156
J. Eur. Opt. Society-Rapid Publ. 21, 14( 2025)
Fig. 6. First line: Proportion of false nearest neighbours versus the dimension of the reconstructed vectors in the case( a) stable( b) intermittences, and( c) chaotic. Second and Third lines: 4D projection of the reconstructed phase space for( d) and( f) intermittent regime and( e) and( g) chaotic regime.
of the rate equations do not, for the moment, provide results like those obtained experimentally, perhaps indicating the need to take into account variables whose impact on dynamics is not yet understood. Hence, to explore more deeply the system dynamics, we realised a phase space reconstruction using a model based on Taken’ s theorem [ 22 ]. To do so, we construct the vectors Y [ n ] suchas:
Y ½ nŠ ¼ fIn ½ Š; Inþ ½ sf e Š;...; Inþ ½ ðd e � 1Þsf e Šg
where I [ n ] represents the n-th measured point of the output intensity at 2.3 lm, s is the time lag, f e is the sampling frequency of our scope and d e is the embedding dimension.
The main difficulty of this reconstruction lies on the choice of the embedding dimension and the right time lag to ensure that the trajectory unfolding is complete. For the time lag, we have to consider a sufficiently long time to ensure the independence of the variables used to reconstruct the system’ s phase space. But, since the system is chaotic, the exponential growth of small perturbations, obliges us to have a sufficiently short time lag to ensure connection between the variables([ 23 ], p. 25). Hence, we chose as our time lag, the natural emission time of laser impulsions, by working directly with the maximal intensities extracted from the time track. Here, we consider that the repetition rate is sufficiently long to let the system explore