## 1 Introduction  
Since the issue of bridge flutter was put forward, numerous studies have been conducted on aerodynamic vibrations in bridges. Flutter, as a form of aerodynamic instability, is the most destructive type of vibration induced by wind. When wind speed is low, structural vibrations caused by the interaction between the structure and the flow field are dissipated by system damping, keeping the bridge in a stationary state. However, when wind speed exceeds a critical value, the motion energy generated by the interaction between the structure and the flow field cannot be completely dissipated by system damping, resulting in flutter. Flutter is a self-excited vibration phenomenon of elastic structures induced by their interaction with fluid in the flow field. This study investigates the nonlinear flutter response when wind speed exceeds the critical value. The occurrence of self-limiting flutter, known as limit cycle oscillations in bridges, has been indicated by the Tacoma Narrows Bridge flutter event and recent research and experiments. This type of limit cycle oscillation, often referred to as \"hard flutter,\" differs from linear divergence-type flutter theories and does not always lead to divergence [1; 2; 3; 4; 5]. Notably, nonlinearity in the aerodynamic self-excited forces of bluff body bridges is significant [6; 7; 8; 9], and the nonlinear phenomenon stands out under angle of attack conditions [10; 11]. Moreover, there exists a significant potential for advancing our assessment of the vibrational performance of bridges through machine learning algorithms based on artificial neural networks. Thesealgorithms employ geometric information and dynamic parameters as inputs to construct vibrational models of bridges. The integration of computational fluid dynamics (CFD) with machine learning models is used to predict wind loads, and it has demonstrated satisfactory results in forecasting and evaluating the aerodynamic elastic responses of bridges [12; 13; 14]. This underscores the widespread attention garnered by the study of nonlinear flutter in long-span bridges. Although bridge design still does not allow for situations beyond the flutter critical wind speed, studying flutter behavior beyond the critical state contributes to a deeper understanding of the structural responses of bridges under flutter conditions. The flutter instability process caused by aeroelastic effects is generally analyzed using the aerodynamic self-excitation theory proposed by [PERSON] et al. [15]. Traditional studies have established linear flutter equations based on this theory, but the limitations of linear systems are evident. They can only solve the flutter critical state of the aerodynamic self-excited system. For example, [PERSON] et al. proposed stability conditions and unified linear variables for bluff bridge deck sections under unsteady flow [16; 17]. However, as the span of suspension bridges increases, the vibration response of bridges after flutter becomes increasingly important. Therefore, it is crucial to establish nonlinear motion equations for self-excited flutter in suspension bridges. In the conventional [PERSON] aerodynamic self-excited force model, the interaction between the bridge cross-section and aerodynamic forces is represented by a linear multiplication of flutter derivatives and motion components. The higher-order terms of flutter derivatives are disregarded. However, the aerodynamic self-excited force is inherently nonlinear. Considering the critical post-flutter of bridges, it becomes essential to formulate the appropriate equations to describe this nonlinear motion. Numerous scholars have delved into the study of nonlinear flutter. Building upon [PERSON]'s research, they have demonstrated a strong correlation between the flutter derivatives related to torsional motion and nonlinearity. They have determined a functional relationship between nonlinear flutter derivatives, reduced wind speed, and flutter amplitudes [2; 11].