## 3 Results and Discussion
### Time Integration Results
Figure 9: Phase plot of the wheel–rail contact point for all modal-derived initial conditions from Table 3. Grey: complete history, blue: intermediate limit cycle, black: final limit cycle reached, ‘x’: initial condition. Figure 10 highlights an interesting result, namely, that the two families of limit cycles identified before (the low-energy and the high-energy ones) can be obtained for both physical and modal initial conditions. Depending on their value, physically-derived initial conditions can lead to either of the two, but in the first case, the same periodic limit cycle, plotted in black in Figure 10a, is always obtained. Test cases with modal initial conditions can lead to more than one unique periodic cycle, as can be seen in Figure 10b: in total, four unique periodic cycles have been identified. The quasi-periodic cycles reached for both categories of test cases correspond in reality to a unique limit cycle that tends to color the same surface in the phase plot, with each individual instance having a different translation in time. Figures 11 and 12 show the velocity spectrum and the friction force spectrum, respectively, for all the periodic limit-cycles that were identified among all test cases regardless of the type of initial conditions used. The aforementioned unique periodic cycles have different fundamental frequencies: 1855 Hz, 2775 Hz, 3750 Hz for test cases \"0L4\", \"0L5\", \"0L6\", respectively, which are close to the natural frequencies of the corresponding modes, and 385 Hz for all other periodic cycles, which is close to the natural frequency of mode 0L2. Figure 11: FFT of the velocity of the periodic limit cycles obtained among all test cases (red: “0L4”, magenta: “0L5”, green: “0L6”, black (curves superpose): all other cases leading to a periodic limit cycle). Figure 10: Phase plot of the final limit cycles obtained for the physically-derived initial conditions (**a**) and the modal-derived ones (**b**). Creep and slip zones are below and above the dashed black line, respectively. (**a**): red: “V_-1.0”, blue: “V_-0.5”, black: all other physically-derived initial conditions. (**b**): red: “0L4”, magenta: “0L5”, green: “0L6”, black (curves superpose): “0L3”/ “0L7”/ “0L8”/“All_unstable_V1.0”/ “All_unstable_V2.0”, blue (curves superpose): all other modal-derived initial conditions. The spectrum in Figure 13 can be considered as quasi-periodic if \(f_{0}=384.6\) Hz, \(f_{1}=4763.2\) Hz and \(f_{2}=5792.8\) Hz are accepted as its fundamental, incommensurable frequencies (i.e., that have no common integer divisor). In spite of the low amplitude of the peak at \(f_{0}\), the latter is considered as a fundamental frequency because it explains some of the harmonic modulations (i.e., linear combinations of the fundamental frequencies) present in the spectrum. Notice that any two of the three high-amplitude peaks at \(f_{2}-f_{1}\), \(f_{1}\) and \(f_{2}\) could have been labeled as fundamental frequencies and an equivalent understanding of the spectrum would have been obtained. For the reader's comprehension, this spectrum can be qualified as bi-periodic with fundamental frequencies \(f_{1}\) and \(f_{2}\) and a secondary contribution of \(f_{0}\). Regardless, these three frequencies are close to the natural frequencies of modes 0L7, 0L8 and 0L2, respectively, and the harmonic modulations at \(f_{2}-f_{1}=1029.6\) Hz and \(2f_{1}-f_{2}=3733.6\) Hz are very close to the natural frequency of modes 0L3 and 0L6, respectively. Indeed, the strong harmonic correlation between the unstable modes 0L2, 0L3, 0L6, 0L7 and 0L8 observed in this limit cycle could explain its quasi-periodic nature. The existence of this particular quasi-periodic limit cycle illustrates the limitation of the stability analysis previously proposed in Section 3.2, which is not able to predict the appearance of some harmonic components based on the combination of the fundamental frequency of unstable modes and that end up having a strong contribution on the self-sustained oscillations.