Key Words: Differentil topology; smooth manifolds; smooth maps; singularities; matrix computations; regularization.
Abstract. This survey briefly reviews some fundamental concepts of differential topology and discusses their application to the solution of problems arising in engineering disciplines. The use of manifolds to represent families of mathematical models makes it possible to describe parameter-dependent problems and to study their global properties. The study of mappings between manifolds includes the investigation of singular properties of these mappings at certain points of the manifolds. Although such points are not typical, they can have a significant influence on the behavior of the corresponding mapping. The application of manifolds and the associated bifurcation diagrams in parameter space to the analysis of parameter-dependent problems is presented in brief. Examples are provided to illustrate the introduced concepts.
