Milan Journal of Mathematics

, Volume 81, Issue 1, pp 153-169

First online:

Lyapunov Functions for General Nonuniform Dichotomies

  • Luis BarreiraAffiliated withDepartamento de Matemática, Instituto Superior Técnico Email author 
  • , Jifeng ChuAffiliated withDepartment of Mathematics, College of Science, Hohai University
  • , Claudia VallsAffiliated withDepartamento de Matemática, Instituto Superior Técnico

Rent the article at a discount

Rent now

* Final gross prices may vary according to local VAT.

Get Access


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: 34D09 34D10


Lyapunov functions nonuniform dichotomies robustness