Workshop on Nonautonomous dynamical systems: from theory to applications – July 5th, 2021
We are planning a one day workshop on nonautonomous dynamical systems with particular focus on climate change applications. It will be held at University of Exeter (Streatham Campus) on the 5th of July, with further opportunities to participate in informal discussions on 6^{th} and 7^{th} July. This is planned to be available for in-person (Exeter) and virtual participants.
Exeter Workshop on Deterministic Extremes and Recurrence 21-23rd June 2021
This is a three-day workshop on deterministic extremes and recurrence in dynamical systems. This meeting will take place over Zoom, and access links/codes will be provided on this site closer to the time of the meeting (see below).
Contact Mark Holland, Surabhi Desai, Peter Ashwin, and Tomas Persson if you are interested and come back to here for more information in due course. Details are as follows.
Links to notes/slides from talks:
Borel Cantelli notes (Persson)
Shrinking/moving targets (Koivusalo)
Extremes/maxima notes (Holland)
Extremes/energy-observables (Carney)
Monday 21st June
Theme: Shrinking targets & recurrence
13:45 | Meeting opens | |
13.50 | Introduction | |
14:00-14.45 | Tomas Persson (Lund): Dynamical Borel–Cantelli lemmata, shrinking targets and recurrence | |
15:00-15.50 | Henna Koivusalo (Bristol): Path-dependent, shrinking, moving targets and beyond, on generic self-affine sets | |
16:15-17:00 | Tomas Persson. (2nd lecture). | |
17:00-17:30 | Informal discussion |
Tuesday 22nd June
Theme: Extremes, recurrence and limit laws in dynamical systems
13:00-13.45 | Mark Holland (Exeter): On the distribution of extreme events for dynamical systems | |
14:00-14.45 | Mark Holland. (2nd lecture). | |
15:00-15.50 | Meagan Carney (University of Queensland): Extremes for Energy-like Observables on Hyperbolic Systems. | |
16:15-17.05 | Mike Todd (St Andrews): Capturing clustering in extreme values | |
17:05-17:30 | Informal discussion |
Wednesday 23rd June
Theme: Extremes and applications
13:30-14.15 | Tobias Kuna (Reading): A qualitative aspect of extreme value theory for dynamical systems | |
14:30-15.15 | Tobias Kuna. (2nd lecture). | |
15.45-16.35 | Julian Newman (Exeter): Attractors and attracting measures | |
16:35-17:00 | Informal discussion |
The 2×45 min sessions are mini-courses accessible to PhD students/post-doc researchers. The 50 minute talks are on general current research in the relevant themes.
Abstracts
Meagan Carney (University of Queensland)
Extremes for Energy-like Observables on Hyperbolic Systems.
We consider an ergodic, measure-preserving dynamical system $(T, X, \mu)$ equipped with an observable $\phi: X\rightarrow R$. Given the stochastic process $X_n(x) = \phi(T^n(x))$, we establish an extreme value law for the sequence of maxima $M_n = max_{k\le n} X_k$ where $\phi$ is an energy-like observable and $(T, X, \mu)$ is hyperbolic. Observables of this form have the property that the set of maximization is a curve rather than a single point. We will discuss results in the case of Anosov diffeomorphisms, Sinai dispersing billiards, and coupled expanding maps. We will highlight the dependence of the extremal index on the set of maximization and discuss some numerical results for these systems.
Mark Holland (Exeter)
On the distribution of extreme events for dynamical systems
This lecture will cover almost sure growth bounds for maxima (extremes), using links to the theory of dynamical Borel Cantelli Lemmas. We also review distributional convergence results of extremes for dynamical systems, contrasting to the classical theory of extremes for i.i.d random variables.
Henna Koivusalo (Bristol)
Path-dependent, shrinking, moving targets and beyond, on generic self-affine sets
The classical shrinking target problem concerns the following set-up: Given a dynamical system (T, X) and a sequence of targets (B_n) of X, we investigate the size of the set of points x of X for which T^n(x) hits the target B_n for infinitely many n. In this talk I will study shrinking target problems in the context of fractal geometry. I will first recall the symbolic and geometric dynamical systems associated with iterated function systems, fundamental constructions from fractal geometry. I will then briefly cover the Hausdorff dimension theory of generic self-affine sets; that is, sets invariant under affine iterated function systems with generic translations. Finally, I will show how to calculate the Hausdorff dimension of shrinking target-type sets on generic self-affine sets. The target sets that I will consider move and shrink at a speed that depends on the path of x. Time permitting, further problems of similar flavour and refinements of the dimension result might also be explored. This talk is based on a joint work with Lingmin Liao and Michal Rams.
