Mathematical Sciences: Global Bifurcations of Forced Oscillator Systems

由周期强迫振子生成的映射族经历了丰富而迷人的分岔集。研究者建议开发新的数值和分析工具来进一步研究这些强迫振子系统的分岔。目前的数值“延拓”程序能够求出低周期轨道（通常小于4或5个周期）的固定点和周期点局部分岔的单参数曲线。拟议的工作将包括开发新的程序，预计将追踪高周期轨道的分岔曲线。这些新的“高周期”例程将用于定位“全局分岔”，其中不变流形，通常是稳定流形和不稳定流形，在交叉之前成为切线。一旦开发，这些例程将使一个更自动和详细的分岔研究，不仅是由强迫振荡产生的映射，而且是任何双参数的映射族。科学和工程中的许多过程可以在数学上归类为“强迫振荡器系统”。振荡器的一个例子是钟摆，因为它来回摆动，或者振荡。我们通常认为钟摆有一个固定的支撑物（墙上的钉子，上面系着重物的绳子），但如果我们不断移动支撑物的位置，我们就说钟摆受到了“强迫”；如果我们以重复的方式移动支撑物，比如来回移动，那么我们就有了一个“周期性强迫振荡器”。结果表明，最终的运动主要取决于支撑前后移动的速度（强迫频率）和前后移动的距离（强迫幅度）。相当简单的数学模型也表现出同样的行为，包括对类似于强迫频率和振幅的“参数”变化的敏感性。随着参数的变化，系统的定性行为（摆锤如何摆动）发生变化，这被称为“分岔”。我们对这些分歧了解很多，但仍有很多是未知的。研究者建议在这方面填补一些缺失的空白。结果将有助于解释在任何可以在数学上归类为强迫振荡器系统的过程中观察到的行为。

Families of maps generated by periodically forced oscillators undergo a rich and fascinating set of bifurcations. The investigator proposes to develop new numerical and analytical tools to further investigate the bifurcations of these forced oscillator systems. Current numerical "continuation" routines are able to trace out one-parameter curves of local bifurcations of fixed and periodic points for orbits of low period, typically less than four or five. Proposed work will include development of new routines that are expected to trace out bifurcations curves for higher period orbits. These new "high period" routines will then be used for locating "global bifurcations," where invariant manifolds, typically a stable one and an unstable one, become tangent before crossing. Once developed, these routines will enable a much more automatic and detailed bifurcation study, not only of maps generated by forced oscillators, but also of any two-parameter family of maps. Many processes in science and engineering can be mathematically categorized as "forced oscillator systems." One example of an oscillator is a pendulum, because it swings back and forth, or oscillates. We usually think of a pendulum as having a fixed support (the nail in the wall to which the string with a weight is tied), but if we keep moving the location of the support, we say the pendulum is being "forced;" if we move the support in a repeating pattern, such as back and forth, then we have a "periodically forced oscillator." It turns out that the resulting motion depends critically on how fast the support is moved back and forth (the forcing frequency) and how far it is moved back and forth (the forcing amplitude). Fairly simple mathematical models also exhibit this same behavior, including the sensitivity to changes in "parameters" analogous to forcing frequency and amplitude. Changes in the qualitative behavior of the system (how the pendulum bob moves) that occur as the parameters are varied are called "bifurcations." Much is known about these bifurcations, but much is still unknown. The investigator proposes to fill in some of the missing gaps in this area. Results will help explain observed behavior in any process that can be classified mathematically as a forced oscillator system.