Generic bifurcation structures in piecewise-smooth maps with extremely high numberof borders in theory and applications for power converter systems

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The first steps in the development of theory of nonlinear dynamical systems have been made mainly for models with a smooth system function. However, various applications both in natural sciences and in engineering may operate in different regimes, which lead us to consider piecewise smooth (PWS) models. In such systems, the phase space of the model is subdivided into partitions where the dynamics is governed by different vector fields, separated from each other by switching manifolds. Interactions of invariant sets with these manifolds lead to numerous effects both interesting from a mathematical point of view and important for a desired behavior of the modeled system. At present, the theory of PWS systems provides a detailed description for many of such phenomena, but mainly for systems with a single switching manifold. This is a necessary but intermediate step and the goal of the proposed project is to outcome its limitations.Power electronics, dealing with switching circuits controlling the flow of electrical energy, is an established application of PWS systems theory. For a special class of power converters (DC/DC converters), the existing bifurcation theory of PWS systems is sufficient, helping to select parameter settings leading to desired mode of operation and to predict possible undesired dynamic effects. By contrast, for DC/AC converters, which are an inherent part of several applications related to renewable energy sources and electric cars, such kind of analysis is more difficult. Previously, we have shown that these systems lead to a novel class of PWS models characterized by an extremely high number of switching manifolds. The specific property of power converters leading to this class of models is that their dynamics is governed by two vastly different fixed frequencies.In the proposed continuation of the project, we will extend our research field, focusing on investigation of intriguing bifurcation phenomena occurring in models of DC/AC and AC/DC converters. We will describe the generic organizing principles of the bifurcation structures occurring in their multi-dimensional parameter spaces. Using the framework of maps with an extremely high number of switching manifolds, we will focus our research on the most challenging phenomena, such as border collision bifurcations, bubbling, and transformation of closed invariant curves with resonant and quasiperiodic dynamics. We will extend the modeling approach, making it applicable for systems with a variable high frequency. We will apply this extension to systems with hysteresis control and investigate the role of border collisions in formation of bifurcations structures occurring in DC/AC and AC/DC converters with this type of control. In this way, the proposed continuation of project will contribute to the progress of the theory of PWS dynamical systems and to development of power electronics for such applications as renewable energy sources and autonomous systems.