Mathematical Sciences: Global Bifurcations of Forced Oscillator Systems

由周期性强迫振子产生的映射族经历了丰富而有趣的一组分支。研究者建议开发新的数值和分析工具来进一步研究这些受迫振子系统的分叉。目前的数值“延拓”程序能够在低周期轨道(通常小于四、五个周期)上跟踪固定点和周期点局部分叉的单参数曲线。拟议的工作将包括开发新的例程，这些例程有望绘制出较高周期轨道的分叉曲线。然后，这些新的“高周期”例程将被用于定位“全局分支”，其中不变流形，通常是稳定流形和不稳定流形，在相交之前变得相切。一旦开发完成，这些例程将使更自动化和更详细的分叉研究成为可能，不仅是对强迫振荡器产生的映射，也是对任何两参数映射族。科学和工程中的许多过程可以在数学上归类为“受迫振子系统”。振荡器的一个例子是钟摆，因为它来回摆动或摆动。我们通常认为钟摆有一个固定的支撑物(墙上的钉子，有重量的绳子被绑在上面)，但如果我们不断移动支撑物的位置，我们说钟摆是被“强迫”的；如果我们以重复的方式移动支撑物，例如来回移动，那么我们就有了一个“周期性受迫振荡器”。结果表明，由此产生的运动关键取决于支承来回移动的速度(强迫频率)和它来回移动的距离(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幅度的“参数”变化的敏感性。随着参数的变化，系统的定性行为(摆锤如何运动)发生的变化称为“分叉”。关于这些分叉我们已经知道了很多，但还有很多是未知的。调查人员建议填补这一领域中缺失的一些空白。结果将有助于解释在任何可以被数学归类为强迫振子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.