Home » Activities » Nonautonomous dynamical systems: from theory to applications – July 2021

Nonautonomous dynamical systems: from theory to applications – July 2021

This event will take place on July 5, 2021 at the University of Exeter (Streatham Campus, IAIS Building LT1 and LT2) in Exeter, with further opportunities to participate in informal discussions on 6th and 7th July.


Please register here by June 30, 2021. There is no registration fee.


This research workshop aims to discuss the latest developments of theory and application of nonautonomous dynamical systems, i.e. dynamical systems that evolve in time but where parameters also change with time, with specific reference to application of climate modelling. There is an international consensus on the importance of understanding the response of the climate system to anthropogenic, and potentially fast, change in forcing, so that negative impacts on society can be mitigated. Of particular focus will be notions of attractors of nonautonomous dynamical systems and new methods to understand “typical behaviour”.

The workshop will take place physically (subject to any changing restrictions due to Covid) but virtual attendance will also be possible. Please indicate how you would like to attend during registration.

If you would like to contribute a poster (also to be made available physically and virtually), please let us know in the registration.

Organisation and funding

This workshop is organised by, Peter Ashwin (Exeter) and Lea Oljaca (Exeter), under the EPSRC funded project “Applied Nonautonomous Dynamical Systems: Theory, Methods and Examples” and together with principal researchers; Martin Rasmussen (Imperial) and Valerio Lucarini,(Reading) and project partners; Richard Wood (Met office), TiPES and S Wieczorek (Cork).

For more information, contact Lea Oljaca

Speakers include

Prospective Schedule

  • Monday July 5, 2021
    10:30-11:00 Arrival, registration, coffee
    11:00-11:40 I Longo
    11:45-12:25 R Wood
    12:25-14:00 Lunch
    14:00-14:40 J Newman
    14:45-15:25 S Wieczorek
    15:25-16:00 Coffee Break
    16:00-16:40 Niccolo Zagli
    16:40-17:50 Poster session with a drinks reception

    Evening: Workshop dinner (limited slots available, please indicated interest in registration form)


I Longo
Title: Rate-induced tipping as a nonautonomous bifurcation in a class of scalar quadratic ODEs

J Newman
Title: Tipping probabilities
Abstract: Whereas the basic approach to studying critical transitions, such as climate tipping points, is to look at parameter-dependence of attractors and their bifurcations, a more physically accurate approach is to take a real-time parameter shift between two parameter values, giving rise to a nonautonomous dynamical system. This approach has led to the study of rate-induced tipping. We build on this by defining “tipping probabilities” for deterministic parameter-shifting systems and illustrate with numerical examples, including a chaotically forced Stommel model that can be viewed as a toy model for the AMOC. In order to define these “tipping probabilities”, we first define “natural” or “physical” measures for the parameter-shifting system. We will also discuss future directions for our work, particularly in regards to practical application for the study of tipping elements in the Earth system.
Richard Wood
Title: The Atlantic Meridional Overturning Circulation: an example of nonautonomous tipping?
The Atlantic Meridional Overturning Circulation (AMOC) is a global scale system of ocean currents which plays a key role in regulating global climate by transporting heat around the planet. It has long been known from both theoretical modelling studies and palaeoclimatic evidence that the AMOC has the potential to exhibit threshold or tipping behaviour, with the possibility of reaching a weakened or collapsed AMOC state that would have a profound impact on regional climates. It is natural to ask whether the ongoing climate change that we expect to continue over the coming decades will increase the risks of such an AMOC tipping event. Such a problem is a nice example of a nonautonomous dynamical system, because the inherent timescales of the tipping, while not fully understood, are of a similar order to the timescale on which we are currently modifying the parameters of the climate system through increasing greenhouse gases, i.e. decades to centuries.In this talk I shall review the AMOC and its role in climate, as well as what is known about AMOC tipping, from a climate modeller’s perspective. I shall then discuss possible practical responses to the risks posed by AMOC tipping (probability estimation, early warning and safe operating spaces), and describe some recent progress towards these goals. A key tool to study AMOC tipping is to use a *traceable* hierarchy of models: simple dynamical models can be analysed mathematically to provide insight into tipping behaviour, but it is important to demonstrate that the simple models used are capturing the essential dynamics of more comprehensive climate models, and, by extension, of the real world.
Sebastian Wieczorek
Tiltle: Rate-Induced Tipping Points: Beyond Classical Bifurcations
Abstract: Many systems are subject to external disturbances or changing external conditions. For a system near a stable state (an attractor) we might expect that, as external conditions change over time, the stable state will change too.  In many cases the system may adapt to changing external conditions and track the moving stable state.  However, tracking may not always be possible owing to nonlinearities and feedbacks in the system. We consider systems that are particularly sensitive to how fast the external conditions change: such systems suddenly and unexpectedly move to a different state if the external input changes too fast. This happens even though the moving stable state never loses stability in the classical autonomous sense. We describe this phenomenon as rate-induced tipping or R-tipping. Being a genuine non-autonomous bifurcation, R-tipping is not captured by the classical bifurcation theory and requires an alternative framework.In the first part of the talk, we demonstrate R-tipping in a simple ecosystem model where environmental changes are represented by time-varying parameters. We then introduce the concept of basin instability and show how to complement the classical bifurcation diagram with information on nonautonomous R-tipping that cannot be captured by the classical bifurcation analysis.  In the second part of the talk, we present a general mathematical framework for R-tipping with decaying inputs based on the concepts of thresholds, edge states and special compactification. This allows us to transform the R-tipping problem into a connecting heteroclinic orbit problem in the compactified system. This transformation simplifies the analysis and allows us to give rigorous testable criteria for the occurrence of R-tipping due to regular thresholds in arbitrary dimension. In the third part of the talk, we discuss the so-called “compost-bomb instability”, which is an example of R-tipping with quaithresholds. We use geometric singular perturbation theory to reveal non-obvious R-tipping thresholds and edge states associated with quasithresholds.

