Chaos theory, the butterfly effect, and strange attractors

The first to observe this peculiarity was Edward Lorenz, who studied the phenomenon of atmospheric convection. He discovered that the smallest disruption in the initial conditions can lead to unproportionately large changes after a time. This explains why weather forecasting remains such a difficult task to date – even the air movement caused by the flapping wings of an insect can gain momentum and become a hurricane, a phenomenon commonly known as the butterfly effect. The American scientist’s research gave birth to an entirely new branch of knowledge called chaos theory, which has been the focus of our mathematicians. They investigate geometric objects known as chaotic attractors, which include the Lorenz attractor.

Mathematical adventures

AGH UST Professor Boroński obtained his doctor’s degree in the United States, under the supervision of Professor Krystyna Kuperberg, a famous Polish-born American mathematician, who had studied under the eyes of the representatives of the Warsaw School of Mathematics. During his stay, our researcher had the opportunity to meet James Yorke, a pioneer chaos theorist, who was depicted as a mathematician in Jurrasic Park, where the butterfly effect was referenced. After he had returned from the US, Dr Boroński continued his research on chaotic attractors, which resulted in his postdoctoral dissertation pertaining to the inverse boundary methods that can be used to study complex mathematical objects.

The Mathematical Genealogy Project by Dr Jan Boroński

Currently, within a project funded by NCN SONATA BIS, the scientist leads a research team that deals with parameterised families of non-hyperbolic attractors. The Hénon attractor, belonging to one of such families, constitutes – to put it briefly – a two-dimensional simplification of the 3D Lorenz attractor. Dr Boroński and his co-workers are trying to take a step further and compare the isomorphism of the dynamics of these objects and the dynamics of one-dimensional topological graphs, as well as dendrites and their constant transformations. The project also includes the investigation of bifurcations, that is, parameter-bound sudden changes of the state of an attractor. One of the main issues that the team focuses on is the problem introduced by Professor Yorke towards the end of the 1980s. It pertains to the achievable rotation numbers of Hénon attractors, which remains unresolved for three decades.