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

非线性动力系统理论发展的第一步主要是针对具有光滑系统函数的模型。然而，在自然科学和工程中的各种应用可能在不同的制度下运行，这导致我们考虑分段平滑（PWS）模型。在这样的系统中，模型的相空间被细分为多个分区，其中动力学由不同的向量场控制，通过切换流形相互分离。不变量集与这些流形的相互作用导致了许多从数学角度来看很有趣的效应，并且对于建模系统的期望行为很重要。目前，PWS系统理论对许多此类现象提供了详细的描述，但主要是针对具有单个开关流形的系统。这是一个必要的中间步骤，拟议项目的目标是消除其局限性。电力电子学，处理控制电能流动的开关电路，是PWS系统理论的一个既定应用。对于一类特殊的功率变换器（DC/DC变换器），现有的PWS系统分岔理论是足够的，有助于选择参数设置，从而达到期望的工作模式，并预测可能出现的不期望的动态效应。相比之下，对于与可再生能源和电动汽车相关的一些应用中所固有的DC/AC转换器，这种分析就更加困难了。之前，我们已经证明了这些系统导致了一类以极高数量的开关流形为特征的新型PWS模型。导致这类模型的功率转换器的特定属性是它们的动态由两个截然不同的固定频率控制。在项目的后续研究中，我们将扩展我们的研究领域，重点研究在DC/AC和AC/DC转换器模型中发生的有趣的分岔现象。我们将描述在其多维参数空间中发生的分岔结构的一般组织原则。利用具有极高数量开关流形的映射框架，我们将重点研究最具挑战性的现象，如边界碰撞分岔，冒泡和具有共振和准周期动力学的闭不变曲线的变换。我们将扩展建模方法，使其适用于具有可变高频的系统。我们将把这种扩展应用到具有滞后控制的系统中，并研究边界碰撞在具有这种控制的DC/AC和AC/DC变换器中发生的分岔结构形成中的作用。因此，本项目的继续研究将有助于PWS动力系统理论的进步，以及可再生能源和自主系统等应用的电力电子技术的发展。

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.