About the Shadowing Theorem for piecewise smooth vector fields with sliding motion

Abstract

Since every modeling process in real-world situations is subject to errors, the study of the so-called shadowing property becomes highly relevant. This property allows for the identification of true orbits that closely follow a chain of trajectories in which some degree of approximation is introduced at each step. The objective of this paper is to establish a version of the Shadowing Theorem for vector fields governed by ordinary differential equations in the piecewise-smooth setting, where the dynamics alternate between distinct regimes, or on-off stages. First, we present an example showing that such a result can not be obtained under the same hypotheses commonly assumed in the C2 scenario. Consequently, under appropriate assumptions, we prove a Shadowing-like Theorem specifically tailored to this framework. Furthermore, we propose several extensions of the main result aimed at capturing characteristic phenomena intrinsic to piecewise-smooth vector fields - features that do not arise in the C2 setting.