Applications in Partial Differential Equations

Many ecosystems and physical systems, such as the climate, can display strongly nonlinear behaviour, where the properties of the system suddenly change with changes in environmental conditions. A well-known and urgent example of this is the discussion about ‘tipping’ of the climate, thought to set off a dramatic change in the World’s climate if global temperatures start to exceed some threshold. Mathematically, ‘tipping’ is an example of a so-called bifurcation of the underlying system of equations. We will focus on the bifurcations in systems of ordinary differential equations (ODEs) and partial differential equations (PDEs) representing real-world systems, which can often only be solved numerically. Hence, this course focusses on numerical bifurcation theory: the study of understanding parameter sensitivity and bifurcations in numerical solutions.
The course covers mathematical theory and papers applying the theory to real-life applications including climate science and ecosystems. We build on theory of ODEs, numerical methods and mathematical modelling. Basic programming experience is required for the practical assignments.