Characterization of homoclinic chaos through double-valued return time maps

For a laser with a saturable absorber (LSA) and for a subnormal glow discharge (GD), both displaying homoclinic chaos, it is shown that double-valued curves in the return time maps, reconstructed from the time evolution of appropriate variables, are related to different time scales associated with the two mechanisms present in the respective chaotic attractors: the escape from an unstable saddle cycle and the reinjection process. For the LSA the investigation is performed numerically on a 3-2 molecular level model and the results are compared with experimental ones obtained from a ${\mathrm{CO}}_2-{\mathrm{OsO}}_4$ LSA system having the laser frequency detuning as the control parameter. The analysis is complemented with experimental results from the GD. For both systems we show how to obtain single-valued multibranched return time maps starting from double-valued return time maps, enabling the characterization of homoclinic chaos.