## 5 Conclusions  
1. Comparing the saturation correlation dimension \(D_{2}\) among the vibration responses of the pumping station pipeline under different conditions, the \(D_{2}\) of the measuring points are distributed in the range of 1.156-5.283, and all are fractions, which show that the vibration of the pumping station pipeline has chaotic characteristics. The  
Figure 12: Calculation of chaotic eigenvalues of typical IMFs (**a**) Relation between D\({}_{2}\) and m of IMF1 (**b**) \(\lambda_{1}\) Separation factor function of IMF1 (**c**) Slope of double logarithmic curve of IMF2.  
axial vibration of the pipeline presents a chaotic attractor with a lower dimension (1.156-2.569), and the vibration form is relatively simple. At the same time, the \(D_{2}\) of conditions and points which are greatly affected by the unit's operation have a larger value (3.021-5.283), and the vibration form is more complex;
2. The Lyapunov exponents \(\lambda_{1}\) of measuring points under different conditions are between 0.0513 and 0.0774. With the opening of two units, the largest Lyapunov exponent \(\lambda_{1}\) decreases accordingly, suggesting that the unit's operation weakens the chaotic characteristics of the pipeline vibration. The \(\lambda_{1}\) of points at the bifurcation are larger than those of other points under the same condition. The chaotic characteristics of the vibration at the bifurcation are enhanced by the sudden expansion of the pipe diameter at the bifurcation and the impact of water heads at different flow velocities;
3. After the IVMD decomposition of the vibration response of specific points under the unit's operation conditions, the chaotic characteristics of the IMFs are analysed. The results show that the saturation correlation dimension D\({}_{2}\) of IMF1 representing water pulsation excitation in the pipeline is 1.115, and the largest Lyapunov exponent is 0.0774. The IMF2 to IMF4 representing the blade frequency, the rotation frequency, and the frequency doubling vibration excitation generated by the unit's operation do not have chaotic characteristics. It indicates that the chaotic character of the pumping station pipeline is mainly caused by water pulsation in the pipeline, and the vibration caused by the unit masks the chaotic characteristic of the pipeline, which makes the pipeline vibration system more complex.  
In this paper, the chaotic characteristics of the vibration system of the pumping station pipeline are shown by the analysis of the measured vibration responses, and the chaotic excitation is found by combination with IVMD, which provides a theoretical basis for the complete description of the vibration characteristics of the pumping station pipeline. A new way of chaotic characteristics analysis based on IVMD decomposition is also proposed.  
Conceptualization, L.J. and Z.M.; methodology, L.J.; software, [PERSON] and [PERSON]; validation, [PERSON] and [PERSON]; formal analysis, [PERSON] and M.C.; investigation, [PERSON]; resources, [PERSON]; data curation, [PERSON]; writing--original draft preparation, [PERSON]; writing--review and editing, [PERSON]; visualization, [PERSON]; supervision, [PERSON]; funding acquisition, [PERSON] All authors have read and agreed to the published version of the manuscript.  
This research was funded by Program for Science & Water conservancy science and technology innovation project in GuangDong Province, grant number 2020-18.  
Not applicable.  
The data provided in this study are available from the corresponding author.  
The authors declare no conflict of interest.