The effects of slowly-varying wave drift forces on the nonlinear dynamics of mooring systems have been studied extensively in the past 30 years. It has been concluded that slowly-varying wave drift may resonate with mooring system natural frequencies. In recent work, we have shown that this resonance phenomenon is only one of several possible nonlinear dynamic interactions between slowly-varying wave drifts and mooring systems. We were able to reveal new phenomena, based on the design methodology developed at the University of Michigan for autonomous mooring systems, and treating slowly-varying wave drift as an external time-varying force in systematic simulations. This methodology involves exhaustive search regarding the nonautonomous excitation, however, and approximations in defining response bifurcations. In this paper, a new approach is developed, based on the harmonic balance method, where the response to the slowly-varying wave drift spectrum is modeled by limit cycles of frequency, estimated from a limited number of simulations. Thus, it becomes possible to rewrite the nonautonomous system as autonomous and reveal stability properties of the nonautonomous response. Catastrophe sets of the symmetric principal equilibrium, serving as design charts, define regions in the design space where the trajectories of the mooring system are asymptotically stable, limit cycles, or nonperiodic. This methodology reveals and proves that mooring systems subjected to slowly-varying wave drift exhibit many nonlinear phenomena, which lead to motions with amplitudes two to three orders of magnitude larger than those resulting from linear resonance. A turret mooring system (TMS) is used to demonstrate the harmonic balance methodology developed. The produced catastrophe sets are then compared with numerical results obtained from systematic simulations of the TMS dynamics.