Title: The stability properties of Hill's linear periodic ODE for large
parameters

Author: Carles Sim\'o

Address: Department de Matem\`atica Aplicada i An\`alisi,
 Universitat de Barcelona,
 Gran Via de les Corts Catalanes 585,
 08007 Barcelona, Catalunya

Email: carles@maia.ub.es

Abstract

The goal is to study the parameter plane in the large for Hill-like
equations, that is, of the form $\ddot{x}+(a+bp(t))x=0$, $p$ being
$1$-periodic (or $2\pi$-periodic) with zero average. 

Asymptotic estimates of the density of the stability regions in the
$(a,b)$-plane for lines of the form $a=\omega^2\cos(\psi),
b=\omega^2\sin(\psi)$ when $\omega\to\infty$ are provided.

This density changes in a discontinuous way at some critical values
of $\psi$ and the fine structure across these critical directions is
investigated.

Furthermore an explanation is given for the web-like structure of the
exponentially narrow stability channels, for large $a,b$, together with
asymptotic estimates of the lines forming that web.

The talk is partly based on works with H. Broer, M. Levi, J. Puig and
R. Vitolo.