Extending the theory of almost stable systems 

Bassam Fayad is currently professor at Université Paris Diderot, France. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will be a visiting professor at the Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden. 

In 1889, the Swedish king Oscar II awarded a prize to French mathematician Henri Poincaré for his work on the three-body problem: what are the trajectories of three heavenly bodies, like the Sun, the Earth and the Moon, when they are only governed by gravity. Poincaré first claimed that a solution existed, but the reviewer at Acta Mathematica, Edvard Phragmen, discovered an error in the manuscript. In his subsequent work Poincaré proved that it was impossible to determine whether such a system remains stable for an infinite time to come. This was the starting point for modern chaos theory. 

Stable systems are rarely encountered in nature, while almost stable systems are much more common, including the trio of the Sun, Earth, and Moon. These systems can be described using the KAM theory (after its originators – Kolmogorov, Arnold, and Moser), which, since the 1960s, has become an essential tool in studies of dynamical systems. According to the theory, the evolution over time of almost stable systems maintains certain stability characteristics of stable systems.  

However, classic KAM theory rests on assumptions, which are violated in many natural applications. The aim of the project is to describe possible dynamics, especially chaotic, in almost stable systems both within the framework of KAM theory and beyond it, when some of its more common assumptions no longer hold.  

When and in what way can a system prove unstable? For example, can a planet leave our solar system? Can a system become chaotic? The invited visiting professor Bassam Fayad is a top-level researcher in the theory of dynamical systems, and has worked on several past and present projects with Associate Professor Maria Saprykina and her colleagues at KTH Royal Institute of Technology in Stockholm

Photo: Université Paris Diderot