Here proper orthogonal decomposition (POD) modal decomposition are performed for flow past a circular cylinder at supercritical Reynolds numbers by projecting this onto instability modes. The important task of modeling a cylinder wake by Stuart-Landau (SL) and the Stuart-Landau-Eckhaus (SLE) equation for instability modes is discussed, with the latter shown to be more consistent with multimodal pictures of POD and instability modes. The difficult task of finding the coefficients of the SLE equation is reported by taking a least squares approach for the reduced order model (ROM). The important aspect of the ROM is the choice of initial condition for the developed SLE equations, as these are stiff ordinary differential equations which are very sensitive to the choice of initial conditions. An accurate representation of enstrophy-based POD also reveals the presence of modes which occur in isolation (in comparison to modes that come in pairs) and the traditional approach of treating instability modes by SL or SLE equations does not work directly, which also reveals higher frequency variations. Quantifying effects of this mode by time-averaged Navier-Stokes equation (NSE) fail to show the variation of the phase of these isolated time-varying modes and this is captured here using direct numerical simulation (DNS) data by a multitime scale approach. A reconstructed 3-mode ROM solution and the disturbance vorticity from DNS match globally in the flow. The agreement between 3-mode SLE reconstruction and DNS also proves the consistency of the proposed method and helps explain the physical nature of the ensuing Hopf bifurcation following an instability.