Theory and applications of symmetric bifurcation theory

对称分岔理论的理论与应用

• 批准号: RGPIN-2015-06396
• 负责人: Buono, PietroLuciano
• 金额: $ 1.24万
• 依托单位: University of Ontario Institute of Technology
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675857
• 关键词: Theory; applications; symmetric; bifurcation; theory

The research program I am pursuing consists in investigating the interaction between theory and applications of nonlinear dynamics with symmetry, in particular the use of bifurcation theory. Symmetry is often an assumption in mathematical modelling using differential equations. Its origin is sometimes intrinsic to the phenomena/equations under study or symmetry appears as part of simplifying assumptions on the model. I am planning to explore the interaction between symmetry and nonlinear dynamics in the context of three major application areas: pattern formation in animal aggregation models, symmetrically coupled devices (e.g lasers, gyroscopes) and symmetric periodic orbits in Hamiltonian systems. The goals of my research program will be achieved by using application areas as a springboard to develop new tools in nonlinear dynamics with symmetry, and by using existing and new theory to contribute significantly to the application area.***The phenomenon of animal aggregation of individuals of the same species are observed from cells to large mammals and is recently the focus of major mathematical modelling efforts to complement a large body of experimental and empirical data. However, the theoretical basis and methods for the study of several of the patterns emerging from the models follows at a much slower pace and I am committed in the coming years to make important contributions in this area. In particular, symmetry has been shown to be a major factor in the patterns observed numerically in some partial differential equation models and my research expertise is ideally suited to continue making significant contributions in this exciting area of knowledge. ***As the need for more powerful devices grows, networks have become popular alternatives to advance the fundamental limits of performance of an individual unit. Symmetrically coupling the devices is a convenient way to reduce the complexity of the system and favours the emergence of synchronization. The analysis of those models calls for the use of symmetric nonlinear dynamics and it is an important part of my research program to use my expertise in this domain to further the understanding of these coupled systems. Moreover, I expect that these application studies will lead to challenges which will stimulate the emergence of new theoretical results. ******The search for collisionless periodic orbits in the Newtonian N-body problem has been revived by the finding of new orbits (e.g. Hip-Hop, Figure-Eight). The use of time-reversing and spatial symmetries has been an important aspect of this renewed activity. The topic of stability and bifurcations of periodic orbits with symmetry in the N-body problem received less interest and many questions still lie open. Part of my research program seeks to characterize linear and nonlinear stability and explore bifurcation issues using symmetric, topological and numerical tools.  ********

我正在从事的研究项目包括研究对称性非线性动力学的理论和应用之间的相互作用，特别是分叉理论的使用。对称性通常是使用微分方程进行数学建模的假设。它的起源有时是内在的现象/方程的研究或对称性出现的一部分，简化假设的model. I计划探索对称性和非线性动力学之间的相互作用在三个主要应用领域的背景下：图案形成在动物聚集模型，对称耦合设备（如激光，陀螺仪）和对称周期轨道的哈密顿系统。我的研究计划的目标将通过使用应用领域作为跳板来开发具有对称性的非线性动力学的新工具，并通过使用现有的和新的理论来为应用领域做出重大贡献来实现。从细胞到大型哺乳动物，都可以观察到同一物种个体的动物聚集现象，这是最近主要数学建模工作的重点，以补充大量的实验和经验数据。然而，从模型中出现的几种模式的研究的理论基础和方法的速度要慢得多，我致力于在未来几年在这一领域作出重要贡献。特别是，对称性已被证明是在一些偏微分方程模型中数值观察到的模式的一个主要因素，我的研究专长非常适合继续在这一令人兴奋的知识领域做出重大贡献。* 随着对更强大设备的需求不断增长，网络已成为推动单个单元性能基本极限的流行替代方案。对称耦合的设备是一种方便的方式，以减少系统的复杂性，有利于同步的出现。这些模型的分析要求使用对称非线性动力学，这是我的研究计划的一个重要组成部分，利用我在这一领域的专业知识，以进一步了解这些耦合系统。此外，我预计这些应用研究将导致挑战，这将刺激新的理论结果的出现。** 在牛顿N体问题中寻找无碰撞周期轨道已经恢复了新轨道的发现（例如，Hip-Hop，图8）。时间反演和空间对称性的使用是这一新活动的一个重要方面。在N体问题中具有对称性的周期轨道的稳定性和分叉的主题受到较少的关注，许多问题仍然悬而未决。我的研究计划的一部分，旨在描述线性和非线性稳定性，并探讨分叉问题，使用对称，拓扑和数值工具。 ********