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

We consider a differential equation p over dot = X(p), p is an element of R-3, with discontinuous right-hand side and discontinuities occurring on a set Sigma. We discuss the dynamics of the sliding mode which occurs when, for any initial condition near p is an element of Sigma, the corresponding solution trajectories are attracted to Sigma. Firstly we suppose that Sigma = H-1(0), where H is a smooth function and 0 is an element of R is a regular value. In this case Sigma is locally diffeomorphic to the set F = {(x, y, z) is an element of R-3; z = 0}. Secondly we suppose that Sigma 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 R-3 in terms of implicit functions and immersions are broken in a stable manner. In this case Sigma is locally diffeomorphic to one of the following algebraic varieties: D = {(x, y, z) is an element of R-3; xy = 0} (double crossing); T = {(x, y, z) is an element of R-3; xyz = 0} (triple crossing); C = {(x, y, z) is an element of R-3; z(2) -x(2)-y(2) = 0} (cone) or W = {(x, y, z) is an element of R-3; zx(2)-y(2) = 0} (Whitney's umbrella).