Control-based bifurcation analysis for experiments

Lead Research Organisation: University of Exeter
Department Name: Engineering Computer Science and Maths

Abstract

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.

Planned Impact

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.

Publications

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Krauskopf B (2014) Bifurcation analysis of delay-induced resonances of the El-Niño Southern Oscillation. in Proceedings. Mathematical, physical, and engineering sciences

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Sieber J (2014) Controlling unstable chaos: stabilizing chimera states by feedback. in Physical review letters

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Thompson J (2015) Nonlinear dynamic Interactions between flow-induced galloping and shell-like buckling in International Journal of Mechanical Sciences

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Barton DA (2013) Systematic experimental exploration of bifurcations with noninvasive control. in Physical review. E, Statistical, nonlinear, and soft matter physics

 
Description The research developed a method that can find phenomena in controlled laboratory experiments that would be invisible otherwise (because they are unstable).
Exploitation Route DAW Barton (Univ. Bristol) developed a low-cost open-source real-time feedback control box, partially inspired by the needs of experiments related to the Grant's research (also supported by EPSRC funding).

Frank Schilder developed a software toolbox (CONTINEX), which is able to cope with the low accuracy typical for experiments (in contrast to purely numerical computations).

DAW Barton has also started collaboration with Schlumberger (Industrial CASE studentship) to explore the potential of methods derived from my research for drillstring problems. Barton, Rezgui and Neild will start a grant (EP/P019323/1) based on research outcomes from this grant.
Ludovic Renson (Bristol) started a 5-year RAEng Fellowship "Developing next generation testing methods for nonlinear mechanical structures" (2017-2021), which incorporates methods from my research.
Sectors Aerospace, Defence and Marine,Electronics,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology

 
Description Collaboration INRA 
Organisation INRA (UMR-MISTEA) Montpellier, France
Country France 
Sector Academic/University 
PI Contribution Collaboration on control of chemostats with INRA (UMR-MISTEA) Montpellier, France. The collaboration was started on initiative of Alain Rapaport to explore control problems in industrial chemostats.
Collaborator Contribution Collaboration on joint publication and joint conference paper.
Impact Joint paper J. Sieber, A. Rapaport, S. Rodrigues, M. Desroches, A new method for the reconstruction of unknown non-monotonic growth functions in the chemostat. Bioprocess and Biosystems Engineering 36(10) pp. 1497-1507, 2013.
Start Year 2012
 
Description Collaboration WIAS 
Organisation Weierstrass Institute for Applied Analysis and Stochastics WIAS
Country Germany 
Sector Academic/University 
PI Contribution Ongoing collaboration with Weierstrass Institute for Applied Analysis and Stochastics, Berlin (Germany) on effects of time delay on stability of dynamical systems.
Collaborator Contribution Invitation to research visit to Berlin during which work on joint publication was carried out by collaborators and me.
Impact Contributed to joint publications.
 
Description Collaboration with DTU Lyngby 
Organisation Technical University of Denmark
Country Denmark 
Sector Academic/University 
PI Contribution Collaboration with J Starke, C. Marschler, P Hjoerth from DTU Lyngby. One paper is currently in revision, a conference proceedings review paper has been accepted. Colleagues at DTU applied for funding related to the topic of this grant (to perform experiments) with the Danish Research Council. Currently, I am collaborating with Jens Starke and Christian Marschler on a paper (which is in revision). F Schilder and E. Bureau (also DTU) performed expermients using some of the methods developed as part of the grant, developing new methods for estimating eigenvalues. of unstable periodic orbits in experiments.
Collaborator Contribution Development of CONTINEX by collaborators.
Impact Development of CONTINEX by collaborators. Collaboration with Harry Dankowicz (Urbana-Champaign).
Start Year 2012
 
Description Collaboration with P Verrier 
Organisation University of Strathclyde
Country United Kingdom 
Sector Academic/University 
PI Contribution Contributed to paper listed in outcomes
Collaborator Contribution Main authorship of paper listed in outcomes
Impact P. Verrier, T. Waters, J. Sieber, Evolution of the L1 Halo family in the radial solar sail CRTBP. Celestial Mechanics and Dynamical Astronomy
Start Year 2012
 
Description Collaboration with University of Auckland 
Organisation University of Auckland
Department Department of Mathematics
Country New Zealand 
Sector Academic/University 
PI Contribution Contributed to joint paper listed in outcome.
Collaborator Contribution Contributed to joint paper listed in outcome. Paid for my travel and accommodation during three-week research visit. Hosted two further long term visits to their research group
Impact B. Krauskopf, J. Sieber, Bifurcation analysis of delay-induced resonances of the El-Niño Southern Oscillation. Proceedings of the Royal Society A 470(2169), 20140348, 18 pages, 2014.
Start Year 2013
 
Title DDE-Biftool 
Description Bifurcation analysis library for dynamical systems with delay (originally developed by KU Leuven). I extended the library, took over maintenance and published a new version. The licence is currently being changed to make it more open-source compatible. 
Type Of Technology Software 
Year Produced 2014 
Open Source License? Yes  
Impact The program is used in the academic community of researchers in physics, applied mathematics and life sciences, investigating effects of delay. Am major new contribution by Yuri Kuznetsov and his group has been included, enhancing the capabilities for nonlinear analysis.