Studying the Essence of Chaos

Björn Winckler received his Ph.D. in Mathematics at the KTH Royal Institute of Technology in Stockholm in 2011. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will hold a postdoctoral position in Professor Marco Martens’ research group at the State University of New York, Stony Brook, USA.

While the meteorologist Edward Lorenz in the 1960’s was studying heat transfer in the atmosphere he made a surprising discovery. It changed our way of thinking about determinism and predictability. The discovery originated a field of study called chaos theory.

In chaotic systems small deviations of initial conditions amplify quickly in time. Lorenz named such a phenomenon the butterfly effect: A flap of the wings of a butterfly in Brazil can result in a tornado in Texas. This means for example that a weather system could be chaotic, and that to make long-term weather predictions is practically impossible.

A mathematical example of the butterfly effect is the Lorenz attractor. It consists of solutions of deterministic equations which can be geometrically described as a spiral pattern which is called the attractor. The attractor does not reach a stable mode; it never repeats its flow. It is an example of a deterministic chaotic system since its pattern appears random forever.

The spiral pattern, or the attractor, lies at the center of mathematical theory of chaos. Contrary to its name, there exist an order and a structure in chaotic systems.

The goal of Björn Winckler’s research project is in depth understanding of the so-called Lorenz maps which are closely related to Lorenz attractors. Understanding the transition from periodic to chaotic behavior is of importance in chaotic dynamical systems, where Lorenz maps play a significant role.