Horseshoe templates with global torsion in a driven laser

Abstract

We perform a topological analysis of chaotic signals from a Nd-doped fiber laser with pump modulation at different values of the modulation frequency. In this experiment, the system displays chaotic behavior in three regions C 1/4 , C 1/3 , and C 1/2 of parameter space, located around the subharmonics r /4, r /3, and r /2 of the relaxation frequency r . We observe that the topological structures of the chaotic regimes inside a given region C 1/n are described by the same template. However, templates corresponding to different regions display different global torsions g , which we find to be related to the order of the subharmonics by g (C 1/n )ϭnϪ1. ͓S1063-651X͑97͒09804-8͔ PACS number͑s͒: 05.45.ϩb, 42.65.Sf, 42.55.Wd *Electronic address: Guillaume.Boulant@univ-lille1.fr † Electronic address: Marc.Lefranc@univ-lille1.fr ‡ Electronic address: Serge.Bielawski@univ-lille1.fr § Electronic address: Dominique.Derozier@univ-lille1.fr

Topological analysis of chaos in neural spike train bursts

“…Such scrolls have previously been identified in the Duffing attractor 14 and have been observed in the YAG laser. 15 As the temperature is increased, the first return map, shown in Fig. 2, moves to the left.…”

“…One of these, the one which rolls up ''from outside to inside,'' has been identified in the Duffing oscillator 14 and has been observed in experiments performed on a YAG laser. 15 We provide the algebraic descriptions for these two templates in Tables II and III. In these tables, the branches are labeled by integers 0,1,2,... which describe the local torsion of the period one orbits in these branches, T(i,i)ϭi.…”

“…This systematic behavior has also previously been observed in the Duffing attractor 14 and the YAG laser. 15 …”

Analysis of a Nonsynchronized Sinusoidally Driven Dynamical System

“…Examples are provided by the Duffing equations [Gilmore & McCallum, 1995] or by experimental driven systems such as a driven laser [Boulant et al, 1997] or a driven thermoionic plasma diode [Mansbach et al, 1999]. When the driving force is identified, the phase space R m of the undriven system is usually extended to R m+1 to describe the driven system.…”

“…When the driving force is identified, the phase space R m of the undriven system is usually extended to R m+1 to describe the driven system. In such a case, the analysis relies on the use of stroboscopic sections to compute Poincaré sections and to extract periodic orbits [Gilmore & McCallum, 1995;Boulant et al, 1997]. The system is therefore implicitly considered as being phase synchronized with the driving force.…”