The Differential Stability Problem. Part II: the Dynamical Spectral Decomposition

This work develops a unified framework for the analysis of stability and contractivity of nonlinear timevarying dynamical systems. After introducing the eigenproblems in the general settings of operators, we formalize system dynamics on manifolds with differentiable timevarying vector fields and their associates variational and prolonged dynamics in this general setting.Motivated by the limitations of standard eigenvalue approach to study the stability analysis of nonlinear systems, we propose a dynamical spectral decomposition based on an operator eigenproblem, generalizing the classical Jacobian-based analysis for equilibrium points and enabling the treatment of systems that are nondiagonalizable. The resulting spectral framework yields criteria for differential asymptotic and differential exponential stability and provides constructive methods for designing contractive metrics on prescribed regions of the state space. Several examples illustrate the applicability of the approach to linear time-varying, linear time-invariant, and nonlinear systems.Overall, the proposed framework bridges operatortheoretic and spectral perspectives on stability, offering new tools for the study of a broad class of nonlinear timevarying dynamical systems.