of stable and unstable manifolds during transition

Abstract

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.