VOLUME 68, NUMBER 8 PHYSICAL REVIEW LETTERS 24 FEBRUARY 1992 Structure of Chaos in the Laser with Saturable Absorber F. PapofT, (,) A. Fioretti, (2) E. Arimondo, (2) G. B. Mindlin, (3Ma) H. SoIari, OMb) and R. Gilmore (3) (]) Scuola IS ormale Superiore, Piazza dei Cavalieri 2, 56100 Pisa, Italy (2) Dipartimento di Fisica, Piazza Torricelli 6 56100 Pisa, Italy <3) Department of Physics and Atmospheric Science, Drexel University, Philadelphia, Pennsylvania 19104 (Received 7 August 1991) We carry out a topological analysis on an experimental data set from the laser with saturable ab- sorber. This analysis is based on the topological organization of low period orbits extracted from chaotic time series data. This allows us to determine for the first time that previously proposed models are com- patible with the data. PACS numbers: 42.50.Lc, 05.45.+b, 42.55.-f It has recently become possible to provide a topological classification of strange attractors. In this classification scheme the topological structure of a strange attractor is given by a set of integers. These integers describe the structure of the "knot-holder" [1] or "template" which supports the strange attractor [2]. The template de- scribes the stretching and compressing mechanisms re- sponsible for creating the strange attractor. These stretching and compressing mechanisms are also responsi- ble for organizing the unstable periodic orbits which are embedded in the strange attractor in a unique way. This topological classification is in contrast to the classification of strange attractors according to their metric properties (e.g., Lyapunov exponents, various di- mensions). Metric properties are invariant under a coor- dinate transformation but not under control parameter variation. Topological properties remain invariant under both coordinate transformations and control parameter variations, or change in experimental conditions. This means, in particular, that chaotic data sets taken for a physical system under different experimental conditions will exhibit the same topological classification. Furthermore, it is possible to determine the topological classification of a strange attractor by carrying out an analysis ("topological analysis") on scalar time series data [3]. This provides, for the first time, a test to deter- mine whether a model which is proposed to describe a chaotic process is in fact compatible with that process. Topological analyses are carried out on the experimental time series data and data generated by the model. If the topological analyses identify different templates (sets of integers), the model can be rejected as not compatible with the data. Otherwise, the model is compatible with the data. In this Letter we apply this topological analysis, for the first time, to determine whether models proposed to de- scribed the laser with saturable absorber (LSA) [4-7] are compatible with a number of experimental data sets which have been taken from the LSA [6-9]. The experimental setup consists of an infrared cavity containing a discharge CO2 amplifier and an absorber cell. We have used CH3l:He and OsC^iHe in the ratio 1:20 as absorbers. The laser output intensity / is digitized and discretely sampled at a rate of about 80 samples per period for fixed values of the control parameters. These include the discharge current, the absorber pressure, and the laser frequency detuning. A number of long time series, up to 32x]0 3 8-bit data, were stored in a micro- computer by use of a digital oscilloscope. In the region of the LSA where instabilities and chaos are found, the laser intensity pulse starts below threshold, developing a large peak L followed by a variable number of small peaks S. Each minimum is followed by a max- imum; the deeper the minimum, the more intense the fol- lowing maximum. A segment of a typical data set is shown in Fig. 1. Three [4,6], four [5], and five [7] variable models have been proposed to describe the LSA. The mechanism re- sponsible for the existence of chaotic behavior in these models is the following. An unstable limit cycle (saddle cycle) or its degenerate limit, an unstable focus, has stable and unstable invariant manifolds which approach tangency [10] [cf. Fig. 2(a)]. In the three-, four-, and five-dimensional models the unstable manifold of the sad- dle cycle is two dimensional, while the stable manifold is 2D, 3D, and 4D, respectively. As the tangency is ap- proached, a number of stable periodic orbits are created by saddle-node bifurcations, which then undergo period- doubling cascades. This generates a complicated dynam- ics even before the tangency occurs [10,11]. When the manifolds behave as shown in Fig. 2(b) the flow is hyper- bolic and exhibits a Smale horseshoe [10] with zero glo- bal torsion. A point in the strange attractor near the un- stable invariant manifold W u will evolve during one period along W u (stretching) while at the same time be- ing compressed along the direction of the stable manifold IV s towards the invariant set W u (folding). The stretch- ing and folding mechanisms, responsible for the creation of chaos, are represented schematically by the template for this flow, shown in Fig. 2(c). To determine whether these models are compatible with the experimental data, we must identify the tem- plate underlying the experimental strange attractor, and compare it with the template shown in Fig. 2(c). The to- pological analysis of the chaotic data was carried out in a number of simple steps. First, the unstable periodic orbits embedded in the strange attractor were determined by the method of close 1128 © 1992 The American Physical Society