The Dynamics of Driven Nonlinear Systems

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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.