Model reconstruction for nonlinear dynamical systems on Poincaré section

This paper focuses on the model reconstruction of nonlinear dynamical systems and proposes a novel modeling method on the Poincaré section of the system. Traditional identification methods for dynamical systems rely on long time-series data, facing bottlenecks of low computational efficiency in high-dimensional complex systems, especially struggling with the non-differentiability issue of non-smooth impact systems. When studying dynamical systems, we usually analyze the motion characteristics by examining the response on the Poincaré section, where the data response can fully reflect the system's dynamic properties. Therefore, this study borrows the model equation identification idea from the Sparse Identification of Nonlinear Dynamics (SINDy) method and innovatively constructs the mapping equation using only the discrete data collected on the Poincaré section. This design not only significantly reduces the data demand, but also enables accurate reconstruction of the dynamic characteristics of the original system while remarkably improving computational efficiency. Numerical experiments show that the method exhibits good adaptability in both smooth and non-smooth impact systems, accurately capturing the dynamic characteristics of periodic and chaotic motions. Notably, for vibro-impact systems, the constructed equation is a smooth mapping, which effectively circumvents the non-differentiability problem caused by the impact effect. The proposed method achieves the identification of the key motion characteristics of the system in a short time with a small amount of data, successfully addresses the limitations and shortcomings of traditional methods, expands the technical dimension of dynamical system model identification, provides a new data-driven approach for the study of multi-scale problems, and holds important application value in fields such as mechanical engineering and nonlinear vibration.