Rock-Paper-Scissors (RPS) games have been widely studied, while research on asymmetric multiplayer cases has remained relatively limited. In this paper, we constructed a three-player asymmetric RPS model by introducing a novel payoff matrix, which then serves as the basis for developing a reverse-cyclic mutation logit system. Analysis of the interior equilibrium showed that both mutation rate and selection intensity can destabilize the equilibrium and generate oscillatory dynamics through Hopf bifurcations. When the first Lyapunov coefficient vanished, the second Lyapunov coefficient was derived to classify the bifurcation type, enabling us to distinguish supercritical and subcritical cases. Numerical simulations confirmed the theoretical predictions. These results revealed stability conditions and bifurcation mechanisms in asymmetric multiplayer interactions and demonstrated how higher-order Lyapunov coefficients enhance the analysis of complex dynamics.