This research introduces a novel model that examines the movements of powerful cyclones that interact with topography on a β-plane. The model reveals two velocity components that emerge from conserving potential vorticity: the meridional adjustment velocity (MAV) and the topographic adjustment velocity (TAV). Thorough computations involving strong vortices interacting with a model mountain have revealed intriguing track patterns and various intricate and unsteady movements. The prevalent tracks predominantly exhibit an ‘S-shaped’ configuration. This configuration is characterized by a deviation southward on the front (windward) side, followed by a northern realignment as the vortex transitions to the leeward side. This distinctive S-shaped pathway elucidates three specific types of fluctuating movement, labeled as the decelerate–accelerate motion (Type I), the decelerate–accelerate–decelerate–accelerate motion (Type II), and the accelerate–decelerate motion (Type III). We also calculate the tracks for strong vortices passing over a 1-km resolution Taiwan topography. The results show interesting similarities with some historical typhoons encountered in Taiwan. In conclusion, the introduced dynamic tracking model offers a fresh perspective into the multifaceted trajectories and erratic behaviors of strong cyclonic vortices navigating terrains. It beckons an optimistic avenue in its applicability, hinting at the possibility of integration with advanced vortex models and considering the nuances of environmental currents.