We analyse local aspects of chaos for nonautonomous periodic dynamical systems in the context of generating autonomous dynamical systems and the possibility of disturbing them.

In this paper we prove a realizability theorem for Quinn’s mapping cylinder obstructions for stratified spaces. We prove a continuously controlled version of the s-cobordism theorem which we further use to prove the relation between the torsion of an h-cobordism and the mapping cylinder obstructions. This states that the image of the torsion of an h-cobordism is the mapping cylinder obstruction of the lower stratum of one end of the h-cobordism in the top filtration. These results are further used to prove a theorem about the realizability of end obstructions.

Strata in manifold stratified spaces are shown to have neighborhoods that are teardrops of manifold stratified approximate fibrations (under dimension and compactness assumptions). This is the best possible version of the tubular neighborhood theorem for strata in the topological setting. Applications are given to replacement of singularities, to the structure of neighborhoods of points in manifold stratified spaces, and to spaces of manifold stratified approximate fibrations.