Exploring instability in complex systems - simulations in no-man's land

Lead Research Organisation: University of Exeter
Department Name: Mathematics

Abstract

The prediction and analysis of sudden changes in complex systems and models (called tipping) are a current topic in science and an urgent problem for society. A few hotly debated examples are the possible collapse of the Gulf Stream, the sudden loss of vegetation in nutrient-polluted lakes, or the change between the vegetated and desert state in dry regions. While these cases of tipping are well understood in idealised mathematical models, their analysis is restricted in field observations, laboratory experiments and complex model simulations by the impossibility to systematically explore dynamically unstable phenomena. For many complex systems the notion of equilibrium is only defined in a statistical mechanics sense as an emerging phenomenon, which, by its definition, must be stable.

The proposed research will develop general mathematical methods that will remove these restrictions: they will enable experimenters and modellers to discover and track unstable phenomena in laboratory experiments and complex model simulations, or, more generally, in any situation where one can provide input into the system depending on its output in real time. Two features distinguish the systems under study from the idealised models.

* (Limited input only) One has input into the system but may not be able to set the entire internal state at will.
* (Variability) The experiment or model run is repeatable, but the system has internal variability such that outputs are affected by randomness or disturbances.

Several areas will serve as testbeds and springboards to the wider scientific community: individual-based models in ecology, epidemiology and social science, vibration tests in engineering, models of climate subsystems, and abstract spatially extended systems (such as used for neuron population models).

Planned Impact

The research will have immediate impact on several other scientific disciplines such as ecology, epidemiology, neuroscience and climate modelling. Engineering research on new experimental techniques and model validation will also benefit directly from achievement of the objectives. The methods developed during the project will enable modellers and experimenters in both areas to discover new evidence for or against hypothesised underlying nonlinear mechanisms (such as tipping) and make new predictions, for example, about transition probabilities between different states. Dissemination of the results will be through publications in journals and presentations at conferences in the respective application areas.

Impact on society and economy will come mainly through uptake of the research in the above-mentioned disciplines. The new mathematical and experimental tools for gathering evidence and validating models will be used in science and engineering to inform policy-making and industrial testing.
To broaden the potential impact of the proposed research the PI will initiate collaboration with two groups that have a proven track record of research with high impact beyond academia: the Aquatic Ecology group at TU Wageningen and the Mechanical Engineering group of JJ Thomsen and I Santos at DTU Lyngby. The TUW group's research on tipping (to which the proposed research is directly relevant) spans modelling, controlled laboratory and large-scale experiments, field work and policy advice. The DTU group has already invested in prototype experiments to which the newly developed methods can be directly applied. Their development of hardware for non-contact real-time feedback control and software for continuation of unstable vibrations in experiments is directly guided by industry needs. The PI has secured support from external collaborators with experimental know-how and industry contacts (David Barton and Jens Starke, see letters of support), which supports knowledge transfer between the DTU group and the PI's project.

The knowledge transfer between DTU and TUW and the PI's research group will flow both ways since the PI will also dedicate time to spend on problems and models specific to the collaborators' research.

To ensure that the outcomes of the research are accessible to end users in science and engineering (within and outside of academia) all general-purpose software elements will be made publicly available on open-source software repositories (such as Sourceforge, where the PI currently maintains DDE-Biftool).

Several application areas of the proposed research (climate, ecology, epidemiology) and the underlying mathematical themes (tipping, instability) lend themselves naturally to popularisation and public presentation. The PI will visit local schools and contribute to science popularisation initiatives such as Pint of Science.

Publications

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Falkena S (2021) A delay equation model for the Atlantic Multidecadal Oscillation in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

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Falkena SKJ (2019) Derivation of delay equation climate models using the Mori-Zwanzig formalism. in Proceedings. Mathematical, physical, and engineering sciences

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Quinn C (2019) Effects of Periodic Forcing on a Paleoclimate Delay Model in SIAM Journal on Applied Dynamical Systems

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Quinn C (2018) The Mid-Pleistocene Transition induced by delayed feedback and bistability in Dynamics and Statistics of the Climate System

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Ritchie P (2017) Probability of noise- and rate-induced tipping. in Physical review. E

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Ritchie P (2019) Inverse-square law between time and amplitude for crossing tipping thresholds. in Proceedings. Mathematical, physical, and engineering sciences

 
Description A joint paper (in review) gives the relation between length and severity of overshooting a tipping point (in, e.g., climate) that needs to be kept to avoid tipping. A new potential mechanism for the transition from small frequent ice ages to large amplitude longer ice ages was discovered in a joint paper with C. Quinn. A joint paper with Renson et al shows how stability boundaries can be directly traced in experiments.
Exploitation Route (Quinn paper) The relation will be tested on data from large-model runs and from palaeoclimate data.
Sectors Education,Environment,Government, Democracy and Justice,Manufacturing, including Industrial Biotechology

 
Description Developing strategies to prevent collapse of the Amazon rainforest
Amount £202,300 (GBP)
Funding ID EP/V04687X/1 
Organisation Engineering and Physical Sciences Research Council (EPSRC) 
Sector Public
Country United Kingdom
Start 03/2021 
End 01/2023
 
Description Newton Bhabha PhD placement programme
Amount £10,200 (GBP)
Funding ID 340891876 
Organisation British Council 
Sector Charity/Non Profit
Country United Kingdom
Start 02/2018 
End 03/2019
 
