N. Petrov, R. de la Llave, J. Vano
Torus Maps and the Problem
of One-Dimensional Optical Resonator
with a Quasiperiodically Moving Wall
(5275K, PS)
ABSTRACT. We study the problem of the asymptotic behavior
of the electromagnetic field in an optical resonator
one of whose walls is at rest and the other is moving quasiperiodically
(with $d\geq 2$ incommensurate frequencies).
We show that this problem can be reduced to a problem about the behavior
of the iterates of a map of the $d$-dimensional torus
that preserves a foliation by irrational straight lines.
In particular, the Jacobian of this map
has $(d-1)$ eigenvalues equal to~$1$.
We present rigorous and numerical results about
several dynamical features of such maps.
We also show how these dynamical features
translate into properties for the field in the cavity.
In particular, we show that when the torus map satisfies a KAM theorem
-- which happens for a Cantor set of positive measure of parameters --
the energy of the electromagnetic field remains bounded.
When the torus map is in a resonant region -- which happens in
open sets of parameters inside the gaps of the previous Cantor set --
the energy grows exponentially.