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

分段光滑系统为许多实际应用提供了充分的数学描述，从微机械系统上的电力电子电路和A/D转换器，具有粘滑行为的冲击和摩擦主导系统到控制和通信系统。为了理解这些系统的可能行为，并通过适当的系统设计来实现期望的行为，有必要了解多维参数空间中发生的分叉结构。研究这些结构最有效的方法是检测它们的组织中心，组织中心由高余维分支给出。在分段光滑系统的情况下，这些分叉通常代表余维2大爆炸分叉，其中有无穷多个余维1分叉曲线。通过检测和识别它们，人们能够理解、预测并因此控制参数空间的扩展区域中的系统行为。在以前的项目中，我们检测和解释了几种类型的这种分叉在piecewisesmooth系统与一个不连续性，并证明了他们的共性。在拟议的项目中，它的目的是推导出导致这种分叉发生的一般条件。此外，它的目的是扩展的研究，在2D分段光滑系统的组织中心，因为许多应用导致这类系统。

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.