Milan Journal of Mathematics

, Volume 81, Issue 1, pp 153–169

Lyapunov Functions for General Nonuniform Dichotomies


DOI: 10.1007/s00032-013-0198-y

Cite this article as:
Barreira, L., Chu, J. & Valls, C. Milan J. Math. (2013) 81: 153. doi:10.1007/s00032-013-0198-y


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.

Mathematics Subject Classification (2010)

Primary: 34D0934D10


Lyapunov functionsnonuniform dichotomiesrobustness

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Departamento de MatemáticaInstituto Superior TécnicoLisboaPortugal
  2. 2.Department of Mathematics, College of ScienceHohai UniversityNanjingChina