Stable and unstable sets of saddles in continuous and discontinuous piecewise smooth maps: Structure and computation

• Algorithm computing stable/unstable 1D sets of saddles in piecewise-smooth maps • Stable/unstable sets in discontinuous maps consist of disconnected segments • Virtual values detect segment endpoints and arbitrarily sharp kink points precisely • Handles continuous/discontinuous and invertible/non-invertible maps uniformly • Discontinuous invariant curves exhibit unusual geometric and dynamic properties Stable and unstable one-dimensional manifolds (or sets) of saddles play a fundamental role in understanding global dynamics in both smooth and piecewise-smooth systems. They are essential for analyzing basin boundaries and chaotic dynamics. Several algorithms developed for smooth maps compute these manifolds numerically and yield sufficiently accurate results for this class of systems. However, when applied to continuous piecewise-smooth maps, these methods often suffer accuracy problems near sharp kink points of the stable and unstable sets, as well as near critical curves in non-invertible maps. Moreover, for discontinuous piecewise-smooth maps, these methods typically fail due to the presence of gaps and disconnected segments in the sets. In this work, we propose an algorithm specifically designed for computing stable and unstable sets in piecewise-smooth maps. Our method extends traditional approaches by incorporating virtual values of the smooth functions that compose the piecewise-smooth map, i.e., values obtained by applying these functions beyond their designated domains. For maps with several borders, regardless of whether these borders are defined explicitly or implicitly, and regardless of whether the map is continuous or discontinuous along these borders, we achieve results with arbitrary low longitudinal error near kink points or segment endpoints of stable and unstable sets. Moreover, the proposed extension unifies the calculation procedures for unstable and stable sets in both invertible and non-invertible maps. We discuss key implementation aspects, parameter tuning strategies, and challenges related to computational accuracy and efficiency. We illustrate the algorithm by discussing several examples of discontinuous invariant curves, i.e., invariant sets in discontinuous maps which play a role analogous to that of closed invariant curves in smooth and piecewise-smooth continuous maps.