To address the difficulty of balancing diversity and smoothness in gait phase manifold modulation for quadruped robots, this study proposes a modulation mechanism based on nonlinear differential geometry and Hopf bifurcation analysis. A central pattern generator (CPG)-Hopf oscillator network with environmental force feedback is established, and a phase perturbation dynamics model is constructed using polar coordinate transformation and contact force feedback. Nonlinear differential geometry defines the limiting circulation shape, while geodesic distance and Floquet theory analyze gait stability and switching paths. The manifold topological transition is driven by Hopf bifurcation conditions, with parameter optimization and Lyapunov functions enabling smooth gait modulation. This method assumes rigid grounding and ideal phase tracking. The differential geometry framework is applicable to low dimensional limit cycle manifolds and locally smooth gait transitions. Experimental results show that the frequency error is concentrated in the 0.4 Hz-1.1 Hz range, with a peak density of 0.8 Hz. The overall gait switching success rate across different terrain categories exceeds 96.0%, and the stability margin during the gait cycle exceeds 7.9 cm. When the time increases from 1.0 s to 5.0 s, the forward speed rises from 0.60 m/s to 1.20 m/s, an increase of 0.60 m/s. The proposed method improves gait adaptability and motion stability of quadruped robots in unstructured environments.
