A generalized nonuniform contraction and Lyapunov function (Q1938250)

Summary: For nonautonomous linear equations \(x' = A(t)x\), we give a complete characterization of general nonuniform contractions in terms of Lyapunov functions. We consider the general case of nonuniform contractions, which corresponds to the existence of what we call nonuniform \((D, \mu)\)-contractions. As an application, we establish the robustness of the nonuniform contraction under sufficiently small linear perturbations. Moreover, we show that the stability of a nonuniform contraction persists under sufficiently small nonlinear perturbations.