Global torsions of the template in a subharmonic driven fiber laser
G. Boulant, M. Lefranc, S. Bielawski and D. Derozier
Laboratoire de spectroscopie hertzienne, Centre d'Etudes et de
Recherches Lasers et Applications,
Université de Lille I, F-59655 Villeneuve d'Ascq Cedex
France
Template analysis provides a relevant classification and comparison of chaotic attractors by means of their topological properties. This method has been successfully applied to experiments in various fields [1]. In these investigations, most attractors have been shown to be topologically equivalent, as they are associated to the same template, the horseshoe with zero global torsion. Here, our aim is to use topological analysis in order to study and compare the attractors of a single system (the modulated Nd-doped fiber laser), for different values of a control parameter (the modulation frequency).
Low-dimensional chaos (with Lyapunov dimension less than 3) may appear
in this system when the modulation frequency ranges near the
subharmonics 1/2, 1/3
and 1/4 of the laser relaxation frequency. We will call the associated
regimes 
, 
, and 
, respectively. The
time-series analysis is performed in the following way. First, we
extract the periodic orbits embedded in the chaotic attractor by the
well-known close-return technique [2]. Then, we characterize the
topological organization of these orbits in a three-dimensional
embedding space, by computing their linking numbers and relative
rotation rates. From these invariants, we deduce the underlying
template (or knot-holder) [3]. This latter is a branched manifold on
which one can place all periodic orbits, while keeping their invariants
unchanged.
Inside each parameter region ( , , or ),
the template does not undergo any changes, since the invariants are
robust to parameter modifications. However, the and templates differ each other by their global torsions.
This phenomenon stems from the resonance of modulation harmonics with
the relaxation frequency.