Mathematical Sciences: Bifurcation and Symmetry

研究人员继续研究对称性对微分方程解的影响。三个特殊的主题是异宿周期的稳定性(这与间歇和爆发现象的存在有关)，混沌吸引子的对称性(这可能与在湍流中观察到的一些独特的模式有关)，以及规范和空间对称性的耦合(如在Ginzburg-Landau超导模型中发生的)。他们还将尝试将他们的结果与具体的实验联系起来。他们认为，在Couette-Taylor实验中已经观察到了与异宿循环相关的爆发现象和混沌吸引子的对称性增加分叉；他们的部分工作将使这些联系更加精确。许多物理现象的模型，包括各种主题，如流体动力学中从传导到对流的转变，以及动物行走的方式，在它们的公式中都包含对称性。众所周知，这些对称性会影响在这类模型中找到的解的种类。大多数以前的工作都集中在出现的各种均衡和周期解上。这个项目致力于扩大所研究的解决方案的类别，以包括对称系统中的混沌和间歇等主题。最近的研究和实验表明，这既是合理的，也是容易处理的。

The investigators continue studies of the effect that symmetry has on the solutions of differential equations. Three particular topics are the stability of heteroclinic cycles (which is related to the existence of intermittency and bursting phenomena), the symmetry of chaotic attractors (which may be related to some of the distinctive patterns observed in turbulent fluid flow), and the coupling of gauge and spatial symmetries (such as occurs in the Ginzburg-Landau model for superconductivity). They will also attempt to relate their results to specific experiments. They believe that both the bursting phenomena associated with heteroclinic cycles and the symmetry-increasing bifurcations of chaotic attractors have been observed in the Couette-Taylor experiment; part of their work will be to make these connections more precise. Many models of physical phenomena including topics as diverse as the transition from conduction to convection in fluid dynamics and the ways in which animals walk contain symmetries in their formulation. These symmetries are known to influence the kinds of solutions that are found in such models. Most previous work has focused on the kinds of equilibria and periodic solutions that appear. This project undertakes to broaden the class of solutions that are studied to include such topics as chaos and intermittency in symmetric systems. Recent research and experiments indicate that this is both reasonable and tractable.