The Puzzle of Integrable Circle Maps
Talk by Dr. Jarek Kwapisz (Mathematical Sciences, MSU)
03/01/2021 4:10-5:15pm Zoom Talk
The simplest model of a non-linear oscillator is given by certain maps of the circle to itself. The asymptotic frequency of the oscillator, called the rotation number, depends continuously on the parameter. In natural families, including physical systems like the Josephson junction, the rotation number has the Devil's staircase-like graph, with the steps corresponding to (mode-locked) resonances.
I will discuss the existence of non-trivial maps for which all the steps vanish. Some informal insight can be gained from a novel system consisting of a transport PDE and a functional equation. However, beating the Devil by a rigorous argument seems hard. This is a failure report.