## 3 Chaotic Characteristics Analysis of Pipeline Vibration Response
The slope fitted by the near line segment of the curve is the correlation dimension under the corresponding embedding dimension. As the embedding dimension increases, it is the saturation dimension \(D_{2}\) when the correlation dimension reaches saturation. Figure 6 is the diagram representing the calculation of the saturation correlation dimension of specific points. Figure 5: \(m\) Calculation of point 1 Z-axis vibration under condition 4. Figure 6: Calculation of point 1 z-axis vibration under condition 4 (**a**) Double logarithmic curve, (**b**) Slope of double logarithmic curve, and (**c**) Relation between \(D_{2}\) and m.  
To reveal the distribution law of the saturation correlation dimension, the \(D_{2}\) variation curves of points in each direction under different conditions are shown in Figure 7.  
As can be seen from Figure 7:  
1. In general, \(D_{2}\) ranges from 1.156 to 5.283, and they are fractional, indicating that the responses of the pipeline in all directions are chaotic;
2. Compared with the other two directions, the correlation dimension of the axial measurement points (Y-axis) of the main pipe is obviously smaller than that of the other two directions. It shows that the axial vibration of the pipeline has a smaller dimension chaotic attractor and requires fewer independent control variables to describe the dynamic system. This is mainly because the direction of the centrifugal force generated by the centrifugal pump of units is not in the axial direction of the main pipeline;
3. At the same points, the \(D_{2}\) of each point in condition 4 (No. 4 and 5 units in stable operation) is greater than in other conditions, while the corresponding \(D_{2}\) in condition 3 (No. 4 unit in closing) is less. It indicates that the pipeline vibration is more complicated in the stable conditions of the two units and the complexity of the pipeline vibration is relatively weak in the closed condition. Unit operation increases the uncertainty of the pipeline vibration;
4. In the same condition, the points near the units (point 1 and 5) and the bifurcated pipe (point 4) reach a relatively larger \(D_{2}\), indicating that the vibration complexity of the pumping station pipe is greatly affected by the unit's vibration and the flow pattern stability. Figure 2:0 Correlation dimension curves of points in different directions (**a**) X-axis points, (**b**) Y-axis points, and (**c**) \(Z\)-axis points. To verify the validity of the above analysis results, the chaotic characteristics of the pumping station pipeline are further analyzed by using the largest Lyapunov exponent \(\lambda_{1}\). According to the time delay \(\tau\) and embedding dimension \(m\), the small data sets calculate the largest Lyapunov exponent. Figure 8 is the \(\lambda_{1}\) calculation diagram of typical points, and the value of the separation factor \(\text{y}(i)\) tends to be stable after nearly linear growth. The linear slope is adjusted by the least square method, and the value is \(\lambda_{1}\). The \(\lambda_{1}\) of each point in different vibration directions are shown in Figure 2:1  
Figure 2:2 Calculation diagram of typical points (**a**) Point 3 Y-axis vibration under condition 1, and (**b**) Point 1 Z-axis vibration under condition 4. Figure 2:3 Largest Lyapunov exponent curves of points in different directions (**a**) X-axis points (**b**) Y-axis points, and (**c**) Z-axis points. As shown in Figure 2:4  
1. The largest Lyapunov exponents \(\lambda_{1}\) of different points are between 0.0323 and 0.0734, greater than 0. It shows that the measured vibration responses of pipelines have obvious chaotic characteristics. Also, the axial points \(\lambda_{1}\) of the main pipeline (Y-axis) are obviously larger than the other two directions in the same condition, indicating the chaotic characteristics of the points separated from the influence of centrifugal force generated by pumping station units are more obvious;
2. The \(\lambda_{1}\) of measuring points under condition 4 (No.