Complex Dynamics and Strange Attractors through Non-Autonomous Bifurcations

Project Information

CDSANAB

Grant agreement ID: 750865

DOI

10.3030/750865

Project closed

EC signature date 17 March 2017

Start date 1 March 2018

End date 31 July 2020

Funded under

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Total cost

€ 183 454,80

EU contribution

€ 183 454,80

Coordinated by

IMPERIAL COLLEGE OF SCIENCE TECHNOLOGY AND MEDICINE
United Kingdom

Periodic Reporting for period 1 - CDSANAB (Complex Dynamics and Strange Attractors through Non-Autonomous Bifurcations)

Reporting period: 2018-03-01 to 2020-02-29

Summary of the context and overall objectives of the project

An autonomous dynamical system is defined by an evolution law which is constant over time and governs the change of a given system along the progression of time. While much of the theory building with regards to dynamical systems takes place in this autonomous setting, practical applications ask for a better understanding of non-autonomous systems, as real-world models—be it in ecology, climatology or other sciences—typically include external forces.

This project accommodated comprehensive rigorous analytical investigations as well as application-oriented research shedding light on the question as to how complex dynamical behaviour—as is visible in the succession of ice ages or other seemingly unpredictable real-world processes—arises. A key notion in this context are so-called bifurcations. These are drastic changes in a system’s behaviour due to minor variations of some parameter(s) of the system. Despite their ubiquity, the prediction of such bifurcations so far simply applied tools whose applicability in a non-autonomous context was hardly investigated prior to this project. Among these tools, the most prominent is to measure so-called recovery rates which indicate how fast a system returns to its equilibrium after having been slightly nudged away from it. The classical theory suggests that when a system approaches a bifurcation, then the time to recover should increase since the system’s stability is less and less ensured the closer we are to a bifurcation.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

Conference talks:
Sep 2019 Dynamics, Equations and Applications, Krakow (PL).
Jun 2019 Visegrad Conference on Dynamical Systems, Budapest (HU).
May 2019 Conference on Dynamical Systems, Krakow (PL).
Mar 2018 Critical Transitions in Complex Systems: Mathematical theory and applications, Wöltingerode (DE).

Seminar talks:
May 2020 Faculty of Mathematics and Computer Science,Jagiellonian University Krakow (PL)—ONLINE EDITION.
Mar 2020 Seminar for Mathematical Statistics, TU München (DE).
Feb 2020 Model Theory and Logic seminar at the University of Lyon (FR).
Feb 2020 Dynamical Systems Seminar, Loughborough University (GB).
Jan 2020 Research seminar Dynamical Systems \& Mathematical Physics FSU Jena (DE).
Nov 2019 Dynamical Systems seminar at the University of Liverpool (UK).
Oct 2019 Dynamical Systems seminar at the Open University (UK).
May 2019 Ergodic Theory seminar at the LPSM Paris (FR).
Apr 2019 Geometry seminar at the ICMAT Madrid (ES).
Dec 2018 Dynamics seminar at the University of Montevideo (UY).
Oct 2018 Department of Mathematics, Imperial College London (UK).
Sep 2018 BudWiSer Seminar, University of Vienna (AT).

At the beginning of the action (in March 2018), I was co-organising a one-week winter school and a one-week workshop “Critical Transitions in Complex Systems: Mathematical theory and applications” which took place in the context of the ITN CRITICS in Wöltingerode, Germany. During the entire action, I further organised the dynamical systems seminar at DynamIC and was an active member of a reading group at DynamIC.

It has to be mentioned that the pandemic’s impact resulted in quite some extra organisational effort with regards to the seminar as we decided to not only shift the seminar online but keep—despite the lack of face-to-face meetings—the interaction in our group going by experimenting with different formats of online meetings and talks. Judging from the feedback of the group, however, this extra effort clearly payed of. It will further help me during the next months when much of my activities (including the teaching on my new position) will have to remain online.

Besides me paying regular visits to research groups, I also invited members of other institutions to visit Imperial. Tobias Jäger (University of Vienna) came to visit me in late March/early April. Maik Gröger (University of Vienna) came for an extended research stay of two weeks in the second half of November 2019. Further invitations got cancelled due to the global pandemic.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

One of the key messages from this project is that this simple mechanism may not be visible if a system is subject to external forces. More precisely, unlike in the unforced case, the answer to the question whether slow recovery rates can be observed prior to a bifurcation hinges on the particular time-scales of the observations of a system and the size of the acquired data sets. Numerical experiments support that this fact is not only an academic artifact but actually of practical significance.
The action further included plenty of in-depth theoretical analysis regarding the creation of complex behaviour both in a chaotic environment and in a non-chaotic setting. Here, one extensively studied mechanism was the blow-up of orbits in so-called almost automorphic systems. This mechanism also occurs in non-smooth bifurcations where it results in so-called strange non-chaotic attractors. It turns out that already this one mechanism can result in a plethora of different behaviours of a dynamical system. Summed up in laymen’s terms, the catch of the theoretical work in this action is that the interface between order and chaos is a rather diverse universe where both antagonists exist in close proximity.
The project’s results are communicated in a total of 5 scientific articles and were disseminated in 16 research talks.

pic-1-cds.jpg