Control-based bifurcation analysis for experiments

Many phenomena that are predicted to exist by mathematical theory
remain invisible in real life. Yet, mathematical theory also predicts
that these hidden phenomena determine our fate when real life is "on
the edge". For example, a small increase of wind strength can abruptly
cause a bridge cable to start swinging violently. Still more puzzling,
the bridge cable may continue to swing strongly even if the wind
strength decreases again. Mathematical theory reveals that the
mechanisms behind these striking sudden changes (often catastrophes
from the point of view of engineering) are universal: they apply to a
bridge cable as well as to an ocean current or a neuron. The observed
change is abrupt only because the missing link between the two
different visible behaviours is typically a phenomenon that is
unstable or too sensitive to be visible. This insight enables
engineers and scientists to predict, and avoid or control, sudden
changes whenever they can rely on a set of equations describing the
motion.

This research will develop a method, "control-based continuation",
that enables experimenters to observe unstable phenomena directly in
controlled laboratory experiments. Control-based continuation uses
control to convert the relation between experimental inputs and
outputs into an equation that can be solved computationally. Every
phenomenon that is natural in the uncontrolled experiment can be found
as a solution of this equation. Mechanical prototype experiments
(using, for example, pendula and beam-magnet arrangements) have shown
that the method is indeed feasible. This project aims to make
control-based continuation applicable to more complex experiments and
more complex phenomena.

The PI will collaborate with experimenters at the Technical University
of Denmark (Lyngby) who investigate vibrations in fast rotating
machinery.

One specific objective of the project is to develop and test the continuation
of the exact boundaries between stability and instability (so-called
bifurcations). Traditional computational methods for determining
bifurcations are not applicable to equations extracted from
measurements because they rely on the ability to solve the equation
with high accuracy (8-16 significant digits), which is not achievable
in most experiments.

The proposed research develops a new experimental technique that
permits experimenters to observe phenomena that are inaccessible in
conventional experiments. The project transfers dynamical systems
methods that have been very successful in computer simulations to a
new environment: experimental laboratories.

The most immediate impact is expected through uptake of the developed
methods by experimenters in engineering (mechanical and electrical)
and neuroscience, initially in academic laboratories. The PI will advance
this uptake through direct collaboration with experimenters in Bristol and
Lyngby (Denmark) and is currently exploring the potential for collaboration
with other experimental groups (visits are planned).

Two directions in which the proposed research will find its way into applications
are currently on the horizon.

(1) High-tech companies require extensive testing of new machinery and
product components. This method extends the range of feasible
laboratory testing conditions.

(2) Neuroscience researches new treatments for conditions such as
epileptic seizures, Parkinson's disease or migraine. New insights
about the dynamics of neurons will help inform this
research. Furthermore, the proposed methods are a new robust way
to make feedback control non-invasive (that is, applying control
without changing the natural behaviour). Non-invasive control is one
of the potential treatments currently discussed in neuroscience.
Thus, the project may contribute to neuroscience by providing a new
avenue for interaction with the neurosystem without disturbing its natural state.