Title DDE-Biftool extension 
Description An open-source software library to analyse mathematical models for delay effects (DDE-Biftool), for which I am the maintainer and lead developer has been improved to be able to deal with problems where the delay depends on the state of the system or where the delay is very large (typical in models for semiconductor lasers). This work was done in collaboration with Maikel Bosschaert (Univ. utrecht, Netherlands), who developed and implemented normal form analysis for the new code. 
Type Of Material Improvements to research infrastructure 
Year Produced 2018 
Provided To Others? Yes  
Impact The software has been downloaded 700 times in the last year. Judging from the number of support requests it is actively used by other communities. 
URL https://sourceforge.net/projects/ddebiftool
 
Description CSIRO Hobart, follow-on from EU CRITICS project 
Organisation Commonwealth Scientific and Industrial Research Organisation
Department CSIRO Hobart
Country Australia 
Sector Public 
PI Contribution Collaboration on research outputs (papers by Falkena et al 2019, and Quinn et al 2019)
Collaborator Contribution Co-authorship by C Quinn
Impact S. K. J. Falkena, C. Quinn, J. Sieber, J. Frank, H. A. Dijkstra, Derivation of delay equation climate models using the Mori-Zwanzig Formalism. Proceedings of the Royal Society A 475(2227):20190075, 2019 (21 pages) C. Quinn, J. Sieber, A. S. von der Heydt, Effects of periodic forcing on a Paleoclimate delay model. SIAM Journal of Applied Dynamical Systems 18(2) pp. 1060-1077, 2019,
Start Year 2019
 
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 John W Hutchinson 
Organisation Harvard University
Department School of Engineering and Applied Sciences
Country United States 
Sector Academic/University 
PI Contribution Simulations performed for study of buckling type instabilities
Collaborator Contribution Modelling and design of considered buckling scenarios
Impact Joint papers, J. Sieber, J. W. Hutchinson, J. M. T. Thompson, Buckling Thresholds for Pre-loaded Spherical Shells Subject to Localized Blasts . ASME Journal of Applied Mechanics 19-1539 (20 pages), 2019. J. Sieber, J. W. Hutchinson, J. M. T. Thompson, Nonlinear Dynamics of Spherical Shells Buckling under Step Pressure. Proceedings of the Royal Society A 475(2223):20180884, 2018. J. M. T. Thompson, J.W. Hutchinson, J. Sieber, Probing Shells Against Buckling: A Nondestructive Technique for Laboratory Testing. Int. J. Bifurcations and Chaos 27(14), 1730048, (15 pages), 2017.
Start Year 2016
 
Description Collaboration with University Udine (Italy) 
Organisation University of Udine
Country Italy 
Sector Academic/University 
PI Contribution Collaboration with Dimitri Breda and Alessia Ando
Collaborator Contribution Collaboration on joint publication about numerical methods for problems with delay
Impact Joint publication (still in preparation)
Start Year 2020
 
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
 
Description Collaboration with University of Rostock (Germany) 
Organisation University of Rostock
Country Germany 
Sector Academic/University 
PI Contribution Collaboration on joint paper in review: J. Sieber, C. Marschler, J. Starke, Convergence of equation-free methods in the case of finite time scale separation with application to stochastic systems. Hosted collaborator on research visit.
Collaborator Contribution Collaboration on joint paper in review: J. Sieber, C. Marschler, J. Starke, Convergence of equation-free methods in the case of finite time scale separation with application to stochastic systems. Member of their research team works on ongoing paper on simulation and experiments of pedestrian flows.
Impact Still in progress
Start Year 2017
 
Description Collaboration with University of Urbana-Champaign (Illinois, US) 
Organisation University of Illinois at Urbana-Champaign
Country United States 
Sector Academic/University 
PI Contribution External collaboration partner on NSF Grant proposal by Prof. Harry Dankowicz (HD), who proposed a collaboration on the application of control-based bifurcation analysis in hybrid testing. Joint organization of ASME graduate school in Urbana-Champaign.
Collaborator Contribution HD has co-developed CONTINEX (a toolbox for his software platform COCO) that can be used for experimental tracking.
Impact still in preparation
Start Year 2016
 
Description Collaboration with University of Utrecht (Netherlands) 
Organisation Utrecht University
Country Netherlands 
Sector Academic/University 
PI Contribution Supervised visiting researcher Swinda Falkena from Utrecht, working on delay effects in Ocean models
Collaborator Contribution AS von der Heydt collaborated on joint paper (in review) with research student Courtney Quinn (under PI's supervision).
Impact Still under Review: C. Quinn, J. Sieber, A. S. von der Heydt, T. M. Lenton, The Mid-Pleistocene Transition induced by delayed feedback and bistability.
Start Year 2017
 
Title DDE-Biftool new version 
Description Bifurcation analysis for systems with delays (parametric and state-dependent). 
Type Of Technology Software 
Year Produced 2017 
Open Source License? Yes  
Impact The software is widely used in the physics and life science community. The update enables the treatment of long delays and of state-dependent delays. The new contribution by Yuri Kuznetsov can now also be applied for the case for state-dependent delays. 
 
Description Co-organizer and Lecturer at ASME funded Advanced Summer School on Continuation Methods for Nonlinear Problems 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact Co-organized a 2-week graduate School on Continuation Methods for Nonlinear Problems. The School was funded by ASME (The American Society of Mechanical Engineers) and took place at the University of Urbana-Champaign. Participants were graduate students and early-career researchers in mechanical and civil engineering. The main topic were methods and their software implementation developed by the main organizer (Harry Dankowicz) and me.
Year(s) Of Engagement Activity 2018
URL http://danko.mechanical.illinois.edu/AdvancedSummerSchool.htm