Periodic Orbits and Bifurcations in a High‐Dimensional Nonautonomous Piecewise Smooth Discontinuous System

ABSTRACT The dry friction coupled vibration system represents a typical nonsmooth system, and the investigation of the dynamic characteristics of nonsmooth systems is of great significance for mechanical control, safe operation, and the formulation of engineering standards. In this paper, the theory of periodic orbits and bifurcations in a class of high‐dimensional piecewise smooth discontinuous systems is explored in response to the need for mathematical theory and practical applications and is applied to study the dynamic characteristics of a two‐degree‐of‐freedom dry friction coupled vibration system. A global Poincaré map is constructed, and the center manifold method for smooth systems is extended to derive sufficient and necessary conditions for the existence of an invariant cone. Based on Floquet theory, the stability of the invariant cone composed of periodic orbits is demonstrated. Additionally, the bifurcation phenomena and the persistence problem of the invariant cone are investigated. The parameter conditions for the periodic orbits generated by Hopf‐like bifurcation induced by nonlinear disturbances are further considered, and the stability of periodic orbit is determined. The distribution configurations of the trajectories are obtained through numerical simulations to validate the accuracy of the theoretical findings. This work provides a solid theoretical foundation for parameter optimization and vibration control in high‐dimensional dry friction coupled vibration systems within engineering practice.