Geometric Desingularization of Higher Codimension Singularities in Fast-Slow Systems

多时间尺度动力系统有着广泛的应用，是奇异摄动微分方程数学分析的基石。在这里，我们建议研究的理论方面的两个时间尺度的动力学，所谓的快-慢系统，侧重于更高的余维奇点。本文提出的工作有两个主要的数学动机：一方面是根据奇异性对快慢系统进行抽象分类的问题，它起源于奇异性理论和Takens关于约束微分方程的工作;另一方面，在自适应网络上有新的和最近的发现，这些发现暗示了超越经典折叠、尖点、跨临界、等等。更准确地说，我们将开发的快速子系统的奇异点的快-慢系统的局部展开的部分，从两个不同的情况下产生：$D_4 ^\pm $-奇点（脐），和一个四维的快-慢Bogdanov-Takens奇点。此外，我们建议将从以前的研究中获得的知识应用到两种模型的快-慢自适应网络：第一种是一个快-慢的共识基序与三个动态权重，而第二种是与竞争网络有关。在这样的模型中，所涉及的奇点需要至少两个慢变量来进行动力学的完整分析，即，与经典的折叠奇点相比，它们具有更高的余维。我们将使用通过爆破的几何去奇异化作为我们的主要工具来理解每个奇点附近的轨迹和不变流形。这将与动力系统技术相结合，如稳定性理论，中心流形，渐近分析的特殊方程在缩放图，分歧理论和变分方程。该项目的成果将是大大扩展的快-慢奇点的展开结果，高维快-慢系统的新简化技术，以及对两个时间尺度自适应网络中奇点的作用和性质的更深入的几何理解。

Multiple time scale dynamical systems occur in a wide variety of applications and are a cornerstone in the mathematical analysis of singularly perturbed differential equations. Here we propose to study the theoretical aspects of two-time scale dynamics, so-called fast-slow systems, focusing on higher-codimension singularities. The work proposed here has two main mathematical motivations: on the one hand there is an abstract classification problem of fast-slow systems according to their singularities, which originates from singularity theory and Takens' work on constrained differential equations; on the other hand there are new and recent discoveries on adaptive networks that hint towards the great relevance of singularities beyond the classical fold, cusp, transcritical, etc\'etera. More precisely, we are going to develop parts of local unfoldings for fast-slow systems with singular points for the fast subsystem, arising from two different scenarios: $D_4^\pm$-singularities (umbilics), and a four-dimensional fast-slow Bogdanov-Takens singularity. Furthermore, we propose to apply the knowledge gained from the previous studies to two models of fast-slow adaptive networks: the first is a fast-slow consensus motif with three dynamic weights, while the second is concerned with a rivalry network. In such models, the singularities involved need at least two slow variables for the full analysis of the dynamics, i.e., they are of higher codimension in comparison to the classical folded singularities. We are going to use geometric desingularization via blow-up as our main tool to understand trajectories and invariant manifolds near each singularity. This will be combined with dynamical systems techniques such as stability theory, center manifolds, asymptotic analysis of special equations in scaling charts, bifurcation theory, and variational equations. The outcomes of the project are going to be substantially extended unfolding results for fast-slow singularities, new reduction techniques for higher-dimensional fast-slow systems, and a deeper geometric understanding of the role and properties of singularities in adaptive networks with two time scales.