Abstract: For girders of large-span bridges, when the incoming wind speed is equal to or surpasses the flutter critical point, the bridge deck exhibits persistent amplitude vibrations, termed "hysteresis flutter" (or soft flutter). Different from hard flutter with rapid increasing amplitudes, soft flutter entails a more intricate coupling with nonlinear effects between motion and aerodynamic forces, resulting in various dynamic states, including step through and bifurcations. Present research comes from the practical extension of classical theories, focusing primarily on the potential equilibrium states which may exist after the flutter critical condition. However, there is little discussion on whether the system can maintain stable amplitude oscillations once it reaches the equilibrium position. This study combines the flutter energy evolution equation with the first-order quasi-static approximation method to explore the intrinsic mathematical principles underlying amplitude stability in hysteretic flutter from a nonlinear perspective of dynamics. It also derives a quantitative indicator, the Hysteretic Flutter Stability Index, which characterizes the amplitude evolution trend of the hysteretic flutter state. By combining the nonlinear structural and aerodynamic characteristics of a streamlined box girder section of a large-span bridge, this study verifies that the analytical method proposed can effectively assess the ability of hysteretic flutter to maintain stable oscillations. Finally, demonstrating by the theory that, even when aerodynamic and structural forces balance within a cycle, stable hysteretic flutter will not occur with an index less than zero in practical environments.