Floquet theory is a fundamental tool for stability analysis of periodic systems, with the monodromy (transition) matrix serving as a critical component. In this study, we propose a novel single-pass time integration method incorporating an adaptive step size controller to enhance the accuracy and efficiency of stability evaluation. Through a comparative analysis of explicit and implicit integrators applied to a cracked Jeffcott rotor, we demonstrate the improved stability behavior and reduced computational costs achieved by our approach. Unlike traditional methods relying on constant step sizes, the proposed step size controller effectively mitigates inaccuracies in eigenvalue computation and artificial instabilities. Our findings not only advance the understanding of asymmetric rotor dynamics but also offer valuable insights for solving periodic ordinary differential equations in broader applications.