The theory or nonlinear dynamical systems was initially developed mainly for models with smooth system function. Since mid of 1900th non-smooth models became a focus or research interest as they provide an adequate description for many systems both in the nature and in engineering sciences. In the last twenty years it was shown that these systems possess many phenomena which can not occur in smooth systems. These phenomena are caused by the presence of so-called switching manifolds in the state space and their interactions with several invariant sets. However, until now mainly systems with one switching manifold were investigated. This represents an important intermediate step but is often not sufficient for explanation of the behaviour of many systems of practical interest. To understand possible behaviours of these systems and to achieve a desired behaviour via a suitable system design, it is necessary to understand the bifurcation structures occurring in multi-dimensional parameter spaces. For the bifurcation structures caused by interactions of invariant sets with one switching manifold, many valuable results were obtained in the last time. The proposed project addresses the following question: What are the basic principles organizing the bifurcation structures in low-dimensional maps with more than one switching manifold? The goal of the proposed project is to explain some generic bifurcation structures characteristic for these systems using a suitable extension of some novel investigation techniques developed for maps with one switching manifold with the concept of organizing centres in multi-dimensional parameter spaces. The practical relevance of the obtained results will be demonstrated by applications not only from the Engineering Science but mainly from the fields of Economics, Financial Modeling and Social Sciences.