Tobias Kuna (Reading)
A qualitative aspect of extreme value theory for dynamical systems
Julian Newman (Exeter)
Attractors and attracting measures
Under mild assumptions, the SRB measure supported on an Axiom A attractor has the following two properties: (i) the empirical measure starting at a typical point near the attractor converges weakly to the SRB measure; (ii) the pushforward of any Lebesgue-absolutely continuous probability measure supported near the attractor converges weakly to the SRB measure. The first property is known as the “physical measure” property, and has been extensively studied and generalised. We will refer to the second property as the “attracting measure” property. It describes “mixing” behaviour, but in a more experimentally accessible way than just saying that the invariant measure itself is mixing: it can be expressed as a decay of “operational correlations” which make reference to the Lebesgue measure, as opposed to decay of “classical” correlations defined purely with respect to the invariant measure. There are various situations in the sciences, such as in climate science, where attractors have zero Lebesgue measure, and the question of whether such attractors support an attracting measure seems to be of high physical relevance. And yet, there appears to be very little literature addressing this question. (For example, is it known whether the physical measure on the classical Lorenz attractor is attracting?) I will present a topological generalisation of the original result of Bowen and Ruelle that establishes the attracting measure property for Axiom A flows.
Tomas Persson (Lund)
Dynamical Borel–Cantelli lemmata, shrinking targets and recurrence
This lecture will give the basics of dynamical Borel–Cantelli lemmata and shrinking targets. Some proofs will be given in at least simple cases. I will also talk about recurrence and other related things.
Mike Todd (St Andrews)
Capturing clustering in extreme values
We build new tools to handle clustering patterns of extreme values in dynamical systems. The main tool I’ll discuss is a new type of point process which fully captures clustering behaviour. I’ll illustrate this with a simple dynamical system. This is joint work with Ana Cristina Freitas and Jorge Freitas.
CriticalEarth – vacancy for one Early Stage Researcher
We are looking for one ESR/PhD position (3-years) in the field of Applied Mathematics, based at University of Exeter (England). Critical Earth is a 4 year project that starts in March 2021 involving 15 early stage researcher (ESR) across the EU and UK. Applicants with the desired skills can, and are encouraged to apply from any country in the world. If you are applying from a location that requires a visa or permit, then we will be able to provide support and advice throughout the process of relocation for you and your family. Feel free to ask us questions in advance if you need more information and reassurance.
PhD ESR9: “Deterministic extremes and climate tipping points” will use geometric and analytic features of dynamical systems to understand generic properties of probability distributions governing extremes, and make predictions on when extremes are likely to occur in the future, working with Prof Mark Holland and partners in CriticalEarth. Click here for details/Application. Deadline: 3rd May 2021
Eligibility: Applicants must not have resided and not have carried out their main activity (work, studies, etc.) in the country of the recruiting beneficiary for more than 12 months in the 3 years immediately before the recruitment date — unless as part of a procedure for obtaining refugee status under the Geneva Convention. The applicant must be an Early Stage Researcher (ESR) i.e. at the time of recruitment you must be in the first 4 years (full-time equivalent research experience) of your research careers and must not have been awarded a doctoral degree.
Research featured in SIAM news
Some work from an article on domino effects and synchrony by Jen Creaser, Peter Ashwin and Krasi Tsaneva-Atanasova of CSDC and the EPSRC centre for Predictive Modelling in Healthcare has been highlighted in a recent SIAM newsletter: https://sinews.siam.org/Details-Page/domino-effects-and-synchrony-in-seizure-initiation
CriticalEarth – vacancies
The Centre will be hosting two positions (3-years) in the field of Applied Mathematics, based at University of Exeter (England). Critical Earth is a 4 year project that starts in March 2021 involving 15 early stage researcher (ESR) across the EU and UK. Applicants with the desired skills can, and are encouraged to apply from any country in the world. If you are applying from a location that requires a visa or permit, then we will be able to provide support and advice throughout the process of relocation for you and your family. Feel free to ask us questions in advance if you need more information and reassurance.
