Organizing centers in discontinuous dynamical systems: bifurcations of higher codimension in theory and applications

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Piecewise-smooth systems provide an adequate mathematical description of many practical applications ranging from power electronic circuits and A/D converters over micro mechanical systems, impacting and friction dominated systems with stick-slip behavior up to control and communication systems. To understand possible behaviors of these systems and to achieve a desired behavior via a suitable system design, it is necessary to understand the bifurcation structures occurring in multi-dimensional parameter spaces. The most e_cient way to investigate these structures is to detect their organizing centers, which are given by bifurcations of higher codimension. In the case of piecewise-smooth systems these bifurcations represent often codimension-2 big bang bifurcations, where an in_nite number of codimension-1 bifurcations curves are issuing from. By detecting and identifying them one is able to understand, to predict and therefore also to control the system behavior in extended regions of the parameter space. In the previous project we detected and explained several types of such bifurcations in piecewisesmooth systems with one discontinuity and demonstrated their generic character. In the proposed project it is intended to derive generic conditions causing such bifurcations to occur. Furthermore it is intended to extend the research to the organizing centers in 2D piecewise-smooth systems because many applications lead to that class of systems.