Control-based bifurcation analysis for experiments

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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.