Mathematical models of chaotic phenomena

Sven Sandfeldt will receive his doctoral degree in mathematics from KTH Royal Institute of Technology in 2025. Thanks to a grant from the Knut and Alice Wal-lenberg Foundation, he will hold a postdoctoral position with Professor Amie Wilkinson, University of Chicago, USA.

The field of dynamical systems studies processes that change over time. One of the areas studied by dynamical systems theory is mathematical models of chaotic phenomena. The aim is to be able to qualitatively understand the system in the long term, even though exact predictions of the system are inconceivable. Examples of the systems that could be modeled include weather, population growth, the spread of disease, and the movement of celestial bodies.

This project deals with extracting information about the system's dynamics from its symmetries. Symmetries in a system are transformations of space that do not change the system. More broadly, in mathematics, symmetries are commonly studied objects. For example, a square has eight symmetries – eight ways in which it can be flipped or rotated to make a square again. However, a circle can be rotated as much as you like – it has an infinite number of symmetries.

Studying symmetries makes it possible to derive algebraic properties of the dynamical system, which leads to a deeper understanding of the dynamics. One might think that chaotic systems would be very difficult to preserve under symmetry operations, but if this happens, it gives rise to a surprising number of additional symmetries. One result is that even a minimal number of symmetries combined with some chaotic behaviour can lead to a complete understanding of the dynamics of the system. This phenomenon is called rigidity, and its local and global variants are the topics of further exploration in the project.