Geometric Stability Analysis of Autonomous Systems via Riemannian Potential Dynamics

This paper proposes a geometric framework for analyzing stability in autonomous systems using Riemannian information geometry. The system dynamics are modeled as a potential-driven flow on a statistical manifold equipped with the Fisher information metric. We introduce a formulation based on Riemannian gradients and derive a Lagrangian representation of the system. A Lyapunov-based analysis is developed to establish stability under bounded stochastic perturbations. A global stability theorem is proved, showing that system trajectories remain bounded and converge toward critical points of the potential function. Experimental results demonstrate improved stability and reduced drift compared to Euclidean and constrained optimization approaches. These results highlight the relevance of geometric structures in designing robust and stable autonomous systems.