Sliding vector fields for non-smooth dynamical systems having intersecting switching manifolds

We consider a differential equation p-dot =X(p), p∈R³, with discontinuous right-hand side and discontinuities occurring on a set Σ. We discuss the dynamics of the sliding mode which occurs when, for any initial condition near p ∈ Σ, the corresponding solution trajectories are attracted to Σ. Firstly we suppose that Σ = H⁻¹(0), where H is a smooth function and 0∈R is a regular value. In this case Σ is locally diffeomorphic to the set F={(x,y,z)∈R³;z=0}. Secondly we suppose that Σ is the inverse image of a non-regular value. We focus our attention to the equations defined around singularities as described in Gutierrez and Sotomayor (1982 Proc. Lond. Math. Soc 45 97–112). More precisely, we restrict the degeneracy of the singularity so as to admit only those which appear when the regularity conditions in the definition of smooth surfaces of R³ in terms of implicit functions and immersions are broken in a stable manner. In this case Σ is locally diffeomorphic to one of the following algebraic varieties: D={(x,y,z)∈R³;xy=0} (double crossing); T={(x,y,z)∈R³;xyz=0} (triple crossing); C={(x,y,z)∈R³;z²−x²−y²=0} (cone) or W={(x,y,z)∈R³;zx²−y²=0} (Whitney's umbrella). (paper)