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Mathematics and Statistics

Dynamical Systems and Analysis

This theme includes exciting research into dynamical systems theory, probability and their applications in life and physical sciences. Our research is divided into three overlapping interests: Analysis of Dynamical Systems, Applications of Dynamical Systems and Complexity and Control.

Research interests

Our group works on mathematical theory underpinning application areas. Our theoretical research involves:

  • recurrence properties of one dimensional maps and coupled differential equations,
  • dynamical networks, their symmetries and control,
  • nonautonomous and random dynamical systems,
  • probabilistic approaches to dynamical systems,
  • pattern formation in spatially extended systems

As a common theme we use advanced mathematical methodologies to understand, predict and control the dynamical behaviour of complex nonlinear systems. Our main application areas are listed below.

Analysis of Dynamical Systems

The mathematical theory of dynamical systems is mature and vital part of modern mathematics, where new theoretical developments have been inspired by applications, just as new developments in pure mathematics have quickly found dynamical applications. Because of this, this area forms a strong bridge between pure and applied mathematics. Research undertaken at the centre is at the forefront of developing and applying new mathematical results in a range of applications. Specific areas of active research in the centre include:

Chaotic and deterministic behaviour, Bifurcation theory, Ergodic theory, Differential equations, Dynamics with symmetries, Applications to number theory and geometry

Dynamical Systems Software

Many members of the Dynamical Systems and Analysis are involved in developing software for analysing dynamical systems and analyzing data from the lens of dynamical systems theory. We list the most notable of such software in this page.

DynamicalSystems.jl

DynamicalSystems.jl is an award-winning open source Julia software library that spans the entire field of nonlinear dynamics and nonlinear timeseries analysis. It is coordinated by and lead-developed by George Datseris. A brief summary of its contents is:

  • A unified infrastructure for defining dynamical systems and developing new algorithm implementations.
  • 500+ complexity measures for analyzing timeseries and data, building upon a framework for rigorously defining probability spaces in code.
  • 21+ recurrence quantification analysis metrics and plots.
  • 27+ algorithms for optimizing and performing a delay embedding reconstruction of observed timeseries, ranging from the most established to the most recent literature.
  • 15+ algorithms for estimating fractal dimensions from data.
  • Fixed point stability, unstable periodic orbit identification in maps.
  • Automatic attractor detection and multistability, global stability analysis, continuation of any kind of attractors and global stability metrics.
  • Numerous algorithms for analyzing chaotic dynamical systems and extreme/rare events, such as Lyapunov exponents or mean return time statistics.

DDE-BIFTOOL

DDE-BIFTOOL is a set of routines for performing numerical bifurcation analysis of delay-differential equations, running in Matlab or Octave. It was originally created by Koen Engelborghs at KU Leuven and it is currently co-developed by Jan Sieber. Features:

  • Bifurcation analysis for dynamical systems with delay (distributed or discrete)
  • numerical continuation of equilibria, periodic orbits and connecting orbits
  • linear stability analysis for equilibria and periodic orbits
  • normal form computation for bifurcations of equilibria

Agents.jl

Agents.jl is an open source Julia framework for agent-based modeling (ABM): a computational simulation methodology where autonomous agents react to their environment (including other agents) given a predefined set of rules. It is coordinated by and lead-developed by George Datseris.

Some major highlights of Agents.jl that distinguish it among other open source ABM frameworks are:

  1. It is fast (faster than MASON, NetLogo, or Mesa)
  2. It is simple: has a very short learning curve and requires writing minimal code
  3. Has an extensive interface of thousands of out-of-the box possible agent actions
  4. Straightforwardly allows simulations on Open Street Maps
  5. Allows both traditional discrete-time ABM simulations as well as continuous time "event queue based" ABM simulations.

ConceptualClimateModels.jl

ConceptualClimateModels.jl is an open source Julia framework for building, and analyzing as dynamical systems, conceptual models of climate. Such conceptual models are simplified representation of basic climate components, and the processes that connect them, such as flows of energy or mass. Examples are: energy balance models, glaciation cycle models, or climate tipping models.

The framework affords accelerated development to the scientist, being simple and fast to create various kinds of models by selecting from a library of existing climate dynamical processes, or implementing their own. This allows easily testing different physical hypotheses regarding how climate variables couple to each other, or how climate processes are represented by equations. Because ConceptualClimateModels.jl directly integrates with DynamicalSystems.jl, it also allows the researchers to efficiently analyze the created models in terms of tipping points and global stability.

