A Generalized Nonuniform Contraction and Lyapunov Function

Abstract EN

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,μ)-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.