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A generalized nonuniform contraction and Lyapunov function. (English) Zbl 1261.34043

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.

MSC:

34D20 Stability of solutions to ordinary differential equations
34A30 Linear ordinary differential equations and systems
34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
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