<p>To analyze the dynamic behavior of the cold rolling mill under the influence of nonlinear constraints, a vertical nonlinear dynamic model of the cold rolling system considering multi-piecewise nonlinear elastic constraints coupled with dynamic rolling force was established. Considering the complexity of solving the kinetic equations under the nonlinear action, this paper introduces a modified Lindstedt-Poincare solution method with high stability and accuracy.&#xa0;Besides, to obtain the critical conditions for the system to diverge from a periodically stable to a chaotic unstable motion state, the Melnikov method is utilized to calculate the chaos threshold in the sense of the existence of Smale's horseshoe. Meanwhile, bifurcation analysis, maximum Lyapunov exponent, and two-parameter vibration response analysis indicate the system's vertical nonlinear dynamical behavior with the impact of parameters. The findings demonstrate that: the system's stability domain drops with increasing nonlinear elastic constraint and dynamic rolling force coupling effect; excessive external excitation will trigger the system to swiftly reach a chaotic state and produce irresistible violent vibration. Moreover, the nonlinear stiffness parameter excitation should be adjusted in the critical chaotic state of the system, and the excessive nonlinear stiffness causes long-term vibration. The effect of damping and linear stiffness is relatively lagging, but too small a damping coefficient affects the energy dissipation of the system, which exacerbates the vibration generation. This study provides a theoretical reference for the nonlinear vibration stability control of the cold rolling mill system and the adjustment of process parameters.</p>