## 4 Results and Discussion
### The HRVD Oscillator Model with a Periodic External Excitation
Comparing Figure 3a with Figure 3b reveals that the trajectory structures in both images exhibit similar vortex shapes and singularity locations, although the axes and scaling of the images are slightly different (this is due to the fact that Figure 3a is plotted based on the time-step data while Figure 3b is plotted based on the raw time data, hence there is a three-order-of-magnitude difference between the two), the embedded flow field's overall correspondence between the trajectory morphology and the phase diagram is evident, implying that the embedded flow field performs well in preserving the global topology. A closer look at the local vortex structure and trajectory spacing, the dense length of the trajectories, and the trend of morphology change shows that the embedded flow field is also more successful in restoring the local features. In addition, the trajectories in the embedded flow field retain a similar smoothness to the original phase diagram, with no obvious discrete points or anomalous jumps, which implies that the flow field is also more accurately restored in terms of continuity and smoothness. Thus, assessed visually, the two maps are almost topologically equivalent. Observe the second subfigure of Figure 3f, where the intermittent periodic bursts observed in the time series \(v_{r}\) of the stressing features indicate that the model (23) under the parameter group Para1 would exhibit bursting oscillatory behavior. If we take \(r=3\) while keeping the embedding dimension \(m=3\) constant, then \(E(k)\) still satisfies the condition of being greater than 99%. Although the MSE in this case increases slightly to 1.0795 \(\times\)\(10^{-15}\), it is still at a relatively high level. Since we have raised the value of \(r\) to 3, we can plot the reconstructed 2D flow field (Figure 4b). As can be seen from the figure, the original 2D phase diagram of the reconstructed 2D flow field as well as the 2D-embedded flow field are likewise almost equivalent in terms of topology. Figure 4: _Cont._  
However, observing Figure 4d we can find that when \(m=3,\ r=3\), the construction of the 3D-embedded flow field fails to maintain the smoothness although we achieve a more accurate and effective 1D and 2D reconstruction (Figure 4e) and capture the emergence of the bursting oscillations by the forced eigenmodes. Therefore, we consider fixing \(r=4\) (to facilitate the reconstruction of the 3D-embedded flow field), and raising the embedding dimension. Fixing \(r=4\) in Figure 5, we record the 3D-embedded flow field plotted when the embedding dimension m takes different values and its reconstructed flow field. Figure 4: Series of experimental plots when the parameter set is taken to be Para1, \(m=3\), \(r=3\). (**a**,**b**) 2D\(-\)embedded flow field and its reconstructed flow field; (**c**) Original 2D phase diagram; (**d**) 3D-embedded flow field; (**e**) Reconstructed fitted diagrams, similar to Figure 2; (**f**) Dominant feature time series vs. forced feature time series. Figure 5: _Cont._  
Based on Figure 5 and the reconstruction error MSE for each dimension, we organize and obtain Table 1.  
From Table 1 we find that when fixing \(r=4\) and increasing the embedding dimension, the MSE has been kept at a relatively low level, which ensures that all our reconstructions have a very low error. However, the performance of different embedding dimensions varies in terms of how smooth the reconstructed flow field is with respect to the time series of the forced features. The embedding dimension \(m=200\) is reasonable in terms of reconstruction