**1.1 Brief description of chaotic motion **

Chaotic motion is a mathematical concept that is difficult to define precisely. Chaos studies deterministic motion. Earlier research only focused on linear models and special integrable nonlinear models. With the popularization of computers, more precise studies have been conducted on complex nonlinear motions including chaotic motions. Compared with other complex phenomena, chaotic motion has its own unique characteristics: the nonlinear dynamic characteristics of chaos determine that chaos is unpredictable, and the sensitivity of chaos to the initial value shows that it is difficult to predict it. For a chaotic dynamic system, the initial conditions can cause significant differences between the numerical trajectories after a long period of simulation. In addition, Lorenz research shows that the Lyapunov exponent changes between negative and positive values even if the time step is small. In the chaotic process of a long-span bridge, when monitoring its sensitivity, it will be found that part of the information will be lost every time it is predicted. As the number of predictions increases, its accuracy is significantly reduced, so chaos is not suitable for long-term prediction. Therefore, Teixeira et al. [12] further studied the time step sensitivity of three nonlinear models using traditional double-precision atmospheric models. Their conclusion shows that "for chaotic systems, numerical convergence cannot be guaranteed forever."

Through some documents, it can be found that the large-span bridge is prone to chaotic motion under the action of wind load. In this paper, the poincare cross-section method and power spectrum method are used to briefly qualitatively analyze the chaotic motion of long-span bridges.

**1.2 Analyze the chaotic motion of the bridge**

**1.2.1 Poincare section method **

The Poincare section method mainly selects a cross-section appropriately in the multi-dimensional phase space. After the system is continuously mapped, different forms of phase points or phase gauge lines can be observed on the Poincare section. According to the topological properties, the periodic motion, quasi-periodic motion or Chaotic movement.

The main idea and analysis method of this kind of method is basically qualitative analysis. It is given by the following state equation. We know that this system has an unstable limit cycle with a radius of 1, as shown in Figure 1.

Consider the nonlinear system. Fig. 1a shows a trajectory of a=1, and Fig. 1b shows the case of a=0.2. In drawing these figures, two sets of initial conditions were used. The first group of conditions is outside the system limit cycle, and the second group of conditions is inside the system limit cycle. In both cases, the trajectory tends to escape the limit loop. The constant a in the equation cannot change the nature of the phase, but its value is valid for the speed at which the flow escapes the limit cycle.

It can be seen that the graph drawn by the phase plan has a complicated form, and the Poincare mapping shows a strangely attractive form. The analysis shows that when a=1, the system has stronger divergence, on the contrary, when a=2, the system has stronger cohesion.

In the same way, the nonlinear flutter problem of long-span bridges is analyzed according to this type of method. Yan Cong [14] used wind tunnel tests to build a scaled-down suspension bridge to study the changes in the flutter of the bridge under different wind conditions. The phase diagrams of the bridge when the wind speed is 98m/s and 94.4m/s are tested respectively. And compare this result with the mid-span displacement under the action of two wind speeds. Under the action of two wind speeds, the phase diagram of the bridge is shown in Figure 2.

According to the judgment method of Poincare section method, it can be seen that when the wind speed is 98m/s, the bridge has obvious divergence characteristics. It shows that when the wind speed reaches 98m/s, the long-span bridge is more prone to damage.

**1.2.2 Power spectrum method**

Spectrum analysis is generally used together with Poincare section method, but the analysis objects of the two are different. The spectrum analysis object is a discrete point sequence generated by point mapping, using Fourier analysis. In addition, when Poincare analyzes chaotic motion, it is distributed as infinite points in a certain area, and the point sequence of the spectrum is continuous. For the analysis of the spectrogram, it mainly analyzes the duration of the sample process and the change of the total number of points. When the extension time of the sample process is longer and the interval is smaller, the spectral lines will be encrypted, and the correctness of the real physical process is better. This is because the interval between the spectra is 2π/T, and as T increases, the spectrum is encrypted.

Sun Qian [14] took Changde Baima Lake Hongqiao as a research project. The site map is shown in Figure 3. An acceleration sensor is set at the fifth span, and the field dynamic experiment is carried out using the frequency spectrum method. The measured frequency spectrum is shown in Figure 4.

According to the spectrum analysis method, it can be seen that the flutter of the bridge is a chaotic motion. In addition, the frequency in the early stage of the test is relatively dense, indicating that the bridge's early stability is better, but as the test progresses, the frequency gradually stabilizes, indicating that the structure is relatively stable during the test period.