Research on extended high dimensional Melnikov and singularity based on double inertial coupling vibration system

The dual inertia flexible transmission system is widely used in fields such as precision manufacturing. However, traditional linear integer order modeling and low-dimensional analysis methods struggle to accurately describe its nonlinear and fractional order viscoelastic characteristics, and fail to effectively capture the evolution laws of chaotic motion and resonance instability. To address this, this paper constructs a two degree of freedom dynamic model for a dual inertia coupled vibration system containing multiple nonlinear and fractional order differential terms, which is more in line with engineering reality. Secondly, the extended high-dimensional Melnikov method, which is suitable for fractional order systems, is used to derive the critical criterion of chaotic motion by taking the fractional derivative as a disturbance term, and its effectiveness is verified by numerical simulation. Then, combined with singularity theory, the characteristics of bifurcation set and lag set of the system are analyzed, and the boundary of stable region and unstable region and the law of multistable path dependence in parameter space are defined. The regulation mechanism of fractional order parameters and nonlinear parameters on the dynamic behavior of the system is further revealed. The research results provide reliable theoretical support for chaos suppression, parameter optimization and stability control of the dual inertia system, and can help to improve the operation accuracy and service life of related industrial equipment.