Solutions leaving a codimension-2 sliding
Nicola Guglielmi and Ernst Hairer
Abstract. Piece-wise smooth differential
equations (their regularization, numerical integration, and classification of
solutions) is the topic of the present work.
The behaviour close to one
discontinuity surface and also the entering into the intersection of two
discontinuity surfaces is well understood. Here, we study the solutions that
exit a codimension-$2$ sliding mode. Some results are expected, others come
as a surprise. We are able to explain situations, where
difficulties in numerical computations are reported in the recent literature.
The analysis is based on asymptotic
expansions for singularly perturbed problems and on the study of a
time-parameterized two-dimensional dynamical system (hidden dynamics).
The various situations are illustrated by examples.
Key Words. Piece-smooth dynamics, discontinuous differential equations,
regularization, singular perturbation, asymptotic expansions.