The Dynamics of Driven Nonlinear Systems

在这个项目中，我们将研究三类参数驱动系统：(1)周期驱动的常微分方程组，(2)含时变边界条件的热对流偏微分方程组和(3)含时变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的行为复杂化的方式。在微分方程组的情况下，我们将考虑一个表现出混沌的周期驱动的非线性振荡器。我们的目标是为这类非自治方程发展与Shil‘nikov理论等价的理论。隐含流动的固定点的缺失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，我们将使用一种允许仔细考虑问题的垂直结构的模式展开过程，并将我们的结果与直接的数值模拟进行比较。这一过程将允许我们包括复杂的效果，如旋转。我们对非自主地图的探索将集中在它们可能表现出的间歇性的性质上。在这样的系统中，盆地边界可能变得千疮百孔，我们将把这种现象的开始与坎托里形成时没有耗散的类似情况进行比较。最简单的混沌系统一遍又一遍地做同样的事情，但永远不会以同样的方式两次。它们行为的细节可以很容易地被外部扰动改变，无论多么微小。然而，当外部扰动持续时，它们会导致受干扰的系统显著地改变其整体行为模式，著名的水滴酷刑就是例证。微弱但混乱的强迫可能会导致不稳定的活动爆发，中间穿插着平静的时期，这一过程被称为断断续续。每隔几个世纪就出现一次、持续数十年的太阳活动极小值就说明了这一点。我们希望深入了解对持续的外部扰动的响应如何根据扰动在时间上是规则的还是不规则的而变化。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。这类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.