## 3 Results
### Qualitative Analysis by RP  
The results for Case 1 at 700 rpm with the SDEMDI method are presented in Figure 13 and the same calculation for Case 2 at 700 rpm is shown in Figure 14. In both figures, it is possible to observe the evolution of the state variables. The behavior is dominated by the trends that remain in each spectrum; however, in the RP with friction, the characteristic diagonal lines of the nonlinearities of the system prevail. According to the results, there is a clear difference when friction is included in the system, and it is reflected in RP. All recurrence graphs were calculated with the same threshold, \(e=0.1\), for comparison purposes.  
The same two cases presented in the previous paragraph are calculated with the proposed PSEMDI method: Case 1 in Figure 15 and Case 2 in Figure 16. At first glance, there are striking visual differences between the results. The phase graphs show a more defined and symmetrical behavior. Similarly, the recurrence graphs are quite different and provide more information about the system. Again, a clear difference can be observed when friction is included in the analysis, although it is more uniform between both cases.  
The results for Case 2 at 700 rpm with the proposed method are presented in Figure 14. Friction has more influence on the main diagonal of the RP.  
The results for the two cases at 1800 rpm are shown in Figure 17. The system behavior at higher speed showed a similar behavior to that at 700 rpm in the phase diagrams for each integration method, and it is shown here. The recurrence plots indicate a decrease in the pattern point density with the inclusion of friction, compared with the behavior at 700 rpm. Again, there is a big difference between the results from both methods. Similarly, friction also generates differences between the main diagonals in both methods.  
Figure 14: (**a**) Evolution of state variables; (**b**) phase plane; (**c**) recurrence plot for Case 2 at 700 rpm.  
Figure 13: (**a**) Evolution of state variables (ESV); (**b**) phase plane (PP); (**c**) recurrence plot for Case 1 at 700 rpm.  
Figure 16: (a) Evolution of state variables; (**b**) phase plane; (**c**) recurrence plot for Case 2 at 700 rpm.  
Figure 15: (a) Evolution of state variables; (**b**) phase plane; (**c**) recurrence plot for Case 1 at 700 rpm.  
Similarly, the results obtained with the proposed PSEMDI method were quantified by RQA and presented in Table 4. In this case, the differences due to friction are low for all parameters. The parameters that increased with friction for the two speeds were only \(DET\) and \(L_{max}\).  
A graphical comparison of all the parameters for the two integration methods is shown in Figure 18. As it can be seen, the results for the PSEMDI method are more constant and uniform. Moreover, the most marked differences between the two methods are shown by the parameters \(L_{max}\) and \(V_{max}\). These differences are due to the double integration process of the SDEMDI method, which has a very significant effect on the values of the maximum vertical and diagonal lines.  
Figure 17: RPs from the two methods for Case 1 and Case 2, at 1800 rpm: without friction (**a,c**); with friction (**b,d**).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **U700** & **UF700** & **U1800** & **UF1800** \\ \hline \(RR\) & 0.06 & 0.06 & 0.06 & 0.06 \\ \hline \(DET\) & 0.77 & 0.78 & 0.63 & 0.74 \\ \hline \(L\) & 3.54 & 3.64 & 3.28 & 3.60 \\ \hline \(L_{max}\) & 72 & 57 & 41 & 92 \\ \hline \(ENTR\) & 1.56 & 1.64 & 1.49 & 1.61 \\ \hline \(LAM\) & 0.89 & 0.90 & 0.78 & 0.85 \\ \hline \(TT\) & 3.64 & 3.44 & 3.39 & 4.02 \\ \hline \(V_{max}\) & 25 & 15 & 15 & 19 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Quantification of SDEMDI results.