Experiments and results

If we wish to use the RQA method to a time series long enough for reliability, we face the usual difficulties, memory capacity and calculus complexity. Most of the codes were developed in C under Linux operating system. This language does not implement the Boolean data type using integer type instead, so the required memory amount to store the recurrence matrix coming from a typical time series of $2^{16}$ data points is 128 Gbits and, the algorithm complexity is quadratic, which implies huge time consuming. In order to solve these problems we propose to reduce the dimension of the matrix $R$, but then the mapping may not be one to one and a whole region of the reconstructed space would have the same points as image. Therefore $R$ as defined in (1) is modified as: $R_{(k,l)} > 0$ if and only if at least a pair of indexes exists $(i,j) \in \{(i,j) / k=[i \frac{N}{m}], \, l=[j \frac{N}{m}] \}$ for $k$ and $l$ given, with $N=\textrm{dim}(R)$ and $m$ the amount data points, such as $\Theta(\delta_h-\vert\vert y_i-y_j\vert\vert _{\infty}) - \Theta(\delta_l-\vert\vert y_i-y_j\vert\vert _{\infty}) >0$ and zero otherwise. Due to hardware limitation we use recurrence matrices 10240 x 10240 values, although this fact was discussed before,and means a loss in the matrix resolution, the obtained results are quite satisfactory taking into account an adjusting of the corridor width and studying the parameters asymptotic behavior as a function of the number of the data. Another problem concers the RQA high sensibility with the number of cycles during a time interval. As is depicted in figure-1 %DET has an asymptotic value when the number of cycles is greater than 30. If we are dealing with a sub-harmonic cascade or in the intermittency region an abrupt change in the number of cycles is observed, giving us false transitions associate to the change in the cycles number. But the number of cycles can not be so big because it produces a graphic saturation then %DET falls on minor values. A reason is an increment in vectors amount mapped to same $(k,l)$ matrix component. So when this amount goes too far a heuristic limit; determinism is lost. Taking this into account the parametric measurements associate to the RQA were done in such a way to use the half cycles on the plateau.

Figure 1: %DET as a function of cycles number for PO chemical reaction. For $k_3=0.028$ a limit cycle periodic behavior is observed. From 50 cycles the %DET decays due to the saturation of the RP.

Figure 2: $S(k_3)$ for 40 cycles, a plateau can be seen as in the chaotic as in the periodic regions, for values in the range $0.033 < k_3 < 0.035$ the plateau observed is an indication of the chaotic state of the system.

Since RP is not quite sensible to the election of the embedding dimension, $d$, we use $d=9$ which this value allow us to eliminate the non diagonal points in the RP plots without affecting its general structure [11]. On the other hand, no difference is observed neither $S(k_3)$, or any RQA parameters within a range $3 \ge d \ge 15$, so we use the value recommended by Takens' theorem [4]. As it can be seen in figures-2and 3 there is no direct proportion found between $S(k_3)$, and the maximum Lyapunov exponent $\lambda (k_3)$, by using the Aurell's algorithm et al. in TISEAN package [12] calculated with a Thieler's window [13] with 500 time units. Different algorithms associated with the evaluation of $\lambda (k_3)$ int TISEAN package were used with unsatisfactory results due to the time consuming for getting a reliable slope in the graphical methods. On the other hand in figure-3 the expected plateau is not observed in the range of values when the system is in the chaotic regime as studied by Geest et al[7]. They have done an exhaustive numerica study of the NADH oxidation reaction in the chaotic regime. It is important to note as depicted in figure-2 the chaos transition characterization since $S(k_3)$ increase its value towards a plateau for the $k_3$ values which correspond to a sub-harmonic cascade, as well as a peak in the neighborhood of the first bifurcation value.

Figure 3: Maximum Lyapunov exponent, $\lambda (k_3)$, found by using Aurell's et al. algorithm for 40 cycles. The observed plateau is apparent since in that range of values the system is aproaching to the onset of a subharmonic cascade and in the chaotic region the algorithm does not show conclusive evidences.

%DET($k_3$) is associated with the existence of stable and unstable periodic orbits. In general the probability density related with periodic orbits is uniform, but for a dense family of unstable orbits, it has an exponential distribution since it is more probable to find our systems orbits with shorter period. For us, %DET$\ge$99% always, and the behavior in the chaotic region is the same as $S(k_3)$, that means the increasing number of periodic orbits due to the sub-harmonic cascade from $k_3=0.031$ up to reach a maximum in the chaotic region densely populated by unstable orbits. For $k_3>0.0355$ the number of orbits is considerably reduced after an intermittent behavior. This is intrinsic to our dynamical system since it is not observed in the Lorentz system, used as a toy model [11]in this trail. On the other hand, by analyzing %REC($k_3$) we may appreciate the scarce population of recurrent points in the RP. %REC holds a constant value and is reduced at the beginning of the cascade with a minimum in the chaotic region. This behavior is different of that observed with %DET and $S$ augmenting the value up to get a lower plateau. We assume this can be due to the changes in the shape of the attractor, since the same results is obtained for the Lorentz system [11]. By a simple inspection in the RP the increasing number of periodic orbits is clear, because more parallel lines to the diagonal appear, but the changes in the attractor shape are less impressive and can only be appreciate as skewed curves when the embedding dimension is near $d=3$.