The Reynolds-averaged Navier-Stokes (RANS) technique enables critical engineering predictions and is widely adopted. However, since this iterative computation relies on the fixed-point iteration, it may converge to unexpected non-physical phase points in practice. We conduct an analysis on the phase-space characteristics and the fixed-point theory underlying the k\text{-}\epsilon turbulence model, and employ the classical Kolmogorov flow as a framework, leveraging its direct numerical simulation (DNS) data to construct a one-dimensional (1D) system under periodic/fixed boundary conditions. The RANS results demonstrate that under periodic boundary conditions, the k\text{-}\epsilon model exhibits only a unique trivial fixed point, with asymptotes capturing the phase portraits. The stability of this trivial fixed point is determined by a mathematically derived stability phase diagram, indicating the fact that the k\text{-}\epsilon model will never converge to correct values under periodic conditions. In contrast, under fixed boundary conditions, the model can yield a stable non-trivial fixed point. The evolutionary mechanisms and their relationship with boundary condition settings systematically explain the inherent limitations of the k\text{-}\epsilon model, i.e., its deficiency in computing the flow field under periodic boundary conditions and sensitivity to boundary-value specifications under fixed boundary conditions. These conclusions are finally validated with the open-source code OpenFOAM.