Slow–fast systems and sliding on codimension 2 switching manifolds

Abstract

In this work, we consider piecewise smooth vector fields X defined in R n \ ∑, where Σ is a self-intersecting switching manifold. A double regularization of X is a 2-parameter family of smooth vector fields X ε.η , ε,η > 0 satisfying that X ε,η converges uniformly to X in each compact subset of R n \ ∑ when ε, η → 0. We define the sliding region on the non-regular part of Σ as a limit of invariant manifolds of X ε.η . Since the double regularization provides a slow–fast system, the GSP-theory (geometric singular perturbation theory) is our main tool.