Dynamics of bifurcations with broken reflection symmetry in finite domains

在这个应用程序中，我们请求支持资助博士后研究助理，以探索在有界系统中发现的复杂动力学，这些系统对具有首选方向的行波不稳定。当系统具有破缺的反射对称性时，这些会发生。这类系统有许多物理实例，包括流体动力学中的剪切流不稳定性、存在流动的反应扩散系统以及旋转系统中的不稳定性，如MHD发电机不稳定性。在过去，我们已经取得了重大进展，了解这样的系统使用技术从动力系统和最新的数值方法。拟议的研究将把我们的理解扩展到三个新的重要案例。第一种情况是当不稳定性以一个优选的非零波数出现时，例如Kuramoto-Sivashinsky方程。在非线性区域存在波数选择机制之间的竞争。第二种情况下着眼于这些行波不稳定性的空间不均匀性的效果，无论是通过包含一个谐振项（如果不均匀性是在载波的规模）或引入一个不同的背景状态（如果不均匀性是在调制的长度尺度）。对于这些情况下，对称性破缺预计会显着改变系统的行为与对称性破缺和钉扎之间的竞争的不均匀性。在最后一种情况下，我们检查系统的全局约束，体现自己通过耦合的上游和下游的边界，以了解这样的耦合可以导致整个域的干扰再循环。

In this application we request support to fund a post-doctoral research assistant to explore the complex dynamics found in bounded systems that are unstable to travelling waves with a preferred direction. These occur when the system has a broken reflection symmetry. There are many physical examples of such systems including shear-flow instabilities in fluid dynamics, reaction diffusion systems in the presence of flow, and instabilities in rotating systems such as the MHD dynamo instability. In the past we have made significant progress in understanding such systems using techniques from dynamical systems and the latest numerical methods. The proposed research would extend our understanding to three new and important cases. The first arises when the instability sets in with a preferred non-zero wavenumber, such as for the Kuramoto-Sivashinsky equation. Here there exists a competition between wavenumber selection mechanisms in the nonlinear regime. The second case looks at the effect of spatial inhomogeneities on these travelling wave instabilities either through the inclusion of a resonant term (if the inhomogeneity is on the scale of the carrier wave) or the introduction of a varying background state (if the inhomogeneity is on the modulational lengthscale). For these cases the symmetry breaking is expected to alter significantly the behaviour of the system with a competition between the symmetry breaking and the pinning by the inhomogeneity. In the final case we examine systems with a global constraint that manifests itself through coupling of the upstream and downstream boundaries to understand how such a coupling can lead to recycling of disturbances throughout the domain.