<p>Bidirectional (along-wind and across-wind) vibration often occurs in high-rise buildings under strong wind, and its mitigation remains to be a critical challenge. This study introduces a bidirectional pendulum tuned mass damper enhanced with an additional stopper to mitigate structural vibrations. The system is based on the unidirectional pendulum tuned mass damper with additional stopper (PTMD-AS) and improves upon it by extending the stopper from a single-point design to a circular ring configuration in the horizontal plane. The nonlinear stiffness of the TMD in two orthogonal directions occurs when the pendulum is in contact with the stopper. The analytical model of this control system is formulated and applied to the study of dynamic behavior of the structural system with the extended incremental harmonic balance method. Simulation results demonstrate that in the symmetric control system, changes in the contact point between the pendulum and stopper yield different TMD performances due to a change in the stiffness hardening property of the pendulum. Stability analysis using Floquet theory demonstrates that an excitation phase difference and stopper position may lead to Saddle-node and Hopf bifurcations of the system with multiple solutions and non-periodic responses.</p>