The Dynamics of Driven Nonlinear Systems

9972022 Spiegel在这个项目中，我们将研究三种类型的参数驱动系统：（1）周期驱动的常微分方程，（2）具有时间依赖边界条件的热对流偏微分方程和（3）具有时间依赖控制参数的直线映射。这项工作的目的是深入了解环境变化使自然系统行为复杂化的方式。 在常微分方程的情况下，我们将考虑一个周期驱动的非线性振荡器，表现出混沌。 我们的目标是发展等价的Shil'nikov理论，这样的非自治方程。 隐含流不动点的缺乏给这种方法带来了有趣的困难。 为了研究边界温度随时间变化的对流方程，我们将使用模态展开程序，该程序可以仔细考虑问题的垂直结构，并且我们将根据直接数值模拟检查我们的结果。 这个过程将允许我们包括复杂的效果，如旋转。 我们对非自治地图的探索将集中在它们所能表现出的不连续性的性质上。 在这样的系统中，盆地的边界可能变得千疮百孔，我们将把这种现象的开始与没有耗散的类似情况进行比较，当康托里形成时，最简单的混沌系统一遍又一遍地做同样的事情，但从来不会以同样的方式做两次。 它们行为的细节很容易被外部扰动所改变，无论多么微小。 然而，当外部扰动持续存在时，它们可以导致受扰系统戏剧性地切换其整体行为模式，如著名的水滴折磨所示。 微弱但混乱的强迫可能会导致不稳定的活动爆发，并在一个被称为“开-关不稳定性”的过程中被平静期打断。 太阳活动的极小值每隔几个世纪发生一次，并持续数十年，说明了这种行为。我们希望深入了解持续的外部扰动的反应如何根据扰动是规则的还是不规则的时间而变化。 我们已经知道，地球气候的简单模型，当受到太阳输出持续轻微波动的影响时，可以显示全球条件的波动。 我们的目标是通过研究扰动混沌系统的易处理模型来加深我们对这种影响的数学理解。 这些研究的结果可能为气候变化和人口动态等复杂现象的大规模数值模拟提供指导。

9972022SpiegelIn this project we shall study three kinds of parametrically driven systems: (1) a periodically driven, ordinary differential equation, (2) the partial differential equations of thermal convection with time-dependent boundary conditions and (3) maps of the line with time-dependent control parameters. The aim of this work is to gain insight into the ways that environmental variations complicate the behavior of natural systems. In the case of the ODE, we shall consider a periodically driven nonlinear oscillator that exhibits chaos. We aim to develop the equivalent of the Shil'nikov theory for such nonautonomous equations. The absence of fixed points of the implied flow presents interesting difficulties for this approach. To investigate the convection equations with time-dependent boundary temperatures, we shall use a modal expansion procedure that allows careful accounting of the vertical structure of the problem and we shall check our results against direct numerical simulation. This procedure will allow us to include complicating effects like rotation. Our exploration of nonautonomous maps will focus on the nature of the intermittency that they can exhibit. In such systems the basin boundaries may become riddled and we shall compare the onset of this phenomenon to the analogous case without dissipation when cantori form.The simplest chaotic systems do the same things over and over again but never the same way twice. Details of their behavior can easily be modified by external perturbations, no matter how small. However, when the external perturbations are persistent, they can cause the disturbed systems to switch their overall behavior patterns dramatically, as illustrated by the famous water-drop torture. Weak but chaotic forcing can lead to erratic outbursts of activity punctuated by calm periods in a process known as on-off intermittency. The minima in solar activity that occur every few centuries and last for decades illustrate this behavior. We wish to gain insight into how responses to persistent external perturbations vary according to whether the perturbations are regular or irregular in time. We know already that simple models of the earth's climate, when subjected to the effect of continued slight fluctuations in solar output can show swings in global conditions. We aim to deepen our mathematical understanding of such effects through the study of tractable models of perturbed chaotic systems. The results from such studies may provide guidance for large-scale numerical modeling of complex phenomena such as climate variations and population dynamics.