Dynamics of bifurcations with broken reflection symmetry in finite domains

在这项申请中，我们请求资助一名博士后研究助理，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是不稳定的。当系统的反射对称性被破坏时，就会发生这种情况。有许多这样的系统的物理例子，包括流体动力学中的剪切流动不稳定性，流动存在时的反应扩散系统，以及旋转系统中的不稳定性，如MHD发电机不稳定性。过去，我们利用动力系统的技术和最新的数值方法，在理解这类系统方面取得了重大进展。拟议的研究将把我们的理解扩展到三个新的和重要的案例。第一种情况发生在波数为非零时的不稳定性，如Kuramoto-Sivashinsky方程。这里存在着波数选择机制在非线性区域中的竞争。第二种情况着眼于空间不均匀性对这些行波不稳定性的影响，要么是通过包含共振项(如果不均匀性是在载波尺度上的)，要么是通过引入不同的背景状态(如果不均匀性是在调制长度尺度上)。在这些情况下，对称性破缺预计将显着改变系统的行为，对称性破缺和钉扎之间的竞争是由不均匀引起的。在最后一种情况下，我们研究具有全局约束的系统，该约束通过上游和下游边界的耦合来表现出来，以理解这种耦合如何导致整个域中扰动的循环。

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.