TOPOLOGY OF RESONANCE ZONES

In the paper [35] with Henk Broer we proved a curious effect: for an open set of Mathieu equations $\ddot x + (a + bp(t) ) x = 0 $ with an even $p$, the $ n $th and the $ n+1 $st stability boundaries meet at $\geq n$ points (counting multiplicity).  This happens for an open set of functions $p(t)=p(t+1) \in L^1$ near the piecewise constant, as well as for all known classical examples. This shows in particular that as $b$ changes from $-\infty $ to $\infty $, the $n$th forbidden gap collapses to a point at least $n$ times! 
In a preprint [60] with Henk Broer and Carles Simo we found and and explained remarkable webs that arise in stability diagrams of Hill’s equations.