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Department of Mathematics

A printed geometry equation on a piece of paper

Dynamical systems

Dynamical systems combines theoretical and numerical approaches to understand the dynamic behaviour of models in mathematics and across most areas of science and engineering.

Since the explosion of interest in chaos some fifty years ago, the idea that simple deterministic rules can generate complicated and unpredictable temporal behaviour has been used to improve our understanding of physical and biological processes. For example, weather forecasts now routinely rely on aggregating the results of many forecasts based on slightly different initial conditions so that it is possible to assess the variability of predictions.

These insights lead to two important directions of research, both of which are investigated at Manchester. First, there are theoretical questions about the general properties of solutions of difference equations and differential equations, including how the nature of solutions changes as external parameters are changed, and how additional structure in the equations (eg: Hamiltonian systems, network connections) changes the behaviour of the systems. Secondly, there are issues around the creation of appropriate models for particular applications and the analysis and interpretation of these models.

We work with colleagues from other universities in the UK and abroad, and in inter-disciplinary groups involving other departments within The University of Manchester, to understand a broad range of problems. We also have close ties with other themes within the Department of Mathematics such as Life Sciences, Data Science and Algebra.

Current research areas include:

  • Applications in Biology
  • Applications in Data Science
  • Applications in Health and Epidemiology
  • Arithmetic dynamics
  • Bifurcation theory
  • Hamiltonian dynamics
  • One-dimensional maps
  • Piecewise smooth dynamics
  • Skew products 
  • Stochastic dynamics

More information about our research, and some papers, can be found by browsing the web pages of the staff members. Potential PhD students may email staff directly to discuss possible projects.