Calculating the Probability of Chaos

Michael Jakobson is at present Professor at the University of Maryland, College Park, MD, USA. Thanks to a grant from the Knut and Alice Wallenberg Foundation, he will be Visiting Professor at the Department of Mathematics, Uppsala University.

The goal of this research project is a development of mathematical methods to research one-dimensional and two-dimensional dynamical systems. Dynamical systems study evolution in time of solutions of equations. The concept has Newtonian celestial mechanics at its origin. It has been successfully applied to many branches such as astronomy, meteorology, and financial markets. Also biologists and medical researchers have applied such methods to a range of problems from studies of fish populations to epidemiology.

Even the simplest dynamical systems may contain surprising degree of mathematical complexity. Some dynamical systems, which are extremely sensitive to initial conditions, are called chaotic systems. Sometimes the chaotic nature of a system is not apparent. The goal of the project is to show that the dynamical system is chaotic under certain conditions and to calculate the probability of chaos. Is the probability that a chosen system is chaotic 1 percent or perhaps 99?

Michael Jacobson proved in the 1980’s that such probability is not zero for a long-studied class of such systems described by quadratic equations in one dimension. One of the most important steps in this area was made by the Swedish mathematicians Michael Benedicks and Lennart Carleson who obtained a now famous more general result, for quadratic equations in one and two dimensions, in 1991. Recently Michael Jakobson’s student found a new powerful method to calculate the probability of a system being chaotic. A further development of the new method is the goal of the project. The study requires a blend of subtle theoretical mathematical methods and precise computational techniques.