PhD ESR8: Multiscale variability of coupled systems is a theoretical project aims to develop novel methods to understand multiple timescale conceptual models, working with Prof Peter Ashwin and partners in the CriticalEarth. Click here for details/ Application. Deadline: 24th Jan 2021
PhD ESR9: “Deterministic extremes and climate tipping points” will use geometric and analytic features of dynamical systems to understand generic properties of probability distributions governing extremes, and make predictions on when extremes are likely to occur in the future, working with Prof Mark Holland and partners in CriticalEarth. Click here for details/Application. Deadline: 24th Jan 2021
Eligibility: Applicants must not have resided and not have carried out their main activity (work, studies, etc.) in the country of the recruiting beneficiary for more than 12 months in the 3 years immediately before the recruitment date — unless as part of a procedure for obtaining refugee status under the Geneva Convention. The applicant must be an Early Stage Researcher (ESR) i.e. at the time of recruitment you must be in the first 4 years (full-time equivalent research experience) of your research careers and must not have been awarded a doctoral degree.
Antipodal heteroclinic workshop, 5th October 2020
There will be a collaborative workshop by Zoom on dynamical systems (with focus on heteroclinic dynamics) on 5th October 2020, 8-10am (UK), 8-10pm (NZ).
- 8:00 Introduction and welcome
- 8:05-8:20 Valerie Jeong (Auckland) “A noisy, perturbed heteroclinic cycle and evolutionary robotics”
- 8:25-8:40 Chris Bick (Exeter) “Heteroclinic dynamics in phase oscillator networks with higher order interactions”
- 8:45-9:00 Gray Manicom (Auckland) “A network model of task-switching”
- 9:05-10:00 Virtual reality poster session (Jen Creaser, Max Voit, Gray Manicom, Valerie Jeong)
All welcome! Please write to one of the organizers to be emailed the links and/or if you have a poster you wish to display.
Peter Ashwin (Exeter)
Claire Postlethwaite (Auckland)
Talks:
Valerie Jeong “A noisy, perturbed heteroclinic cycle and evolutionary robotics”
Abstract: Evolutionary robotics is a methodology for a machine learning type problem. An artificial neural network (the controller of a robot) is evolved so that the robot can complete a given task. In a given environment, a robot receives sensory inputs from objects that are related to the task, and these play a key role when evolving the controller. Previous work has shown that small noise can play a significant role in improving a robot’s performance.
One way to model the controller is to use a continuous dynamical structure called a heteroclinic network. The sensory inputs that a robot receives correspond to perturbations to a heteroclinic network. To analyse the resulting behaviour of a robot after evolution, we need a better understanding of the effects of perturbations and/or noise on a heteroclinic network. A heteroclinic network can exhibit interesting dynamics when small noise and/or perturbations are added. In particular, we expect the residence time near equilibria of a heteroclinic network to monotonically decrease as noise gets larger. However, we observe an increase in the residence time for a certain range of noise when both perturbations and noise are added.
In this talk, I will discuss a heteroclinic cycle called the Guckenheimer-Holmes cycle, and how the addition of small noise and/or perturbations change the dynamics. I will also illustrate a general setting for an Evolutionary Robotics task with examples.
Gray Manicom “A network model of task-switching”
Abstract: Psychologists have long been interested in the delay that occurs when people switch from performing one task to performing another task, called a switch cost. In this talk I will propose a model of task-switching that uses a mixed heteroclinic and excitable network.
The time it takes to complete a task is modeled by the time it takes to complete one of the cycles within the network. Input is added to the network so that there are transitions along the excitable connections and so that an appropriate sequence of cycles is followed. This construction allows the network to have memory such that the time it takes to complete a cycle is dependent on which cycle was most recently traversed. Thus, the model to reproduce the characteristic patterns associated with the switch cost observed in experiments.
Posters:
Jen Creaser (Exeter)
Valerie Jeong (Auckland) “Evolving robots that have heteroclinic brains”
Gray Manicom (Auckland) “Noisy heteroclinic networks”
Max Voit (Bremen) “Learning (in) heteroclinic networks”
Workshop: Network Approaches for Healthcare Applications, 30-31 March 2020
This workshop in Exeter will involve the Centre for Systems, Dynamics and Control as well as the Centre for Predictive Modelling in Healthcare: for more details and to register, see: