Control-based bifurcation analysis for experiments

许多被数学理论预言存在的现象在现实生活中是看不见的。然而，数学理论也预言，这些隐藏的现象决定了我们的命运，而现实生活却处于“边缘”。例如，风力的小幅增加可能会突然导致桥梁缆索开始剧烈摆动。更令人费解的是，即使风力再次减弱，桥梁缆索仍可能继续强劲摆动。数学理论表明，这些惊人的突然变化(从工程学的角度看往往是灾难)背后的机制是普遍存在的：它们不仅适用于大桥电缆，也适用于洋流或神经元。观察到的变化是突然的，只是因为这两种不同的可见行为之间缺失的联系通常是一种不稳定或过于敏感的现象，以至于看不见。这种洞察力使工程师和科学家能够预测、避免或控制突然的变化，只要他们能够依靠一组描述运动的方程。这项研究将开发一种方法--“基于控制的延续”，使实验者能够在受控的实验室实验中直接观察不稳定现象。基于控制的延拓利用控制将实验输入和输出之间的关系转化为可以通过计算求解的方程。在不受控制的实验中，任何自然现象都可以作为这个方程的解。机械原型实验(例如，使用摆动和束流-磁铁布置)表明，该方法确实是可行的。这个项目旨在使基于控制的延续适用于更复杂的实验和更复杂的现象。PI将与丹麦技术大学(Lyngby)的实验人员合作，他们研究快速旋转机械中的振动。该项目的一个具体目标是开发和测试稳定和不稳定之间的确切边界(所谓的边界分支)的连续性。传统的确定分叉的计算方法不适用于从测量中提取的方程，因为它们依赖于高精度(8-16位有效数字)求解方程的能力，而这在大多数实验中是无法实现的。

Many phenomena that are predicted to exist by mathematical theoryremain invisible in real life. Yet, mathematical theory also predictsthat these hidden phenomena determine our fate when real life is "onthe edge". For example, a small increase of wind strength can abruptlycause a bridge cable to start swinging violently. Still more puzzling,the bridge cable may continue to swing strongly even if the windstrength decreases again. Mathematical theory reveals that themechanisms behind these striking sudden changes (often catastrophesfrom the point of view of engineering) are universal: they apply to abridge cable as well as to an ocean current or a neuron. The observedchange is abrupt only because the missing link between the twodifferent visible behaviours is typically a phenomenon that isunstable or too sensitive to be visible. This insight enablesengineers and scientists to predict, and avoid or control, suddenchanges whenever they can rely on a set of equations describing themotion.This research will develop a method, "control-based continuation",that enables experimenters to observe unstable phenomena directly incontrolled laboratory experiments. Control-based continuation usescontrol to convert the relation between experimental inputs andoutputs into an equation that can be solved computationally. Everyphenomenon that is natural in the uncontrolled experiment can be foundas a solution of this equation. Mechanical prototype experiments(using, for example, pendula and beam-magnet arrangements) have shownthat the method is indeed feasible. This project aims to makecontrol-based continuation applicable to more complex experiments andmore complex phenomena.The PI will collaborate with experimenters at the Technical Universityof Denmark (Lyngby) who investigate vibrations in fast rotatingmachinery. One specific objective of the project is to develop and test the continuationof the exact boundaries between stability and instability (so-calledbifurcations). Traditional computational methods for determiningbifurcations are not applicable to equations extracted frommeasurements because they rely on the ability to solve the equationwith high accuracy (8-16 significant digits), which is not achievablein most experiments.