A Generalized Nonuniform Contraction and Lyapunov Function

Abstract

For nonautonomous linear equations $x^{\prime }=A(t)x$, 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.