TransitionsInTimeseries.jl

TransitionsInTimeseries.jl is an open source Julia package for estimating transitions (from one dynamic regime or stable state to another) in timeseries and testing the statistical significance of found transitions. It is a by a unifying software for detecting transitions that is accessible, performant, reproducible, reliable and extensible. It is co-developed by  by George Datseris and Jan Swierczek-Jereczek.

TransitionsInTimeseries.jl integrates with the entire Julia ecosystem of timeseries analysis, and hence offers thousands of metrics that can indicate transitions right out of the box. It also offers a variety of analysis pipelines for identifying transitions and a variety of statistical pipelines for testing for significance. In contrast to other existing software with similar target application, TransitionsInTimeseries.jl defines a generic interface for how to find transitions and how to test for significance. Within this interface, it is easy to expand the software by adding more indicators or adding new analysis pipelines to find transitions or to test for their significance.

Beatbox

Vadim N Biktashev has developed and actively maintains the open-source HPC Environment for Biophysically and Anatomically Realistic Cardiac Simulations BeatBox (development supported by EPSRC EP/I029664/1).

Complexity and Control

For larger complex systems there are great challenges in finding the appropriate mathematical framework to understand and (if possible) control their behaviour. In addition to structural complexity and heterogeneity, many real world system questions require a range of mathematical and computational skills in order to find answers. Specific areas of active research in the centre include the following:

Control theory, Hybrid testing of biological and mechanical systems, Multiscale systems and computational modelling, Complex networks and emergent phenomena, Coupled and delayed dynamical systems, Applications to engineering, physical, earth and life sciences

Applications of Dynamics

This group is actively involved in a collaborations that apply interdisciplinary approaches to a wide variety of systems in the life, earth and physical science.  Our group provides underpinning research for and applications of nonlinear dynamical systems and complex networks theory.

This includes the Living Systems Institute which uses interdisciplinary approaches to understand living systems and disease. Peter Ashwin is an investigator with the EPSRC Hub for Quantitative Modelling in Healthcare (supported by EPSRC EP/T017856/1) which aims to promote quantitative modelling in various aspects of healthcare, especially in neuroscience and endocrine modelling.

People and partners

See the drop-down menus below to find out more about the staff members who work within the Dynamical Systems and Analysis theme, and which modules they teach.

Academic staff
Dr Demi Allen Lecturer in Pure Mathematics
Professor Peter Ashwin Professor
Professor Vadim N Biktashev Professor
Professor Mark Holland Associate Professor
Dr Frank Kwasniok Senior Lecturer
Dr Houry Melkonian Senior Lecturer in Mathematics
Dr Julian Newman Lecturer Mathematics and Statistics (E&S)
Professor Ana Rodrigues Associate Professor (Education and Research)
Professor Jan Sieber Associate Professor
Post-doctoral research fellows
Andrea Ceni Postdoctoral Research Fellow
Dr George Datseris Marie Sklodowska-Curie Postdoctoral Fellow
Nicolas Verschueren Van Rees Postdoctoral Research Fellow
Dr Bert Wuyts Postdoctoral Research Fellow
Visitors, emeritus and honorary fellows
Dr Irina V. Biktasheva Honorary Senior Lecturer
Postgraduates
Ahlam Alhadbani Postgraduate
Saleh Alharthi Postgraduate
Ruth Chapman Postgraduate
Noah Cockram Postgraduate
Muhammed Fadera
Abrar Hayat Postgraduate
Raphael Roemer Postgraduate
Gaby Wang Postgraduate

Members of the Dynamical Systems and Analysis theme are happy to discuss potential PhD opportunities. Jobs within this group will be posted on the University job portal.

Postgraduate research opportunities

The Dynamical Systems and Analysis group at Exeter maintains an international reputation for research and we continue to invest in top-quality academics and offer a range of projects to research students to enhance this expertise. For more details of our facilities and training programmes, see our pages for postgraduate taught and postgraduate research degrees.

Apply for a research degree in mathematics

Details of research opportunities within mathematics are listed on our webpage. A list of projects for which funding is currently available can be found on our dedicated studentship webpage.

If you are interested in doing a PhD in pure mathematics, please contact potential supervisors to discuss possible projects. You can find out more and apply on our website.

For questions and enquiries please contact Prof Jan Sieber