Theory and applications of symmetric bifurcation theory

我的研究项目是研究非线性动力学的理论与应用之间的相互作用，特别是分岔理论的应用。对称通常是使用微分方程进行数学建模的一个假设。它的起源有时是所研究的现象/方程所固有的，或者对称是模型上简化假设的一部分。我计划在三个主要应用领域中探索对称与非线性动力学之间的相互作用：动物聚集模型中的模式形成，对称耦合设备（如激光，陀螺仪）和哈密顿系统中的对称周期轨道。我的研究计划将以应用领域为跳板，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，从而实现我的研究目标。***从细胞到大型哺乳动物，都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遵循的速度要慢得多，我致力于在未来几年在这一领域做出重要贡献。特别是，对称性已经被证明是在一些偏微分方程模型中观察到的数值模式中的一个主要因素，我的研究专长非常适合继续在这个令人兴奋的知识领域做出重大贡献。随着对更强大设备需求的增长，网络已成为提高单个设备性能基本极限的流行替代方案。对称耦合是降低系统复杂性的一种方便的方法，有利于同步的出现。这些模型的分析需要使用对称非线性动力学，这是我研究计划的重要组成部分，利用我在这一领域的专业知识来进一步了解这些耦合系统。此外，我期望这些应用研究将带来挑战，从而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的研究已经被发现的新轨道（例如Hip-Hop，图8）所复兴。时间反转和空间对称的使用是这一新活动的一个重要方面。在n体问题中，对称周期轨道的稳定性和分岔问题很少引起人们的兴趣，还有许多问题有待解决。我的部分研究计划旨在描述线性和非线性稳定性，并利用对称、拓扑和数值工具探索分岔问题。* * * * * * * *

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.  ********