Mathematical Sciences: Stability and Dynamics of Parametrically Driven Waves

在这个项目中，首席研究员将研究在一些物理系统中由于时间周期性驱动力的作用而产生的波的结构。这项研究将沿着两个方向进行。首先，首席研究员将研究如何通过空间对称性的影响来平衡时间驱动在激励系统中的作用。这将需要使用一些最新的对称分岔理论方法来推导和分析振幅方程。其次，首席研究员将研究大纵横比系统中参数驱动波的动力学。这些波表现出不同类型的不稳定性，可能导致时空混乱。对这种混沌行为的过渡将进行分析和数值研究。

In this project the principal investigator will study the structure of waves that arise in a number of physical systems through the action of temporally periodic driving forces. The research will follow two directions. First of all, the principal investigator will study how the effect of temporal driving in exciting the system is balanced by the influence of spatial symmetries. This will entail using some recent methods of bifurcation theory with symmetry to derive and analyze the amplitude equations. Secondly, the principal investigator will study the dynamics of parametrically driven waves in large aspect-ratio systems. Such waves exhibit different types of instabilities that can lead to spatio-temporal chaos. The transitions to this chaotic behavior will be studied analytically as well as numerically.