Niccolo Zagli
Title: Response theory and critical phenomena for the thermodynamic limit of interacting identical systems.
In this talk I will present our latest results about linear response theory for the thermodynamic limit of interacting multi-agent systems. In particular, I will show how these systems exhibit pathologies in their linear response that correspond to two different physical processes of criticality, that we named critical transitions and phase transitions. One one hand, critical transitions are characterised by a divergence of correlation properties and can be related to the vanishing of the spectral gap of a suitable transfer operator. On the other hand, phase transitions arise only in the thermodynamic limit, originate from the interaction among the agents and do not show any divergence of correlation properties. At the end of the talk, I will present numerical experiments on both equilibrium and non-equilibrium phase transitions showing that a clear loss of analyticity of the linear response, due to a pole crossing the real axis of frequencies, can be detected even in finite systems.


Thoraya Alharthi
Finding critical rates for a parameter shifts to periodic forcing

Ruth Chapman
Stochastic data adapted AMOC box models

Irene Malmierca-Vallet
Dansgaard-Oeschger Tipping Events (TEs): Towards determining if IPCC-relevant models represent these TEs.

Lea Oljaca
Milnor attractors for nonautonomous dynamical systems


IAIS Building LT1 and LT2

Travel Information

For general advice on how to get to the University of Exeter, see the University maps and directions pages.

By  plane:

  • Fly to Exeter (EXT): Onward travel from Exeter Airport is available by bus or taxi. The bus is cheap (a few pounds), but does not run late. The taxi costs about £25. There are direct flights between Exeter many British and European airports, including London City AirportSchipol (Amsterdam), and Paris CDG; in particular the latter two are options to connect to other flights.
  • Fly to Bristol (BRS): Bristol airport is the next closest airport (total travel time approx. 1.5 hrs). Follow the signs for the Bristol Flyer bus and purchase a ticket to Bristol Temple Meads Rail Station (journey time is 20-30 minutes). From Bristol Temple Meads take a train to Exeter St David’s train station (journey time approximately one hour).
  • Fly to London Heathrow (LHR): From Heathrow by bus (approx. 3.5hrs, nonstop), you can take the National Express Bus which runs fairly frequently to Exeter Bus and Coach station right in the city centre and is inexpensive compared to trains. From Heathrow by rail (approx. 3 hrs, 1 stop), you can take the Railair bus service from Heathrow to Reading train station and then take a train from there to Exeter St. David’s (book a through ticket at Heathrow Bus Station). Alternatively, you can take either the Heathrow Express (fast but expensive) or Heathrow Connect (slower but much cheaper) to London Paddington train station and then take the train from Paddington to Exeter St. David’s; yet another option is to take a bus from Heathrow to Woking and take the train from there to Exeter Central.
  • Fly to London Gatwick (LGW)From Gatwick by rail, you can travel by train either via Reading to Exeter St. Davids (approx. 3.5 hrs, 1 stop) or via Clapham Junction (cheaper, approx. 4 hrs) to Exeter Central train station. There are National Express Bus services as well.

By train: 

  • The closest train station to the university is Exeter St. Davids. The closest train station to the city center is Exeter Central. Rail prices and timetables (including for onward travel from airports other than Exeter) can be found at National Rail Enquiries. Generally,  Advance or Return tickets come with significant discounts.

By car: 

  • Pay and Display parking on campus is incredibly limited. Visitors may park in Car Park C and 15 marked bays in Car Park A. Their locations are marked on the Streatham Campus map. Please note that you must arrive early to find a space. Parking costs £6 per day.


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