The Differential Stability Problem. Part I: Geometric Contraction for Nonlinear Time-Varying Systems

This paper develops a unified framework for the analysis of stability, convergence, and contractivity of nonlinear time-varying dynamical systems from a geometric perspective. After introducing the required mathematical background (including metric notions, parametrized curves on differentiable manifolds, and Riemannian length), we formalize system dynamics on manifolds with time-varying vector fields and revisit classical variational and prolonged dynamics in this general setting. Motivated by the limitations of standard variational equations in time-varying contexts, we generalize the notion of variational state by accounting for perturbations in both the initial conditions and the time argument. The variational state concept is further extended through a doubleparameterized family of curves connecting system trajectories, which enables a rigorous characterization of contraction via time-varying, state-dependent metrics. Within this framework, we establish regional contraction conditions for nonlinear time-varying systems and reinterpret Lyapunov functions as geometric objects induced by the underlying metric space associated with the system. Overall, the proposed approach provides a coherent geometric foundation for trajectory-based stability analysis for a broad class of nonlinear time-varying systems.