, Volume 81, Issue 1, pp 153-169
Date: 19 Mar 2013

Lyapunov Functions for General Nonuniform Dichotomies

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For nonautonomous linear equations x′ = A(t)x, we give a complete characterization of the existence of exponential behavior in terms of Lyapunov functions. In particular, we obtain an inverse theorem giving explicitly Lyapunov functions for each exponential dichotomy. The main novelty of our work is that we consider a very general type of nonuniform exponential dichotomy. This includes for example uniform exponential dichotomies, nonuniform exponential dichotomies and polynomial dichotomies. We also consider the case of different growth rates for the uniform and the nonuniform parts of the dichotomy. As an application of our work, we establish in a very direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations.

Luis Barreira and Claudia Valls were partially supported by FCT through CAMGSD, Lisbon. Jifeng Chu was supported by the National Natural Science Foundation of China (Grants No. 10, No. 2 and No. 21333), the Program for New Century Excellent Talents in University (Grant No. NCET–032), China Postdoctoral Science Foundation funded project (Grant No. 20T0431).